Fixed Point Theorems in Quaternion-Valued Metric Spaces

10
Research Article Fixed Point Theorems in Quaternion-Valued Metric Spaces Ahmed El-Sayed Ahmed, 1,2 Saleh Omran, 1,3 and Abdalla J. Asad 3 1 Mathematics Department, Faculty of Science, Taif University, Taif 888, Saudi Arabia 2 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt 3 Mathematics Department, Faculty of Science, South Valley University, Qena, Egypt Correspondence should be addressed to Ahmed El-Sayed Ahmed; [email protected] Received 8 October 2013; Revised 14 January 2014; Accepted 23 January 2014; Published 30 March 2014 Academic Editor: Naseer Shahzad Copyright © 2014 Ahmed El-Sayed Ahmed et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e aim of this paper is twofold. First, we introduce the concept of quaternion metric spaces which generalizes both real and complex metric spaces. Further, we establish some fixed point theorems in quaternion setting. Secondly, we prove a fixed point theorem in normal cone metric spaces for four self-maps satisfying a general contraction condition. 1. Introduction and Preliminaries A metric space can be thought as very basic space having a geometry, with only a few axioms. In this paper we introduce the concept of quaternion metric spaces. e paper treats material concerning quaternion metric spaces that is important for the study of fixed point theory in Clifford analysis. We introduce the basic ideas of quaternion metric spaces and Cauchy sequences and discuss the completion of a quaternion metric space. In what follows we will work on H, the skew field of quaternions. is means we can write each element ∈ H in the form = 0 + 1 + 2 + 3 , ∈ R, where 1, , , are the basis elements of H and = 1, 2, 3. For these elements we have the multiplication rules 2 = 2 = 2 = −1, = − = , = − = −, and = − = . e conjugate element is given by = 0 − 1 − 2 − 3 . e quaternion modulus has the form of || = √ 2 0 + 2 1 + 2 2 + 2 3 . Quaternions can be defined in several different equiva- lent ways. Notice the noncommutative multiplication, their novel feature; otherwise, quaternion arithmetic has spe- cial properties. ere is also more abstract possibilty of treating quaternions as simply quadruples of real numbers [ 0 , 1 , 2 , 3 ], with operation of addition and multiplication suitably defined. e components naturally group into the imaginary part ( 1 , 2 , 3 ), for which we take this part as a vector and the purely real part, 0 , which called a scalar. Sometimes, we write a quaternion as [V, 0 ] with V = ( 1 , 2 , 3 ). Here, we give the following forms: := [V, 0 ], V ∈ R 3 ; 0 ∈ R = [( 1 , 2 , 3 ), 0 ]; 0 , 1 , 2 , 3 ∈ R = 0 + 1 + 2 + 3 . (1) us a quaternion may be viewed as a four-dimensional vector ( 0 , 1 , 2 , 3 ). For more information about quaternion analysis, we refer to [1–4] and others. Define a partial order ≾ on H as follows. 1 ≾ 2 if and only if Re( 1 )≤ Re( 2 ), Im ( 1 )≤ Im ( 2 ), 1 , 2 ∈ H; = ,,, where Im = 1 ; Im = 2 ; Im = 3 . It follows that 1 ≾ 2 , if one of the following conditions is satisfied. (i) Re( 1 )= Re( 2 ); Im 1 ( 1 )= Im 1 ( 2 ), where 1 = , ; Im ( 1 )< Im ( 2 ). (ii) Re( 1 )= Re( 2 ); Im 2 ( 1 )= Im 2 ( 2 ), where 2 = , ; Im ( 1 )< Im ( 2 ). (iii) Re( 1 )= Re( 2 ); Im 3 ( 1 )= Im 3 ( 2 ), where 3 = , ; Im ( 1 )< Im ( 2 ). (iv) Re( 1 ) = Re( 2 ); Im 1 ( 1 ) < Im 1 ( 2 ); Im ( 1 ) = Im ( 2 ). Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 258985, 9 pages http://dx.doi.org/10.1155/2014/258985

Transcript of Fixed Point Theorems in Quaternion-Valued Metric Spaces

Page 1: Fixed Point Theorems in Quaternion-Valued Metric Spaces

Research ArticleFixed Point Theorems in Quaternion-Valued Metric Spaces

Ahmed El-Sayed Ahmed12 Saleh Omran13 and Abdalla J Asad3

1 Mathematics Department Faculty of Science Taif University Taif 888 Saudi Arabia2Mathematics Department Faculty of Science Sohag University Sohag 82524 Egypt3Mathematics Department Faculty of Science South Valley University Qena Egypt

Correspondence should be addressed to Ahmed El-Sayed Ahmed ahsayed80hotmailcom

Received 8 October 2013 Revised 14 January 2014 Accepted 23 January 2014 Published 30 March 2014

Academic Editor Naseer Shahzad

Copyright copy 2014 Ahmed El-Sayed Ahmed et alThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited

The aim of this paper is twofold First we introduce the concept of quaternion metric spaces which generalizes both real andcomplex metric spaces Further we establish some fixed point theorems in quaternion setting Secondly we prove a fixed pointtheorem in normal cone metric spaces for four self-maps satisfying a general contraction condition

1 Introduction and Preliminaries

A metric space can be thought as very basic space havinga geometry with only a few axioms In this paper weintroduce the concept of quaternionmetric spacesThe papertreats material concerning quaternion metric spaces that isimportant for the study of fixed point theory in Cliffordanalysis We introduce the basic ideas of quaternion metricspaces and Cauchy sequences and discuss the completion ofa quaternion metric space

In what follows we will work on H the skew field ofquaternions This means we can write each element 119909 isin H inthe form 119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are

the basis elements ofH and 119899 = 1 2 3 For these elements wehave the multiplication rules 1198942 = 119895

2= 1198962= minus1 119894119895 = minus119895119894 = 119896

119896119895 = minus119895119896 = minus119894 and 119896119894 = minus119894119896 = 119895 The conjugate element 119909 isgiven by 119902 = 119909

0minus 1199091119894 minus 1199092119895 minus 1199093119896 The quaternion modulus

has the form of |119902| = radic11990920+ 1199092

1+ 1199092

2+ 1199092

3

Quaternions can be defined in several different equiva-lent ways Notice the noncommutative multiplication theirnovel feature otherwise quaternion arithmetic has spe-cial properties There is also more abstract possibilty oftreating quaternions as simply quadruples of real numbers[1199090 1199091 1199092 1199093] with operation of addition and multiplication

suitably defined The components naturally group into theimaginary part (119909

1 1199092 1199093) for which we take this part as

a vector and the purely real part 1199090 which called a scalar

Sometimes we write a quaternion as [V 1199090] with V =

(1199091 1199092 1199093)

Here we give the following forms

119902 = [V 1199090] V isin R

3 1199090isin R

= [(1199091 1199092 1199093) 1199090] 119909

0 1199091 1199092 1199093isin R

= 1199090+ 1199091119894 + 1199092119895 + 1199093119896

(1)

Thus a quaternion 119902 may be viewed as a four-dimensionalvector (119909

0 1199091 1199092 1199093)

Formore information about quaternion analysis we referto [1ndash4] and others

Define a partial order ≾ on H as follows1199021≾ 1199022if and only if Re(119902

1) le Re(119902

2) Im119904(1199021) le Im

119904(1199022)

1199021 1199022isin H 119904 = 119894 119895 119896 where Im

119894= 1199091 Im119895= 1199092 Im119896= 1199093

It follows that 1199021≾ 1199022 if one of the following conditions is

satisfied

(i) Re(1199021) = Re(119902

2) Im1199041

(1199021) = Im

1199041

(1199022) where 119904

1= 119895 119896

Im119894(1199021) lt Im

119894(1199022)

(ii) Re(1199021) = Re(119902

2) Im1199042

(1199021) = Im

1199042

(1199022) where 119904

2= 119894 119896

Im119895(1199021) lt Im

119895(1199022)

(iii) Re(1199021) = Re(119902

2) Im1199043

(1199021) = Im

1199043

(1199022) where 119904

3= 119894 119895

Im119896(1199021) lt Im

119896(1199022)

(iv) Re(1199021) = Re(119902

2) Im1199041

(1199021) lt Im

1199041

(1199022) Im119894(1199021) =

Im119894(1199022)

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 258985 9 pageshttpdxdoiorg1011552014258985

2 Abstract and Applied Analysis

(v) Re(1199021) = Re(119902

2) Im1199042

(1199021) lt Im

1199042

(1199022) Im119895(1199021) =

Im119895(1199022)

(vi) Re(1199021) = Re(119902

2) Im1199043

(1199021) lt Im

1199043

(1199022) Im119896(1199021) =

Im119896(1199022)

(vii) Re(1199021) = Re(119902

2) Im119904(1199021) lt Im

119904(1199022)

(viii) Re(1199021) lt Re(119902

2) Im119904(1199021) = Im

119904(1199022)

(ix) Re(1199021) lt Re(119902

2) Im1199041

(1199021) = Im

1199041

(1199022) Im119894(1199021) lt

Im119894(1199022)

(x) Re(1199021) lt Re(119902

2) Im1199042

(1199021) = Im

1199042

(1199022) Im119895(1199021) lt

Im119895(1199022)

(xi) Re(1199021) lt Re(119902

2) Im1199043

(1199021) = Im

1199043

(1199022) Im119896(1199021) lt

Im119896(1199022)

(xii) Re(1199021) lt Re(119902

2) Im1199041

(1199021) lt Im

1199041

(1199022) Im119894(1199021) =

Im119894(1199022)

(xiii) Re(1199021) lt Re(119902

2) Im1199042

(1199021) lt Im

1199042

(1199022) Im119895(1199021) =

Im119895(1199022)

(xiv) Re(1199021) lt Re(119902

2) Im1199043

(1199021) lt Im

1199043

(1199022) Im119896(1199021) =

Im119896(1199022)

(xv) Re(1199021) lt Re(119902

2) Im119904(1199021) lt Im

119904(1199022)

(xvi) Re(1199021) = Re(119902

2) Im119904(1199021) = Im

119904(1199022)

Remark 1 In particular we will write 1199021⋨ 1199022if 1199021= 1199022and

one from (i) to (xvi) is satisfied Also we will write 1199021≺ 1199022if

only (xv) is satisfied It should be remarked that

1199021≾ 1199022997904rArr

10038161003816100381610038161199021

1003816100381610038161003816le10038161003816100381610038161199022

1003816100381610038161003816 (2)

Remark 2 The conditions from (i) to (xv) look strange butthese conditions are natural generalizations to the corre-sponding conditions in the complex setting (see [5]) Sothe number of these conditions is related to the numberof units in the working space For our quaternion settingwe have four units (one real and three imaginary) then wehave 24 conditions But in the complex setting there were 22conditions

Azam et al in [5] introduced the definition of the complexmetric space as follows

Definition 3 ([5]) Let119883 be a nonempty set and suppose thatthe mapping 119889C 119883 times 119883 rarr C satisfies the following

(d1) 0 lt 119889C(119909 119910) for all 119909 119910 isin 119883 and 119889C(119909 119910) = 0 if andonly if 119909 = 119910

(d2) 119889C(119909 119910) = 119889C(119910 119909) for all 119909 119910 isin 119883

(d3) 119889C(119909 119910) ≾ 119889C(119909 119911) + 119889C(119911 119910) for all 119909 119910 119911 isin 119883

Then (119883 119889C) is called a complex metric spaceNow we extend the above definition to Clifford analysis

Definition 4 Let 119883 be a nonempty set Suppose that themapping 119889H 119883 times 119883 rarr H satisfies

(1) 0 ≾ 119889H(119909 119910) for all 119909 119910 isin 119883 and 119889H(119909 119910) = 0 if andonly if 119909 = 119910

(2) 119889H(119909 119910) = 119889H(119910 119909) for all 119909 119910 isin 119883(3) 119889H(119909 119910) ≾ 119889H(119909 119911) + 119889H(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889H is called a quaternion valued metric on 119883 and(119883 119889H) is called a quaternion valued metric space

Example 5 Let 119889H H times H rarr H be a quaternion valuedfunction defined as 119889H(119901 119902) = |119886

0minus 1198870| + 119894|119886

1minus 1198871| + 119895|119886

2minus

1198872| + 119896|119886

3minus 1198873| where 119901 119902 isin H with

119901 = 1198860+ 1198861119894 + 1198862119895 + 1198863119896 119902 = 119887

0+ 1198871119894 + 1198872119895 + 1198873119896

119886119904 119887119904isin R 119904 = 1 2 3

(3)

Then (119883 119889H) is a quaternion metric spaceNow we give the following definitions

Definition 6 Point 119909 isin 119883 is said to be an interior point of set119860 sube 119883 whenever there exists 0 ≺ 119903 isin H such that

119861 (119909 119903) = 119910 isin 119883 119889H (119909 119910) ≺ 119903 sube 119860 (4)

Definition 7 Point 119909 isin 119883 is said to be a limit point of 119860 sube 119883

whenever for every 0 ≺ 119903 isin H

119861 (119909 119903) cap (119860 minus 119909) = 120601 (5)

Definition 8 Set 119860 is called an open set whenever eachelement of 119860 is an interior point of 119860 Subset 119861 sube 119883 is calleda closed set whenever each limit point of 119861 belongs to 119861 Thefamily

119865 = 119861 (119909 119903) 119909 isin 119883 0 ≺ 119903 (6)

is a subbase for Hausdroff topology 120591 on119883

Definition 9 Let 119909119899 be a sequence in 119883 and 119909 isin 119883 If for

every 119902 isin H with 0 ≺ 119902 there is 1198990isin N such that for all

119899 gt 1198990 119889H(119909119899 119909) ≺ 119902 then 119909

119899 is said to be convergent

if 119909119899 converges to the limit point 119909 that is 119909

119899rarr 119909 as

119899 rarr infin or lim119899119909119899= 119909 If for every 119902 isin H with 0 ≺ 119902 there is

1198990isin N such that for all 119899 gt 119899

0 119889H(119909119899 119909119899+119898) ≺ 119902 then 119909119899 is

called Cauchy sequence in (119883 119889H) If every Cauchy sequenceis convergent in (119883 119889H) then (119883 119889H) is called a completequaternion valued metric space

2 Convergence in Quaternion Metric Spaces

In this section we give some auxiliary lemmas using theconcept of quaternion metric spaces these lemmas will beused to prove some fixed point theorems of contractivemappings

Lemma 10 Let (119883 119889H) be a quaternion valued metric spaceand let 119909

119899 be a sequence in119883 Then 119909

119899 converges to 119909 if and

only if |119889H(119909119899 119909)| rarr 0 as 119899 rarr infin

Abstract and Applied Analysis 3

Proof Suppose that 119909119899 converges to 119909 For a given real

number 120576 gt 0 let

119902 =

120576

2

+ 119894

120576

2

+ 119895

120576

2

+ 119896

120576

2

(7)

Then 0 ≺ 119902 isin H and there is a natural number 119873 such that119889H(119909119899 119909) ≺ 119902 for all 119899 gt 119873

Therefore |119889H(119909119899 119909)| lt |119902| = 120576 for all 119899 gt 119873 Hence|119889H(119909119899 119909)| rarr 0 as 119899 rarr infin

Conversely suppose that |119889H(119909119899 119909)| rarr 0 as 119899 rarr infinThen given 119902 isin H with 0 ≺ 119902 there exists a real number120575 gt 0 such that for ℎ isin H

|ℎ| lt 120575 997904rArr ℎ lt 119902 (8)

For this 120575 there is a natural number119873 such that |119889H(119909119899 119909)| lt120575 for all 119899 gt 119873 Implying that 119889H(119909119899 119909) ≺ 119902 for all 119899 gt 119873hence 119909

119899 converges to 119909

Lemma 11 Let (119883 119889H) be a quaternion valued metric spaceand let 119909

119899 be a sequence in119883Then 119909

119899 is a Cauchy sequence

if and only if |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infin

Proof Suppose that 119909119899 is a Cauchy sequence For a given

real number 120576 gt 0 let

119902 =

120576

2

+ 119894

120576

2

+ 119895

120576

2

+ 119896

120576

2

(9)

Then 0 ≺ 119902 isin H and there is a natural number N such that119889H(119909119899 119909119899+119898) ≺ 119902 for all 119899 gt 119873 Therefore |119889H(119909119899 119909119899+119898)| lt|119902| = 120576 for all 119899 gt 119873 Hence |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infin

Conversely suppose that |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infinThen given 119902 isin Hwith 0 ≺ 119902 there exists a real number120575 gt 0such that for ℎ isin H we have that

|ℎ| lt 120575 997904rArr ℎ ≺ 119902 (10)

For this 120575 there is a natural number 119873 such that|119889H(119909119899 119909119899+119898)| lt 120575 for all 119899 gt 119873 which implies that119889H(119909119899 119909119899+119898) ≺ 119902 for all 119899 gt 119873 Hence 119909

119899 is a Cauchy

sequence This completes the proof of Lemma 11

Definition 12 Let (119883 119889H) be a complete quaternion valuedmetric space For all 119909 119910 isin 119883 119889H(119909 119910) represents 119909 minus

119910 A quaternion valued metric space (119883 119889H) is said to bemetrically convex if119883 has the property that for each 119909 119910 isin 119883with 119909 = 119910 there exists 119911 isin 119883 119909 = 119910 = 119911 such that

1003816100381610038161003816119889H (119909 119911) + 119889H (119911 119910)

1003816100381610038161003816=1003816100381610038161003816119889H (119909 119910)

1003816100381610038161003816 (11)

The following lemma finds immediate applications which isstraightforward from [6]

Lemma 13 Let (119883 119889H) be a metrically convex quaternionvalued metric space and 119870 a nonempty closed subset of 119883 If119909 isin 119870 and 119910 isin 119870 then there exists a point 119911 notin 120597119870 (where 120597119870stands for the boundary of 119870) such that

119889H (119909 119910) = 119889H (119909 119911) + 119889H (119911 119910) (12)

Definition 14 Let 119870 be a nonempty subset of a quaternionvalued metric space (119883 119889H) and let 119865 119879 119870 rarr 119883 satisfy thecondition

120593 [1003816100381610038161003816119889H (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119910 119865119909)

1003816100381610038161003816]

(13)

For all 119909 119910 isin 119870 with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1 and let 120593

R+ rarr R+ be an increasing continuous function for whichthe following property holds

120593 (119905) = 0 lArrrArr 119905 = 0 (14)

We call function 119865 satisfying condition (13) generalized 119879-contractive

Motivated by [7 8] we construct the following definition

Definition 15 Let 119870 be a nonempty subset of a quaternionvalued metric space (119883 119889H) and 119865 119879 119870 rarr 119883 The pair119865 119879 is said to be weakly commuting if for each 119909 119910 isin 119870

such that 119909 = 119865119910 and 119879119910 isin 119870 we have

119889H (119879119909 119865119879119910) ≾ 119889H (119879119910 119865119910) (15)

It follows that1003816100381610038161003816119889H (119879119909 119865119879119910)

1003816100381610038161003816le1003816100381610038161003816119889H (119879119910 119865119910)

1003816100381610038161003816 (16)

Remark 16 It should be remarked that Definition 15 extendsand generalizes the definition of weakly commuting map-pings which are introduced in [7]

3 Common Fixed Point Theorems inQuaternion Analysis

In this section we prove common fixed point theoremsfor two pairs of weakly commuting mappings on completequaternionmetric spacesThe obtained results will be provedusing generalized contractive conditions

Now we give the following theorem

Theorem 17 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are weakly commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof We construct the sequences 119909119899 and 119910

119899 in the

following way

4 Abstract and Applied Analysis

Let 119909 isin 120597119870 Then there exists a point 1199090isin 119870 such that

119909 = 1198791199090as 120597119870 sube 119879119870 From 119879119909

0isin 120597119870 and the implication

119879119909 isin 120597119870 rArr 119865119909 isin 119870 we conclude that 1198651199090isin 119870 cap 119865119870 sube 119879119870

Now let 1199091isin 119870 be such that

1199101= 1198791199091= 1198651199090isin 119870 (17)

Let 1199102= 1198651199091and assume that 119910

2isin 119870 then

1199102isin 119870 cap 119865119870 sube 119879119870 (18)

which implies that there exists a point 1199092isin 119870 such that 119910

2=

1198791199092 Suppose 119910

2notin 119870 then there exists a point 119901 isin 120597119870 (using

Lemma 13) such that

119889H (1198791199091 119901) + 119889H (119901 1199102) = 119889H (1198791199091 1199102) (19)

Since 119901 isin 120597119870 sube 119879119870 there exists a point 1199092isin 119870 such that

119901 = 1198791199092and so

119889H (1198791199091 1198791199092) + 119889H (1198791199092 1199102) = 119889H (1198791199091 1199102) (20)

Let 1199103= 1198651199092 Thus repeating the forgoing arguments we

obtain two sequences 119909119899 and 119910

119899 such that

(i) 119910119899+1

= 119865119909119899

(ii) 119910119899isin 119870 rArr 119910

119899= 119879119909119899 or

(iii) 119910119899notin 119870 rArr 119879119909

119899isin 120597119870

119889H (119879119909119899minus1 119879119909119899) + 119889H (119879119909119899 119910119899) = 119889H (119879119909119899minus1 119910119899) (21)

We denote

119875 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

119876 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

(22)

Obviously the two consecutive terms of 119879119909119899 cannot lie in

119876Let us denote 119905

119899= 119889H(119879119909119899 119879119909119899+1) We have the following

three cases

Case 1 If 119879119909119899 119879119909119899+1

isin 119875 then

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 [

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816]

= 120593 [1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889119867(119879119909119899minus1

119865119909119899minus1

)1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

(23)

Thus

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (24)

Case 2 If 119879119909119899isin 119875 119879119909

119899+1isin 119876 note that

119889H (119879119909119899 119879119909119899+1) + 119889H (119879119909119899+1 119910119899+1) = 119889H (119879119909119899 119910119899+1) (25)

or

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899 119910119899+1) = 119889H (119910119899 119910119899+1) (26)

which implies that1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816 (27)

Hence120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 [

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816] = 120593 [

1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)]

(28)

Therefore

120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (29)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (30)

Case 3 If 119909119899isin 119876 119879119909

119899+1isin 119875 so 119879119909

119899minus1isin 119875 Since 119879119909

119899is a

convex linear combination of 119879119909119899minus1

and 119910119899 it follows that

119889H (119879119909119899 119879119909119899+1) ≾ max 119889H (119879119909119899minus1 119879119909119899+1) 119889H (119910119899 119879119909119899+1) (31)

If 119889H(119879119909119899minus1 119879119909119899+1) ≾ 119889H(119910119899 119879119909119899+1) then 119889H(119879119909119899 119879119909119899+1) ≾119889H(119910119899 119879119909119899+1) implying that

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816 (32)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816]

(33)

Abstract and Applied Analysis 5

It follows that

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 119887120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 119888120593 (

1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816)

(34)

Since

119889H (119879119909119899minus1 119910119899) ≿ 119889H (119879119909119899 119910119899) as 119879119909119899isin 119876 (35)

then

120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) ge 120593 (119889H (119879119909119899 119910119899)) (36)

Therefore

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (119887 + 119888) 120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

997904rArr 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

(37)

Now proceeding as in Case 2 (because 119879119909119899minus1

isin 119875 119879119909119899isin 119876)

we obtain that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)(

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (38)

Also from (31) if 119889H(119910119899 119879119909119899+1) ≾ 119889H(119879119909119899minus1 119879119909119899+1) then

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899minus1 119879119909119899+1) (39)

which implies that |119889H(119879119909119899 119879119909119899+1)| le |119889H(119879119909119899minus1 119879119909119899+1)|hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus2 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899minus2)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus2)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)] + 119888120593 (

1003816100381610038161003816119905119899minus1

1003816100381610038161003816)

(40)

Therefore noting that by Case 2 120593(|119905119899minus1

|) lt 120593(|119905119899minus2

|) weconclude that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (41)

Thus in all cases we get either

120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus1|)

or 120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus2|)

for 119899 = 1 we have 120593(|1199051|) le ((119887 + 119888)(1 minus 119887))120593(|119905

0|)

for 119899 = 2 we have 120593(|1199052|) le ((119887 + 119888)(1 minus 119887))120593(|119905

1|) le

((119887 + 119888)(1 minus 119887))2120593(|1199050|)

by induction we get 120593(|119905119899|) le ((119887+ 119888)(1minus119887))

119899120593(|1199050|)

Letting 119899 rarr infin we have 120593(|119905119899|) rarr 0 and by (14)

we have1003816100381610038161003816119905119899

1003816100381610038161003816=1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816997888rarr 0 as 119899 997888rarr infin (42)

so that 119879119909119899 is a Cauchy sequence and hence it converges to

point 119911 in119870 Now there exists a subsequence 119879119909119899119896

of 119879119909119899

such that it is contained in 119875 Without loss of generality wemay denote 119879119909

119899119896

= 119879119909119899 Since 119879 is continuous 119879119879119909

119899

converges to 119879119911We now show that 119879 and 119865 have common fixed point

(119879119911 = 119865119911) Using the weak commutativity of 119879 and 119865 weobtain that

119879119909119899= 119865119909119899minus1

119879119909119899minus1

isin 119870 (43)

then

119889H (119879119879119909119899 119865119879119909119899minus1) ≾ 119889H (119865119909119899minus1 119879119909119899minus1) = 119889H (119879119909119899 119879119909119899minus1)

(44)

This implies that1003816100381610038161003816119889H (119879119879119909119899 119865119879119909119899minus1)

1003816100381610038161003816le1003816100381610038161003816119889119867(119879119909119899 119879119909119899minus1

)1003816100381610038161003816 (45)

On letting 119899 rarr infin we obtain

119889H (119879119911 119865119879119909119899minus1) 997888rarr 0 (46)

which means that

119865119879119909119899minus1

997888rarr 119879119911 as 119899 997888rarr infin (47)

Now consider

120593 (1003816100381610038161003816119889H (119865119879119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119879119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119879119909119899minus1)

1003816100381610038161003816]

(48)

Letting 119899 rarr infin yields

120593 (1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816) le 119887120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816] (49)

a contradiction thus giving 120593(|119889H(119879119911 119865119911)|) = 0 whichimplies |119889H(119879119911 119865119911)| = 0 so that 119889H(119879119911 119865119911) = 0 and hence119879119911 = 119865119911

To show that 119879119911 = 119911 consider

120593 (1003816100381610038161003816119889H (119879119909119899 119879119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119909119899minus1)

1003816100381610038161003816]

(50)

Letting 119899 rarr infin we obtain

120593 (1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816) le 119888120593 [

1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816] (51)

6 Abstract and Applied Analysis

a contradiction thereby giving 120593(|119889H(119911 119879119911)|) = 0 whichimplies |119889H(119911 119879119911)| = 0 so that 119889H(119911 119879119911) = 0 and hence119911 = 119879119911 Thus we have shown that 119911 = 119879119911 = 119865119911 so 119911 is acommon fixed point of 119865 and 119879 To show that 119911 is unique let119908 be another fixed point of 119865 and 119879 then

120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119908 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119908 119865119908)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119908 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119908)

1003816100381610038161003816]

= 119888120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

(52)

a contradiction therefore giving 120593(|119889H(119908 119911)|) = 0 whichimplies that |119889H(119908 119911)| = 0 so that 119889H(119908 119911) = 0 thus 119908 = 119911This completes the proof

Remark 18 If Im119904119889H(119909 119910) = 0 119904 = 119894 119895 119896 then 119889H(119909 119910) =

119889(119909 119910) and therefore we obtain the same results in [9] Soour theorem is more general than Theorem 31 in [9] for apair of weakly commuting mappings

Using the concept of commuting mappings (see eg[10]) we can give the following result

Theorem 19 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof The proof is very similar to the proof of Theorem 17with some simple modifications so it will be omitted

Corollary 20 Let (119883 119889C) be a complete complex valuedmetric space119870lowast a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (14) and

120593 [1003816100381610038161003816119889C (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889C (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889C (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889C (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889C (119879119910 119865119909)

1003816100381610038161003816]

(53)

for all 119909 119910 isin 119870lowast with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1Let 119865 119879 119870

lowastrarr 119883 be such that 119865 is generalized 119879-

contractive satisfying the conditions

(i) 120597119870lowast sube 119879119870lowast 119865119870lowast sube 119879119870lowast(ii) 119879119909 isin 120597119870lowast rArr 119865119909 isin 119870

lowast

(iii) 119865 and 119879 are weakly mappings(iv) 119879 is continuous at 119870lowast

then there exists a unique common fixed point 119911 in 119870lowast suchthat 119911 = 119879119911 = 119865119911

Proof Since each element 119909 isin H can be written in the form119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are the

basis elements of H and 119899 = 1 2 3 Putting 1199092= 1199093= 0 we

obtain an element in C So the proof can be obtained fromTheorem 17 directly

4 Fixed Points in Normal Cone Metric Spaces

In this section we prove a fixed point theorem in normalcone metric spaces including results which generalize aresult due to Huang and Zhang in [11] as well as a resultdue to Abbas and Rhoades [12] The obtained result gives afixed point theorem for four mappings without appealing tocommutativity conditions defined on a cone metric space

Let 119864 be a real Banach space A subset 119875 of 119864 is called acone if and only if

(a) 119875 is closed and nonempty and 119875 = 0(b) 119886 119887 isin R 119886 119887 ge 0 119886 119887 isin 119875 implies that 119886119909 + 119887119910 isin 119875(c) 119875⋂(minus119875) = 0

Given cone 119875 sub 119864 we define a partial ordering lewith respectto 119875 by 119909 le 119910 if and only if 119910minus119909 isin 119875 Cone 119875 is called normalif there is a number 119870

1gt 0 such that for all 119909 119910 isin 119864

0 le 119909 le 119910 implies 119909 le 1198701

10038171003817100381710038171199101003817100381710038171003817 (54)

The least positive number satisfying the above inequality iscalled the normal constant of 119875 while 119909 ≪ 119910 stands for 119910 minus119909 isin int119875 (interior of 119875) We will write 119909 lt 119910 to indicate that119909 ≪ 119910 but 119909 = 119910

Definition 21 (see [11]) Let119883 be a nonempty set Suppose thatthe mapping 119889 119883 times 119883 rarr 119864 satisfies

(d1) 0 le 119889(119909 119910) for all 119909 119910 isin 119883 and 119889(119909 119910) = 0 if and onlyif 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

(d3) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889 is called a conemetric on119883 and (119883 119889) is called a conemetric space

Definition 22 (see [11]) Let (119883 119889) be a conemetric space 119909119899

a sequence in 119883 and 119909 isin 119883 For every 119888 isin 119864 with 0 ≪ 119888 wesay that 119909

119899 is

(1) a Cauchy sequence if there is an 119873 such that for all119899119898 gt 119873 119889(119909

119899 119909119898) ≪ 119888

(2) a convergent sequence if there is an 119873 such that forall 119899 gt 119873 119889(119909

119899 119909) ≪ 119888 for some 119909 isin 119883 119889(119909 119910) le

119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 2: Fixed Point Theorems in Quaternion-Valued Metric Spaces

2 Abstract and Applied Analysis

(v) Re(1199021) = Re(119902

2) Im1199042

(1199021) lt Im

1199042

(1199022) Im119895(1199021) =

Im119895(1199022)

(vi) Re(1199021) = Re(119902

2) Im1199043

(1199021) lt Im

1199043

(1199022) Im119896(1199021) =

Im119896(1199022)

(vii) Re(1199021) = Re(119902

2) Im119904(1199021) lt Im

119904(1199022)

(viii) Re(1199021) lt Re(119902

2) Im119904(1199021) = Im

119904(1199022)

(ix) Re(1199021) lt Re(119902

2) Im1199041

(1199021) = Im

1199041

(1199022) Im119894(1199021) lt

Im119894(1199022)

(x) Re(1199021) lt Re(119902

2) Im1199042

(1199021) = Im

1199042

(1199022) Im119895(1199021) lt

Im119895(1199022)

(xi) Re(1199021) lt Re(119902

2) Im1199043

(1199021) = Im

1199043

(1199022) Im119896(1199021) lt

Im119896(1199022)

(xii) Re(1199021) lt Re(119902

2) Im1199041

(1199021) lt Im

1199041

(1199022) Im119894(1199021) =

Im119894(1199022)

(xiii) Re(1199021) lt Re(119902

2) Im1199042

(1199021) lt Im

1199042

(1199022) Im119895(1199021) =

Im119895(1199022)

(xiv) Re(1199021) lt Re(119902

2) Im1199043

(1199021) lt Im

1199043

(1199022) Im119896(1199021) =

Im119896(1199022)

(xv) Re(1199021) lt Re(119902

2) Im119904(1199021) lt Im

119904(1199022)

(xvi) Re(1199021) = Re(119902

2) Im119904(1199021) = Im

119904(1199022)

Remark 1 In particular we will write 1199021⋨ 1199022if 1199021= 1199022and

one from (i) to (xvi) is satisfied Also we will write 1199021≺ 1199022if

only (xv) is satisfied It should be remarked that

1199021≾ 1199022997904rArr

10038161003816100381610038161199021

1003816100381610038161003816le10038161003816100381610038161199022

1003816100381610038161003816 (2)

Remark 2 The conditions from (i) to (xv) look strange butthese conditions are natural generalizations to the corre-sponding conditions in the complex setting (see [5]) Sothe number of these conditions is related to the numberof units in the working space For our quaternion settingwe have four units (one real and three imaginary) then wehave 24 conditions But in the complex setting there were 22conditions

Azam et al in [5] introduced the definition of the complexmetric space as follows

Definition 3 ([5]) Let119883 be a nonempty set and suppose thatthe mapping 119889C 119883 times 119883 rarr C satisfies the following

(d1) 0 lt 119889C(119909 119910) for all 119909 119910 isin 119883 and 119889C(119909 119910) = 0 if andonly if 119909 = 119910

(d2) 119889C(119909 119910) = 119889C(119910 119909) for all 119909 119910 isin 119883

(d3) 119889C(119909 119910) ≾ 119889C(119909 119911) + 119889C(119911 119910) for all 119909 119910 119911 isin 119883

Then (119883 119889C) is called a complex metric spaceNow we extend the above definition to Clifford analysis

Definition 4 Let 119883 be a nonempty set Suppose that themapping 119889H 119883 times 119883 rarr H satisfies

(1) 0 ≾ 119889H(119909 119910) for all 119909 119910 isin 119883 and 119889H(119909 119910) = 0 if andonly if 119909 = 119910

(2) 119889H(119909 119910) = 119889H(119910 119909) for all 119909 119910 isin 119883(3) 119889H(119909 119910) ≾ 119889H(119909 119911) + 119889H(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889H is called a quaternion valued metric on 119883 and(119883 119889H) is called a quaternion valued metric space

Example 5 Let 119889H H times H rarr H be a quaternion valuedfunction defined as 119889H(119901 119902) = |119886

0minus 1198870| + 119894|119886

1minus 1198871| + 119895|119886

2minus

1198872| + 119896|119886

3minus 1198873| where 119901 119902 isin H with

119901 = 1198860+ 1198861119894 + 1198862119895 + 1198863119896 119902 = 119887

0+ 1198871119894 + 1198872119895 + 1198873119896

119886119904 119887119904isin R 119904 = 1 2 3

(3)

Then (119883 119889H) is a quaternion metric spaceNow we give the following definitions

Definition 6 Point 119909 isin 119883 is said to be an interior point of set119860 sube 119883 whenever there exists 0 ≺ 119903 isin H such that

119861 (119909 119903) = 119910 isin 119883 119889H (119909 119910) ≺ 119903 sube 119860 (4)

Definition 7 Point 119909 isin 119883 is said to be a limit point of 119860 sube 119883

whenever for every 0 ≺ 119903 isin H

119861 (119909 119903) cap (119860 minus 119909) = 120601 (5)

Definition 8 Set 119860 is called an open set whenever eachelement of 119860 is an interior point of 119860 Subset 119861 sube 119883 is calleda closed set whenever each limit point of 119861 belongs to 119861 Thefamily

119865 = 119861 (119909 119903) 119909 isin 119883 0 ≺ 119903 (6)

is a subbase for Hausdroff topology 120591 on119883

Definition 9 Let 119909119899 be a sequence in 119883 and 119909 isin 119883 If for

every 119902 isin H with 0 ≺ 119902 there is 1198990isin N such that for all

119899 gt 1198990 119889H(119909119899 119909) ≺ 119902 then 119909

119899 is said to be convergent

if 119909119899 converges to the limit point 119909 that is 119909

119899rarr 119909 as

119899 rarr infin or lim119899119909119899= 119909 If for every 119902 isin H with 0 ≺ 119902 there is

1198990isin N such that for all 119899 gt 119899

0 119889H(119909119899 119909119899+119898) ≺ 119902 then 119909119899 is

called Cauchy sequence in (119883 119889H) If every Cauchy sequenceis convergent in (119883 119889H) then (119883 119889H) is called a completequaternion valued metric space

2 Convergence in Quaternion Metric Spaces

In this section we give some auxiliary lemmas using theconcept of quaternion metric spaces these lemmas will beused to prove some fixed point theorems of contractivemappings

Lemma 10 Let (119883 119889H) be a quaternion valued metric spaceand let 119909

119899 be a sequence in119883 Then 119909

119899 converges to 119909 if and

only if |119889H(119909119899 119909)| rarr 0 as 119899 rarr infin

Abstract and Applied Analysis 3

Proof Suppose that 119909119899 converges to 119909 For a given real

number 120576 gt 0 let

119902 =

120576

2

+ 119894

120576

2

+ 119895

120576

2

+ 119896

120576

2

(7)

Then 0 ≺ 119902 isin H and there is a natural number 119873 such that119889H(119909119899 119909) ≺ 119902 for all 119899 gt 119873

Therefore |119889H(119909119899 119909)| lt |119902| = 120576 for all 119899 gt 119873 Hence|119889H(119909119899 119909)| rarr 0 as 119899 rarr infin

Conversely suppose that |119889H(119909119899 119909)| rarr 0 as 119899 rarr infinThen given 119902 isin H with 0 ≺ 119902 there exists a real number120575 gt 0 such that for ℎ isin H

|ℎ| lt 120575 997904rArr ℎ lt 119902 (8)

For this 120575 there is a natural number119873 such that |119889H(119909119899 119909)| lt120575 for all 119899 gt 119873 Implying that 119889H(119909119899 119909) ≺ 119902 for all 119899 gt 119873hence 119909

119899 converges to 119909

Lemma 11 Let (119883 119889H) be a quaternion valued metric spaceand let 119909

119899 be a sequence in119883Then 119909

119899 is a Cauchy sequence

if and only if |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infin

Proof Suppose that 119909119899 is a Cauchy sequence For a given

real number 120576 gt 0 let

119902 =

120576

2

+ 119894

120576

2

+ 119895

120576

2

+ 119896

120576

2

(9)

Then 0 ≺ 119902 isin H and there is a natural number N such that119889H(119909119899 119909119899+119898) ≺ 119902 for all 119899 gt 119873 Therefore |119889H(119909119899 119909119899+119898)| lt|119902| = 120576 for all 119899 gt 119873 Hence |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infin

Conversely suppose that |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infinThen given 119902 isin Hwith 0 ≺ 119902 there exists a real number120575 gt 0such that for ℎ isin H we have that

|ℎ| lt 120575 997904rArr ℎ ≺ 119902 (10)

For this 120575 there is a natural number 119873 such that|119889H(119909119899 119909119899+119898)| lt 120575 for all 119899 gt 119873 which implies that119889H(119909119899 119909119899+119898) ≺ 119902 for all 119899 gt 119873 Hence 119909

119899 is a Cauchy

sequence This completes the proof of Lemma 11

Definition 12 Let (119883 119889H) be a complete quaternion valuedmetric space For all 119909 119910 isin 119883 119889H(119909 119910) represents 119909 minus

119910 A quaternion valued metric space (119883 119889H) is said to bemetrically convex if119883 has the property that for each 119909 119910 isin 119883with 119909 = 119910 there exists 119911 isin 119883 119909 = 119910 = 119911 such that

1003816100381610038161003816119889H (119909 119911) + 119889H (119911 119910)

1003816100381610038161003816=1003816100381610038161003816119889H (119909 119910)

1003816100381610038161003816 (11)

The following lemma finds immediate applications which isstraightforward from [6]

Lemma 13 Let (119883 119889H) be a metrically convex quaternionvalued metric space and 119870 a nonempty closed subset of 119883 If119909 isin 119870 and 119910 isin 119870 then there exists a point 119911 notin 120597119870 (where 120597119870stands for the boundary of 119870) such that

119889H (119909 119910) = 119889H (119909 119911) + 119889H (119911 119910) (12)

Definition 14 Let 119870 be a nonempty subset of a quaternionvalued metric space (119883 119889H) and let 119865 119879 119870 rarr 119883 satisfy thecondition

120593 [1003816100381610038161003816119889H (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119910 119865119909)

1003816100381610038161003816]

(13)

For all 119909 119910 isin 119870 with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1 and let 120593

R+ rarr R+ be an increasing continuous function for whichthe following property holds

120593 (119905) = 0 lArrrArr 119905 = 0 (14)

We call function 119865 satisfying condition (13) generalized 119879-contractive

Motivated by [7 8] we construct the following definition

Definition 15 Let 119870 be a nonempty subset of a quaternionvalued metric space (119883 119889H) and 119865 119879 119870 rarr 119883 The pair119865 119879 is said to be weakly commuting if for each 119909 119910 isin 119870

such that 119909 = 119865119910 and 119879119910 isin 119870 we have

119889H (119879119909 119865119879119910) ≾ 119889H (119879119910 119865119910) (15)

It follows that1003816100381610038161003816119889H (119879119909 119865119879119910)

1003816100381610038161003816le1003816100381610038161003816119889H (119879119910 119865119910)

1003816100381610038161003816 (16)

Remark 16 It should be remarked that Definition 15 extendsand generalizes the definition of weakly commuting map-pings which are introduced in [7]

3 Common Fixed Point Theorems inQuaternion Analysis

In this section we prove common fixed point theoremsfor two pairs of weakly commuting mappings on completequaternionmetric spacesThe obtained results will be provedusing generalized contractive conditions

Now we give the following theorem

Theorem 17 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are weakly commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof We construct the sequences 119909119899 and 119910

119899 in the

following way

4 Abstract and Applied Analysis

Let 119909 isin 120597119870 Then there exists a point 1199090isin 119870 such that

119909 = 1198791199090as 120597119870 sube 119879119870 From 119879119909

0isin 120597119870 and the implication

119879119909 isin 120597119870 rArr 119865119909 isin 119870 we conclude that 1198651199090isin 119870 cap 119865119870 sube 119879119870

Now let 1199091isin 119870 be such that

1199101= 1198791199091= 1198651199090isin 119870 (17)

Let 1199102= 1198651199091and assume that 119910

2isin 119870 then

1199102isin 119870 cap 119865119870 sube 119879119870 (18)

which implies that there exists a point 1199092isin 119870 such that 119910

2=

1198791199092 Suppose 119910

2notin 119870 then there exists a point 119901 isin 120597119870 (using

Lemma 13) such that

119889H (1198791199091 119901) + 119889H (119901 1199102) = 119889H (1198791199091 1199102) (19)

Since 119901 isin 120597119870 sube 119879119870 there exists a point 1199092isin 119870 such that

119901 = 1198791199092and so

119889H (1198791199091 1198791199092) + 119889H (1198791199092 1199102) = 119889H (1198791199091 1199102) (20)

Let 1199103= 1198651199092 Thus repeating the forgoing arguments we

obtain two sequences 119909119899 and 119910

119899 such that

(i) 119910119899+1

= 119865119909119899

(ii) 119910119899isin 119870 rArr 119910

119899= 119879119909119899 or

(iii) 119910119899notin 119870 rArr 119879119909

119899isin 120597119870

119889H (119879119909119899minus1 119879119909119899) + 119889H (119879119909119899 119910119899) = 119889H (119879119909119899minus1 119910119899) (21)

We denote

119875 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

119876 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

(22)

Obviously the two consecutive terms of 119879119909119899 cannot lie in

119876Let us denote 119905

119899= 119889H(119879119909119899 119879119909119899+1) We have the following

three cases

Case 1 If 119879119909119899 119879119909119899+1

isin 119875 then

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 [

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816]

= 120593 [1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889119867(119879119909119899minus1

119865119909119899minus1

)1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

(23)

Thus

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (24)

Case 2 If 119879119909119899isin 119875 119879119909

119899+1isin 119876 note that

119889H (119879119909119899 119879119909119899+1) + 119889H (119879119909119899+1 119910119899+1) = 119889H (119879119909119899 119910119899+1) (25)

or

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899 119910119899+1) = 119889H (119910119899 119910119899+1) (26)

which implies that1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816 (27)

Hence120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 [

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816] = 120593 [

1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)]

(28)

Therefore

120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (29)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (30)

Case 3 If 119909119899isin 119876 119879119909

119899+1isin 119875 so 119879119909

119899minus1isin 119875 Since 119879119909

119899is a

convex linear combination of 119879119909119899minus1

and 119910119899 it follows that

119889H (119879119909119899 119879119909119899+1) ≾ max 119889H (119879119909119899minus1 119879119909119899+1) 119889H (119910119899 119879119909119899+1) (31)

If 119889H(119879119909119899minus1 119879119909119899+1) ≾ 119889H(119910119899 119879119909119899+1) then 119889H(119879119909119899 119879119909119899+1) ≾119889H(119910119899 119879119909119899+1) implying that

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816 (32)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816]

(33)

Abstract and Applied Analysis 5

It follows that

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 119887120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 119888120593 (

1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816)

(34)

Since

119889H (119879119909119899minus1 119910119899) ≿ 119889H (119879119909119899 119910119899) as 119879119909119899isin 119876 (35)

then

120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) ge 120593 (119889H (119879119909119899 119910119899)) (36)

Therefore

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (119887 + 119888) 120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

997904rArr 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

(37)

Now proceeding as in Case 2 (because 119879119909119899minus1

isin 119875 119879119909119899isin 119876)

we obtain that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)(

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (38)

Also from (31) if 119889H(119910119899 119879119909119899+1) ≾ 119889H(119879119909119899minus1 119879119909119899+1) then

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899minus1 119879119909119899+1) (39)

which implies that |119889H(119879119909119899 119879119909119899+1)| le |119889H(119879119909119899minus1 119879119909119899+1)|hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus2 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899minus2)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus2)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)] + 119888120593 (

1003816100381610038161003816119905119899minus1

1003816100381610038161003816)

(40)

Therefore noting that by Case 2 120593(|119905119899minus1

|) lt 120593(|119905119899minus2

|) weconclude that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (41)

Thus in all cases we get either

120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus1|)

or 120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus2|)

for 119899 = 1 we have 120593(|1199051|) le ((119887 + 119888)(1 minus 119887))120593(|119905

0|)

for 119899 = 2 we have 120593(|1199052|) le ((119887 + 119888)(1 minus 119887))120593(|119905

1|) le

((119887 + 119888)(1 minus 119887))2120593(|1199050|)

by induction we get 120593(|119905119899|) le ((119887+ 119888)(1minus119887))

119899120593(|1199050|)

Letting 119899 rarr infin we have 120593(|119905119899|) rarr 0 and by (14)

we have1003816100381610038161003816119905119899

1003816100381610038161003816=1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816997888rarr 0 as 119899 997888rarr infin (42)

so that 119879119909119899 is a Cauchy sequence and hence it converges to

point 119911 in119870 Now there exists a subsequence 119879119909119899119896

of 119879119909119899

such that it is contained in 119875 Without loss of generality wemay denote 119879119909

119899119896

= 119879119909119899 Since 119879 is continuous 119879119879119909

119899

converges to 119879119911We now show that 119879 and 119865 have common fixed point

(119879119911 = 119865119911) Using the weak commutativity of 119879 and 119865 weobtain that

119879119909119899= 119865119909119899minus1

119879119909119899minus1

isin 119870 (43)

then

119889H (119879119879119909119899 119865119879119909119899minus1) ≾ 119889H (119865119909119899minus1 119879119909119899minus1) = 119889H (119879119909119899 119879119909119899minus1)

(44)

This implies that1003816100381610038161003816119889H (119879119879119909119899 119865119879119909119899minus1)

1003816100381610038161003816le1003816100381610038161003816119889119867(119879119909119899 119879119909119899minus1

)1003816100381610038161003816 (45)

On letting 119899 rarr infin we obtain

119889H (119879119911 119865119879119909119899minus1) 997888rarr 0 (46)

which means that

119865119879119909119899minus1

997888rarr 119879119911 as 119899 997888rarr infin (47)

Now consider

120593 (1003816100381610038161003816119889H (119865119879119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119879119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119879119909119899minus1)

1003816100381610038161003816]

(48)

Letting 119899 rarr infin yields

120593 (1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816) le 119887120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816] (49)

a contradiction thus giving 120593(|119889H(119879119911 119865119911)|) = 0 whichimplies |119889H(119879119911 119865119911)| = 0 so that 119889H(119879119911 119865119911) = 0 and hence119879119911 = 119865119911

To show that 119879119911 = 119911 consider

120593 (1003816100381610038161003816119889H (119879119909119899 119879119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119909119899minus1)

1003816100381610038161003816]

(50)

Letting 119899 rarr infin we obtain

120593 (1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816) le 119888120593 [

1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816] (51)

6 Abstract and Applied Analysis

a contradiction thereby giving 120593(|119889H(119911 119879119911)|) = 0 whichimplies |119889H(119911 119879119911)| = 0 so that 119889H(119911 119879119911) = 0 and hence119911 = 119879119911 Thus we have shown that 119911 = 119879119911 = 119865119911 so 119911 is acommon fixed point of 119865 and 119879 To show that 119911 is unique let119908 be another fixed point of 119865 and 119879 then

120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119908 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119908 119865119908)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119908 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119908)

1003816100381610038161003816]

= 119888120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

(52)

a contradiction therefore giving 120593(|119889H(119908 119911)|) = 0 whichimplies that |119889H(119908 119911)| = 0 so that 119889H(119908 119911) = 0 thus 119908 = 119911This completes the proof

Remark 18 If Im119904119889H(119909 119910) = 0 119904 = 119894 119895 119896 then 119889H(119909 119910) =

119889(119909 119910) and therefore we obtain the same results in [9] Soour theorem is more general than Theorem 31 in [9] for apair of weakly commuting mappings

Using the concept of commuting mappings (see eg[10]) we can give the following result

Theorem 19 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof The proof is very similar to the proof of Theorem 17with some simple modifications so it will be omitted

Corollary 20 Let (119883 119889C) be a complete complex valuedmetric space119870lowast a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (14) and

120593 [1003816100381610038161003816119889C (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889C (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889C (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889C (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889C (119879119910 119865119909)

1003816100381610038161003816]

(53)

for all 119909 119910 isin 119870lowast with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1Let 119865 119879 119870

lowastrarr 119883 be such that 119865 is generalized 119879-

contractive satisfying the conditions

(i) 120597119870lowast sube 119879119870lowast 119865119870lowast sube 119879119870lowast(ii) 119879119909 isin 120597119870lowast rArr 119865119909 isin 119870

lowast

(iii) 119865 and 119879 are weakly mappings(iv) 119879 is continuous at 119870lowast

then there exists a unique common fixed point 119911 in 119870lowast suchthat 119911 = 119879119911 = 119865119911

Proof Since each element 119909 isin H can be written in the form119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are the

basis elements of H and 119899 = 1 2 3 Putting 1199092= 1199093= 0 we

obtain an element in C So the proof can be obtained fromTheorem 17 directly

4 Fixed Points in Normal Cone Metric Spaces

In this section we prove a fixed point theorem in normalcone metric spaces including results which generalize aresult due to Huang and Zhang in [11] as well as a resultdue to Abbas and Rhoades [12] The obtained result gives afixed point theorem for four mappings without appealing tocommutativity conditions defined on a cone metric space

Let 119864 be a real Banach space A subset 119875 of 119864 is called acone if and only if

(a) 119875 is closed and nonempty and 119875 = 0(b) 119886 119887 isin R 119886 119887 ge 0 119886 119887 isin 119875 implies that 119886119909 + 119887119910 isin 119875(c) 119875⋂(minus119875) = 0

Given cone 119875 sub 119864 we define a partial ordering lewith respectto 119875 by 119909 le 119910 if and only if 119910minus119909 isin 119875 Cone 119875 is called normalif there is a number 119870

1gt 0 such that for all 119909 119910 isin 119864

0 le 119909 le 119910 implies 119909 le 1198701

10038171003817100381710038171199101003817100381710038171003817 (54)

The least positive number satisfying the above inequality iscalled the normal constant of 119875 while 119909 ≪ 119910 stands for 119910 minus119909 isin int119875 (interior of 119875) We will write 119909 lt 119910 to indicate that119909 ≪ 119910 but 119909 = 119910

Definition 21 (see [11]) Let119883 be a nonempty set Suppose thatthe mapping 119889 119883 times 119883 rarr 119864 satisfies

(d1) 0 le 119889(119909 119910) for all 119909 119910 isin 119883 and 119889(119909 119910) = 0 if and onlyif 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

(d3) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889 is called a conemetric on119883 and (119883 119889) is called a conemetric space

Definition 22 (see [11]) Let (119883 119889) be a conemetric space 119909119899

a sequence in 119883 and 119909 isin 119883 For every 119888 isin 119864 with 0 ≪ 119888 wesay that 119909

119899 is

(1) a Cauchy sequence if there is an 119873 such that for all119899119898 gt 119873 119889(119909

119899 119909119898) ≪ 119888

(2) a convergent sequence if there is an 119873 such that forall 119899 gt 119873 119889(119909

119899 119909) ≪ 119888 for some 119909 isin 119883 119889(119909 119910) le

119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 3: Fixed Point Theorems in Quaternion-Valued Metric Spaces

Abstract and Applied Analysis 3

Proof Suppose that 119909119899 converges to 119909 For a given real

number 120576 gt 0 let

119902 =

120576

2

+ 119894

120576

2

+ 119895

120576

2

+ 119896

120576

2

(7)

Then 0 ≺ 119902 isin H and there is a natural number 119873 such that119889H(119909119899 119909) ≺ 119902 for all 119899 gt 119873

Therefore |119889H(119909119899 119909)| lt |119902| = 120576 for all 119899 gt 119873 Hence|119889H(119909119899 119909)| rarr 0 as 119899 rarr infin

Conversely suppose that |119889H(119909119899 119909)| rarr 0 as 119899 rarr infinThen given 119902 isin H with 0 ≺ 119902 there exists a real number120575 gt 0 such that for ℎ isin H

|ℎ| lt 120575 997904rArr ℎ lt 119902 (8)

For this 120575 there is a natural number119873 such that |119889H(119909119899 119909)| lt120575 for all 119899 gt 119873 Implying that 119889H(119909119899 119909) ≺ 119902 for all 119899 gt 119873hence 119909

119899 converges to 119909

Lemma 11 Let (119883 119889H) be a quaternion valued metric spaceand let 119909

119899 be a sequence in119883Then 119909

119899 is a Cauchy sequence

if and only if |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infin

Proof Suppose that 119909119899 is a Cauchy sequence For a given

real number 120576 gt 0 let

119902 =

120576

2

+ 119894

120576

2

+ 119895

120576

2

+ 119896

120576

2

(9)

Then 0 ≺ 119902 isin H and there is a natural number N such that119889H(119909119899 119909119899+119898) ≺ 119902 for all 119899 gt 119873 Therefore |119889H(119909119899 119909119899+119898)| lt|119902| = 120576 for all 119899 gt 119873 Hence |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infin

Conversely suppose that |119889H(119909119899 119909119899+119898)| rarr 0 as 119899 rarr infinThen given 119902 isin Hwith 0 ≺ 119902 there exists a real number120575 gt 0such that for ℎ isin H we have that

|ℎ| lt 120575 997904rArr ℎ ≺ 119902 (10)

For this 120575 there is a natural number 119873 such that|119889H(119909119899 119909119899+119898)| lt 120575 for all 119899 gt 119873 which implies that119889H(119909119899 119909119899+119898) ≺ 119902 for all 119899 gt 119873 Hence 119909

119899 is a Cauchy

sequence This completes the proof of Lemma 11

Definition 12 Let (119883 119889H) be a complete quaternion valuedmetric space For all 119909 119910 isin 119883 119889H(119909 119910) represents 119909 minus

119910 A quaternion valued metric space (119883 119889H) is said to bemetrically convex if119883 has the property that for each 119909 119910 isin 119883with 119909 = 119910 there exists 119911 isin 119883 119909 = 119910 = 119911 such that

1003816100381610038161003816119889H (119909 119911) + 119889H (119911 119910)

1003816100381610038161003816=1003816100381610038161003816119889H (119909 119910)

1003816100381610038161003816 (11)

The following lemma finds immediate applications which isstraightforward from [6]

Lemma 13 Let (119883 119889H) be a metrically convex quaternionvalued metric space and 119870 a nonempty closed subset of 119883 If119909 isin 119870 and 119910 isin 119870 then there exists a point 119911 notin 120597119870 (where 120597119870stands for the boundary of 119870) such that

119889H (119909 119910) = 119889H (119909 119911) + 119889H (119911 119910) (12)

Definition 14 Let 119870 be a nonempty subset of a quaternionvalued metric space (119883 119889H) and let 119865 119879 119870 rarr 119883 satisfy thecondition

120593 [1003816100381610038161003816119889H (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119910 119865119909)

1003816100381610038161003816]

(13)

For all 119909 119910 isin 119870 with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1 and let 120593

R+ rarr R+ be an increasing continuous function for whichthe following property holds

120593 (119905) = 0 lArrrArr 119905 = 0 (14)

We call function 119865 satisfying condition (13) generalized 119879-contractive

Motivated by [7 8] we construct the following definition

Definition 15 Let 119870 be a nonempty subset of a quaternionvalued metric space (119883 119889H) and 119865 119879 119870 rarr 119883 The pair119865 119879 is said to be weakly commuting if for each 119909 119910 isin 119870

such that 119909 = 119865119910 and 119879119910 isin 119870 we have

119889H (119879119909 119865119879119910) ≾ 119889H (119879119910 119865119910) (15)

It follows that1003816100381610038161003816119889H (119879119909 119865119879119910)

1003816100381610038161003816le1003816100381610038161003816119889H (119879119910 119865119910)

1003816100381610038161003816 (16)

Remark 16 It should be remarked that Definition 15 extendsand generalizes the definition of weakly commuting map-pings which are introduced in [7]

3 Common Fixed Point Theorems inQuaternion Analysis

In this section we prove common fixed point theoremsfor two pairs of weakly commuting mappings on completequaternionmetric spacesThe obtained results will be provedusing generalized contractive conditions

Now we give the following theorem

Theorem 17 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are weakly commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof We construct the sequences 119909119899 and 119910

119899 in the

following way

4 Abstract and Applied Analysis

Let 119909 isin 120597119870 Then there exists a point 1199090isin 119870 such that

119909 = 1198791199090as 120597119870 sube 119879119870 From 119879119909

0isin 120597119870 and the implication

119879119909 isin 120597119870 rArr 119865119909 isin 119870 we conclude that 1198651199090isin 119870 cap 119865119870 sube 119879119870

Now let 1199091isin 119870 be such that

1199101= 1198791199091= 1198651199090isin 119870 (17)

Let 1199102= 1198651199091and assume that 119910

2isin 119870 then

1199102isin 119870 cap 119865119870 sube 119879119870 (18)

which implies that there exists a point 1199092isin 119870 such that 119910

2=

1198791199092 Suppose 119910

2notin 119870 then there exists a point 119901 isin 120597119870 (using

Lemma 13) such that

119889H (1198791199091 119901) + 119889H (119901 1199102) = 119889H (1198791199091 1199102) (19)

Since 119901 isin 120597119870 sube 119879119870 there exists a point 1199092isin 119870 such that

119901 = 1198791199092and so

119889H (1198791199091 1198791199092) + 119889H (1198791199092 1199102) = 119889H (1198791199091 1199102) (20)

Let 1199103= 1198651199092 Thus repeating the forgoing arguments we

obtain two sequences 119909119899 and 119910

119899 such that

(i) 119910119899+1

= 119865119909119899

(ii) 119910119899isin 119870 rArr 119910

119899= 119879119909119899 or

(iii) 119910119899notin 119870 rArr 119879119909

119899isin 120597119870

119889H (119879119909119899minus1 119879119909119899) + 119889H (119879119909119899 119910119899) = 119889H (119879119909119899minus1 119910119899) (21)

We denote

119875 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

119876 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

(22)

Obviously the two consecutive terms of 119879119909119899 cannot lie in

119876Let us denote 119905

119899= 119889H(119879119909119899 119879119909119899+1) We have the following

three cases

Case 1 If 119879119909119899 119879119909119899+1

isin 119875 then

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 [

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816]

= 120593 [1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889119867(119879119909119899minus1

119865119909119899minus1

)1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

(23)

Thus

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (24)

Case 2 If 119879119909119899isin 119875 119879119909

119899+1isin 119876 note that

119889H (119879119909119899 119879119909119899+1) + 119889H (119879119909119899+1 119910119899+1) = 119889H (119879119909119899 119910119899+1) (25)

or

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899 119910119899+1) = 119889H (119910119899 119910119899+1) (26)

which implies that1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816 (27)

Hence120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 [

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816] = 120593 [

1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)]

(28)

Therefore

120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (29)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (30)

Case 3 If 119909119899isin 119876 119879119909

119899+1isin 119875 so 119879119909

119899minus1isin 119875 Since 119879119909

119899is a

convex linear combination of 119879119909119899minus1

and 119910119899 it follows that

119889H (119879119909119899 119879119909119899+1) ≾ max 119889H (119879119909119899minus1 119879119909119899+1) 119889H (119910119899 119879119909119899+1) (31)

If 119889H(119879119909119899minus1 119879119909119899+1) ≾ 119889H(119910119899 119879119909119899+1) then 119889H(119879119909119899 119879119909119899+1) ≾119889H(119910119899 119879119909119899+1) implying that

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816 (32)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816]

(33)

Abstract and Applied Analysis 5

It follows that

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 119887120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 119888120593 (

1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816)

(34)

Since

119889H (119879119909119899minus1 119910119899) ≿ 119889H (119879119909119899 119910119899) as 119879119909119899isin 119876 (35)

then

120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) ge 120593 (119889H (119879119909119899 119910119899)) (36)

Therefore

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (119887 + 119888) 120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

997904rArr 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

(37)

Now proceeding as in Case 2 (because 119879119909119899minus1

isin 119875 119879119909119899isin 119876)

we obtain that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)(

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (38)

Also from (31) if 119889H(119910119899 119879119909119899+1) ≾ 119889H(119879119909119899minus1 119879119909119899+1) then

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899minus1 119879119909119899+1) (39)

which implies that |119889H(119879119909119899 119879119909119899+1)| le |119889H(119879119909119899minus1 119879119909119899+1)|hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus2 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899minus2)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus2)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)] + 119888120593 (

1003816100381610038161003816119905119899minus1

1003816100381610038161003816)

(40)

Therefore noting that by Case 2 120593(|119905119899minus1

|) lt 120593(|119905119899minus2

|) weconclude that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (41)

Thus in all cases we get either

120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus1|)

or 120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus2|)

for 119899 = 1 we have 120593(|1199051|) le ((119887 + 119888)(1 minus 119887))120593(|119905

0|)

for 119899 = 2 we have 120593(|1199052|) le ((119887 + 119888)(1 minus 119887))120593(|119905

1|) le

((119887 + 119888)(1 minus 119887))2120593(|1199050|)

by induction we get 120593(|119905119899|) le ((119887+ 119888)(1minus119887))

119899120593(|1199050|)

Letting 119899 rarr infin we have 120593(|119905119899|) rarr 0 and by (14)

we have1003816100381610038161003816119905119899

1003816100381610038161003816=1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816997888rarr 0 as 119899 997888rarr infin (42)

so that 119879119909119899 is a Cauchy sequence and hence it converges to

point 119911 in119870 Now there exists a subsequence 119879119909119899119896

of 119879119909119899

such that it is contained in 119875 Without loss of generality wemay denote 119879119909

119899119896

= 119879119909119899 Since 119879 is continuous 119879119879119909

119899

converges to 119879119911We now show that 119879 and 119865 have common fixed point

(119879119911 = 119865119911) Using the weak commutativity of 119879 and 119865 weobtain that

119879119909119899= 119865119909119899minus1

119879119909119899minus1

isin 119870 (43)

then

119889H (119879119879119909119899 119865119879119909119899minus1) ≾ 119889H (119865119909119899minus1 119879119909119899minus1) = 119889H (119879119909119899 119879119909119899minus1)

(44)

This implies that1003816100381610038161003816119889H (119879119879119909119899 119865119879119909119899minus1)

1003816100381610038161003816le1003816100381610038161003816119889119867(119879119909119899 119879119909119899minus1

)1003816100381610038161003816 (45)

On letting 119899 rarr infin we obtain

119889H (119879119911 119865119879119909119899minus1) 997888rarr 0 (46)

which means that

119865119879119909119899minus1

997888rarr 119879119911 as 119899 997888rarr infin (47)

Now consider

120593 (1003816100381610038161003816119889H (119865119879119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119879119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119879119909119899minus1)

1003816100381610038161003816]

(48)

Letting 119899 rarr infin yields

120593 (1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816) le 119887120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816] (49)

a contradiction thus giving 120593(|119889H(119879119911 119865119911)|) = 0 whichimplies |119889H(119879119911 119865119911)| = 0 so that 119889H(119879119911 119865119911) = 0 and hence119879119911 = 119865119911

To show that 119879119911 = 119911 consider

120593 (1003816100381610038161003816119889H (119879119909119899 119879119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119909119899minus1)

1003816100381610038161003816]

(50)

Letting 119899 rarr infin we obtain

120593 (1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816) le 119888120593 [

1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816] (51)

6 Abstract and Applied Analysis

a contradiction thereby giving 120593(|119889H(119911 119879119911)|) = 0 whichimplies |119889H(119911 119879119911)| = 0 so that 119889H(119911 119879119911) = 0 and hence119911 = 119879119911 Thus we have shown that 119911 = 119879119911 = 119865119911 so 119911 is acommon fixed point of 119865 and 119879 To show that 119911 is unique let119908 be another fixed point of 119865 and 119879 then

120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119908 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119908 119865119908)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119908 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119908)

1003816100381610038161003816]

= 119888120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

(52)

a contradiction therefore giving 120593(|119889H(119908 119911)|) = 0 whichimplies that |119889H(119908 119911)| = 0 so that 119889H(119908 119911) = 0 thus 119908 = 119911This completes the proof

Remark 18 If Im119904119889H(119909 119910) = 0 119904 = 119894 119895 119896 then 119889H(119909 119910) =

119889(119909 119910) and therefore we obtain the same results in [9] Soour theorem is more general than Theorem 31 in [9] for apair of weakly commuting mappings

Using the concept of commuting mappings (see eg[10]) we can give the following result

Theorem 19 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof The proof is very similar to the proof of Theorem 17with some simple modifications so it will be omitted

Corollary 20 Let (119883 119889C) be a complete complex valuedmetric space119870lowast a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (14) and

120593 [1003816100381610038161003816119889C (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889C (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889C (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889C (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889C (119879119910 119865119909)

1003816100381610038161003816]

(53)

for all 119909 119910 isin 119870lowast with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1Let 119865 119879 119870

lowastrarr 119883 be such that 119865 is generalized 119879-

contractive satisfying the conditions

(i) 120597119870lowast sube 119879119870lowast 119865119870lowast sube 119879119870lowast(ii) 119879119909 isin 120597119870lowast rArr 119865119909 isin 119870

lowast

(iii) 119865 and 119879 are weakly mappings(iv) 119879 is continuous at 119870lowast

then there exists a unique common fixed point 119911 in 119870lowast suchthat 119911 = 119879119911 = 119865119911

Proof Since each element 119909 isin H can be written in the form119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are the

basis elements of H and 119899 = 1 2 3 Putting 1199092= 1199093= 0 we

obtain an element in C So the proof can be obtained fromTheorem 17 directly

4 Fixed Points in Normal Cone Metric Spaces

In this section we prove a fixed point theorem in normalcone metric spaces including results which generalize aresult due to Huang and Zhang in [11] as well as a resultdue to Abbas and Rhoades [12] The obtained result gives afixed point theorem for four mappings without appealing tocommutativity conditions defined on a cone metric space

Let 119864 be a real Banach space A subset 119875 of 119864 is called acone if and only if

(a) 119875 is closed and nonempty and 119875 = 0(b) 119886 119887 isin R 119886 119887 ge 0 119886 119887 isin 119875 implies that 119886119909 + 119887119910 isin 119875(c) 119875⋂(minus119875) = 0

Given cone 119875 sub 119864 we define a partial ordering lewith respectto 119875 by 119909 le 119910 if and only if 119910minus119909 isin 119875 Cone 119875 is called normalif there is a number 119870

1gt 0 such that for all 119909 119910 isin 119864

0 le 119909 le 119910 implies 119909 le 1198701

10038171003817100381710038171199101003817100381710038171003817 (54)

The least positive number satisfying the above inequality iscalled the normal constant of 119875 while 119909 ≪ 119910 stands for 119910 minus119909 isin int119875 (interior of 119875) We will write 119909 lt 119910 to indicate that119909 ≪ 119910 but 119909 = 119910

Definition 21 (see [11]) Let119883 be a nonempty set Suppose thatthe mapping 119889 119883 times 119883 rarr 119864 satisfies

(d1) 0 le 119889(119909 119910) for all 119909 119910 isin 119883 and 119889(119909 119910) = 0 if and onlyif 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

(d3) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889 is called a conemetric on119883 and (119883 119889) is called a conemetric space

Definition 22 (see [11]) Let (119883 119889) be a conemetric space 119909119899

a sequence in 119883 and 119909 isin 119883 For every 119888 isin 119864 with 0 ≪ 119888 wesay that 119909

119899 is

(1) a Cauchy sequence if there is an 119873 such that for all119899119898 gt 119873 119889(119909

119899 119909119898) ≪ 119888

(2) a convergent sequence if there is an 119873 such that forall 119899 gt 119873 119889(119909

119899 119909) ≪ 119888 for some 119909 isin 119883 119889(119909 119910) le

119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 4: Fixed Point Theorems in Quaternion-Valued Metric Spaces

4 Abstract and Applied Analysis

Let 119909 isin 120597119870 Then there exists a point 1199090isin 119870 such that

119909 = 1198791199090as 120597119870 sube 119879119870 From 119879119909

0isin 120597119870 and the implication

119879119909 isin 120597119870 rArr 119865119909 isin 119870 we conclude that 1198651199090isin 119870 cap 119865119870 sube 119879119870

Now let 1199091isin 119870 be such that

1199101= 1198791199091= 1198651199090isin 119870 (17)

Let 1199102= 1198651199091and assume that 119910

2isin 119870 then

1199102isin 119870 cap 119865119870 sube 119879119870 (18)

which implies that there exists a point 1199092isin 119870 such that 119910

2=

1198791199092 Suppose 119910

2notin 119870 then there exists a point 119901 isin 120597119870 (using

Lemma 13) such that

119889H (1198791199091 119901) + 119889H (119901 1199102) = 119889H (1198791199091 1199102) (19)

Since 119901 isin 120597119870 sube 119879119870 there exists a point 1199092isin 119870 such that

119901 = 1198791199092and so

119889H (1198791199091 1198791199092) + 119889H (1198791199092 1199102) = 119889H (1198791199091 1199102) (20)

Let 1199103= 1198651199092 Thus repeating the forgoing arguments we

obtain two sequences 119909119899 and 119910

119899 such that

(i) 119910119899+1

= 119865119909119899

(ii) 119910119899isin 119870 rArr 119910

119899= 119879119909119899 or

(iii) 119910119899notin 119870 rArr 119879119909

119899isin 120597119870

119889H (119879119909119899minus1 119879119909119899) + 119889H (119879119909119899 119910119899) = 119889H (119879119909119899minus1 119910119899) (21)

We denote

119875 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

119876 = 119879119909119894isin 119879119909

119899 119879119909

119894= 119910119894

(22)

Obviously the two consecutive terms of 119879119909119899 cannot lie in

119876Let us denote 119905

119899= 119889H(119879119909119899 119879119909119899+1) We have the following

three cases

Case 1 If 119879119909119899 119879119909119899+1

isin 119875 then

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 [

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816]

= 120593 [1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889119867(119879119909119899minus1

119865119909119899minus1

)1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

(23)

Thus

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (24)

Case 2 If 119879119909119899isin 119875 119879119909

119899+1isin 119876 note that

119889H (119879119909119899 119879119909119899+1) + 119889H (119879119909119899+1 119910119899+1) = 119889H (119879119909119899 119910119899+1) (25)

or

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899 119910119899+1) = 119889H (119910119899 119910119899+1) (26)

which implies that1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816 (27)

Hence120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 [

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816] = 120593 [

1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)]

(28)

Therefore

120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (29)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 120593 (

1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816) le (

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus1

1003816100381610038161003816) (30)

Case 3 If 119909119899isin 119876 119879119909

119899+1isin 119875 so 119879119909

119899minus1isin 119875 Since 119879119909

119899is a

convex linear combination of 119879119909119899minus1

and 119910119899 it follows that

119889H (119879119909119899 119879119909119899+1) ≾ max 119889H (119879119909119899minus1 119879119909119899+1) 119889H (119910119899 119879119909119899+1) (31)

If 119889H(119879119909119899minus1 119879119909119899+1) ≾ 119889H(119910119899 119879119909119899+1) then 119889H(119879119909119899 119879119909119899+1) ≾119889H(119910119899 119879119909119899+1) implying that

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816le1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816 (32)

Hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119910119899 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119910119899 119910119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus1)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816]

(33)

Abstract and Applied Analysis 5

It follows that

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 119887120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 119888120593 (

1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816)

(34)

Since

119889H (119879119909119899minus1 119910119899) ≿ 119889H (119879119909119899 119910119899) as 119879119909119899isin 119876 (35)

then

120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) ge 120593 (119889H (119879119909119899 119910119899)) (36)

Therefore

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (119887 + 119888) 120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

997904rArr 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

(37)

Now proceeding as in Case 2 (because 119879119909119899minus1

isin 119875 119879119909119899isin 119876)

we obtain that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)(

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (38)

Also from (31) if 119889H(119910119899 119879119909119899+1) ≾ 119889H(119879119909119899minus1 119879119909119899+1) then

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899minus1 119879119909119899+1) (39)

which implies that |119889H(119879119909119899 119879119909119899+1)| le |119889H(119879119909119899minus1 119879119909119899+1)|hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus2 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899minus2)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus2)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)] + 119888120593 (

1003816100381610038161003816119905119899minus1

1003816100381610038161003816)

(40)

Therefore noting that by Case 2 120593(|119905119899minus1

|) lt 120593(|119905119899minus2

|) weconclude that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (41)

Thus in all cases we get either

120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus1|)

or 120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus2|)

for 119899 = 1 we have 120593(|1199051|) le ((119887 + 119888)(1 minus 119887))120593(|119905

0|)

for 119899 = 2 we have 120593(|1199052|) le ((119887 + 119888)(1 minus 119887))120593(|119905

1|) le

((119887 + 119888)(1 minus 119887))2120593(|1199050|)

by induction we get 120593(|119905119899|) le ((119887+ 119888)(1minus119887))

119899120593(|1199050|)

Letting 119899 rarr infin we have 120593(|119905119899|) rarr 0 and by (14)

we have1003816100381610038161003816119905119899

1003816100381610038161003816=1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816997888rarr 0 as 119899 997888rarr infin (42)

so that 119879119909119899 is a Cauchy sequence and hence it converges to

point 119911 in119870 Now there exists a subsequence 119879119909119899119896

of 119879119909119899

such that it is contained in 119875 Without loss of generality wemay denote 119879119909

119899119896

= 119879119909119899 Since 119879 is continuous 119879119879119909

119899

converges to 119879119911We now show that 119879 and 119865 have common fixed point

(119879119911 = 119865119911) Using the weak commutativity of 119879 and 119865 weobtain that

119879119909119899= 119865119909119899minus1

119879119909119899minus1

isin 119870 (43)

then

119889H (119879119879119909119899 119865119879119909119899minus1) ≾ 119889H (119865119909119899minus1 119879119909119899minus1) = 119889H (119879119909119899 119879119909119899minus1)

(44)

This implies that1003816100381610038161003816119889H (119879119879119909119899 119865119879119909119899minus1)

1003816100381610038161003816le1003816100381610038161003816119889119867(119879119909119899 119879119909119899minus1

)1003816100381610038161003816 (45)

On letting 119899 rarr infin we obtain

119889H (119879119911 119865119879119909119899minus1) 997888rarr 0 (46)

which means that

119865119879119909119899minus1

997888rarr 119879119911 as 119899 997888rarr infin (47)

Now consider

120593 (1003816100381610038161003816119889H (119865119879119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119879119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119879119909119899minus1)

1003816100381610038161003816]

(48)

Letting 119899 rarr infin yields

120593 (1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816) le 119887120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816] (49)

a contradiction thus giving 120593(|119889H(119879119911 119865119911)|) = 0 whichimplies |119889H(119879119911 119865119911)| = 0 so that 119889H(119879119911 119865119911) = 0 and hence119879119911 = 119865119911

To show that 119879119911 = 119911 consider

120593 (1003816100381610038161003816119889H (119879119909119899 119879119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119909119899minus1)

1003816100381610038161003816]

(50)

Letting 119899 rarr infin we obtain

120593 (1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816) le 119888120593 [

1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816] (51)

6 Abstract and Applied Analysis

a contradiction thereby giving 120593(|119889H(119911 119879119911)|) = 0 whichimplies |119889H(119911 119879119911)| = 0 so that 119889H(119911 119879119911) = 0 and hence119911 = 119879119911 Thus we have shown that 119911 = 119879119911 = 119865119911 so 119911 is acommon fixed point of 119865 and 119879 To show that 119911 is unique let119908 be another fixed point of 119865 and 119879 then

120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119908 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119908 119865119908)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119908 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119908)

1003816100381610038161003816]

= 119888120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

(52)

a contradiction therefore giving 120593(|119889H(119908 119911)|) = 0 whichimplies that |119889H(119908 119911)| = 0 so that 119889H(119908 119911) = 0 thus 119908 = 119911This completes the proof

Remark 18 If Im119904119889H(119909 119910) = 0 119904 = 119894 119895 119896 then 119889H(119909 119910) =

119889(119909 119910) and therefore we obtain the same results in [9] Soour theorem is more general than Theorem 31 in [9] for apair of weakly commuting mappings

Using the concept of commuting mappings (see eg[10]) we can give the following result

Theorem 19 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof The proof is very similar to the proof of Theorem 17with some simple modifications so it will be omitted

Corollary 20 Let (119883 119889C) be a complete complex valuedmetric space119870lowast a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (14) and

120593 [1003816100381610038161003816119889C (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889C (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889C (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889C (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889C (119879119910 119865119909)

1003816100381610038161003816]

(53)

for all 119909 119910 isin 119870lowast with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1Let 119865 119879 119870

lowastrarr 119883 be such that 119865 is generalized 119879-

contractive satisfying the conditions

(i) 120597119870lowast sube 119879119870lowast 119865119870lowast sube 119879119870lowast(ii) 119879119909 isin 120597119870lowast rArr 119865119909 isin 119870

lowast

(iii) 119865 and 119879 are weakly mappings(iv) 119879 is continuous at 119870lowast

then there exists a unique common fixed point 119911 in 119870lowast suchthat 119911 = 119879119911 = 119865119911

Proof Since each element 119909 isin H can be written in the form119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are the

basis elements of H and 119899 = 1 2 3 Putting 1199092= 1199093= 0 we

obtain an element in C So the proof can be obtained fromTheorem 17 directly

4 Fixed Points in Normal Cone Metric Spaces

In this section we prove a fixed point theorem in normalcone metric spaces including results which generalize aresult due to Huang and Zhang in [11] as well as a resultdue to Abbas and Rhoades [12] The obtained result gives afixed point theorem for four mappings without appealing tocommutativity conditions defined on a cone metric space

Let 119864 be a real Banach space A subset 119875 of 119864 is called acone if and only if

(a) 119875 is closed and nonempty and 119875 = 0(b) 119886 119887 isin R 119886 119887 ge 0 119886 119887 isin 119875 implies that 119886119909 + 119887119910 isin 119875(c) 119875⋂(minus119875) = 0

Given cone 119875 sub 119864 we define a partial ordering lewith respectto 119875 by 119909 le 119910 if and only if 119910minus119909 isin 119875 Cone 119875 is called normalif there is a number 119870

1gt 0 such that for all 119909 119910 isin 119864

0 le 119909 le 119910 implies 119909 le 1198701

10038171003817100381710038171199101003817100381710038171003817 (54)

The least positive number satisfying the above inequality iscalled the normal constant of 119875 while 119909 ≪ 119910 stands for 119910 minus119909 isin int119875 (interior of 119875) We will write 119909 lt 119910 to indicate that119909 ≪ 119910 but 119909 = 119910

Definition 21 (see [11]) Let119883 be a nonempty set Suppose thatthe mapping 119889 119883 times 119883 rarr 119864 satisfies

(d1) 0 le 119889(119909 119910) for all 119909 119910 isin 119883 and 119889(119909 119910) = 0 if and onlyif 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

(d3) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889 is called a conemetric on119883 and (119883 119889) is called a conemetric space

Definition 22 (see [11]) Let (119883 119889) be a conemetric space 119909119899

a sequence in 119883 and 119909 isin 119883 For every 119888 isin 119864 with 0 ≪ 119888 wesay that 119909

119899 is

(1) a Cauchy sequence if there is an 119873 such that for all119899119898 gt 119873 119889(119909

119899 119909119898) ≪ 119888

(2) a convergent sequence if there is an 119873 such that forall 119899 gt 119873 119889(119909

119899 119909) ≪ 119888 for some 119909 isin 119883 119889(119909 119910) le

119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 5: Fixed Point Theorems in Quaternion-Valued Metric Spaces

Abstract and Applied Analysis 5

It follows that

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le 119887120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) + 119888120593 (

1003816100381610038161003816119889H (119879119909119899 119910119899)

1003816100381610038161003816)

(34)

Since

119889H (119879119909119899minus1 119910119899) ≿ 119889H (119879119909119899 119910119899) as 119879119909119899isin 119876 (35)

then

120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816) ge 120593 (119889H (119879119909119899 119910119899)) (36)

Therefore

(1 minus 119887) 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (119887 + 119888) 120593 (

1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

997904rArr 120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119889H (119879119909119899minus1 119910119899)

1003816100381610038161003816)

(37)

Now proceeding as in Case 2 (because 119879119909119899minus1

isin 119875 119879119909119899isin 119876)

we obtain that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)(

119887

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (38)

Also from (31) if 119889H(119910119899 119879119909119899+1) ≾ 119889H(119879119909119899minus1 119879119909119899+1) then

119889H (119879119909119899 119879119909119899+1) ≾ 119889H (119879119909119899minus1 119879119909119899+1) (39)

which implies that |119889H(119879119909119899 119879119909119899+1)| le |119889H(119879119909119899minus1 119879119909119899+1)|hence

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) = 120593 (

1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816)

le 120593 (1003816100381610038161003816119889H (119879119909119899minus1 119879119909119899+1)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus2 119865119909119899)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899minus2)

1003816100381610038161003816]

+120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus2 119865119909119899)

1003816100381610038161003816]

120593 [1003816100381610038161003816119889H (119879119909119899 119865119909119899minus2)

1003816100381610038161003816]

= 119887 [120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) + 120593 (

1003816100381610038161003816119905119899

1003816100381610038161003816)] + 119888120593 (

1003816100381610038161003816119905119899minus1

1003816100381610038161003816)

(40)

Therefore noting that by Case 2 120593(|119905119899minus1

|) lt 120593(|119905119899minus2

|) weconclude that

120593 (1003816100381610038161003816119905119899

1003816100381610038161003816) le (

119887 + 119888

1 minus 119887

)120593 (1003816100381610038161003816119905119899minus2

1003816100381610038161003816) (41)

Thus in all cases we get either

120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus1|)

or 120593(|119905119899|) le ((119887 + 119888)(1 minus 119887))120593(|119905

119899minus2|)

for 119899 = 1 we have 120593(|1199051|) le ((119887 + 119888)(1 minus 119887))120593(|119905

0|)

for 119899 = 2 we have 120593(|1199052|) le ((119887 + 119888)(1 minus 119887))120593(|119905

1|) le

((119887 + 119888)(1 minus 119887))2120593(|1199050|)

by induction we get 120593(|119905119899|) le ((119887+ 119888)(1minus119887))

119899120593(|1199050|)

Letting 119899 rarr infin we have 120593(|119905119899|) rarr 0 and by (14)

we have1003816100381610038161003816119905119899

1003816100381610038161003816=1003816100381610038161003816119889H (119879119909119899 119879119909119899+1)

1003816100381610038161003816997888rarr 0 as 119899 997888rarr infin (42)

so that 119879119909119899 is a Cauchy sequence and hence it converges to

point 119911 in119870 Now there exists a subsequence 119879119909119899119896

of 119879119909119899

such that it is contained in 119875 Without loss of generality wemay denote 119879119909

119899119896

= 119879119909119899 Since 119879 is continuous 119879119879119909

119899

converges to 119879119911We now show that 119879 and 119865 have common fixed point

(119879119911 = 119865119911) Using the weak commutativity of 119879 and 119865 weobtain that

119879119909119899= 119865119909119899minus1

119879119909119899minus1

isin 119870 (43)

then

119889H (119879119879119909119899 119865119879119909119899minus1) ≾ 119889H (119865119909119899minus1 119879119909119899minus1) = 119889H (119879119909119899 119879119909119899minus1)

(44)

This implies that1003816100381610038161003816119889H (119879119879119909119899 119865119879119909119899minus1)

1003816100381610038161003816le1003816100381610038161003816119889119867(119879119909119899 119879119909119899minus1

)1003816100381610038161003816 (45)

On letting 119899 rarr infin we obtain

119889H (119879119911 119865119879119909119899minus1) 997888rarr 0 (46)

which means that

119865119879119909119899minus1

997888rarr 119879119911 as 119899 997888rarr infin (47)

Now consider

120593 (1003816100381610038161003816119889H (119865119879119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119879119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119879119909119899minus1)

1003816100381610038161003816]

(48)

Letting 119899 rarr infin yields

120593 (1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816) le 119887120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816] (49)

a contradiction thus giving 120593(|119889H(119879119911 119865119911)|) = 0 whichimplies |119889H(119879119911 119865119911)| = 0 so that 119889H(119879119911 119865119911) = 0 and hence119879119911 = 119865119911

To show that 119879119911 = 119911 consider

120593 (1003816100381610038161003816119889H (119879119909119899 119879119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119909119899minus1 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119909119899minus1)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119909119899minus1 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119909119899minus1)

1003816100381610038161003816]

(50)

Letting 119899 rarr infin we obtain

120593 (1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816) le 119888120593 [

1003816100381610038161003816119889H (119911 119879119911)

1003816100381610038161003816] (51)

6 Abstract and Applied Analysis

a contradiction thereby giving 120593(|119889H(119911 119879119911)|) = 0 whichimplies |119889H(119911 119879119911)| = 0 so that 119889H(119911 119879119911) = 0 and hence119911 = 119879119911 Thus we have shown that 119911 = 119879119911 = 119865119911 so 119911 is acommon fixed point of 119865 and 119879 To show that 119911 is unique let119908 be another fixed point of 119865 and 119879 then

120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119908 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119908 119865119908)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119908 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119908)

1003816100381610038161003816]

= 119888120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

(52)

a contradiction therefore giving 120593(|119889H(119908 119911)|) = 0 whichimplies that |119889H(119908 119911)| = 0 so that 119889H(119908 119911) = 0 thus 119908 = 119911This completes the proof

Remark 18 If Im119904119889H(119909 119910) = 0 119904 = 119894 119895 119896 then 119889H(119909 119910) =

119889(119909 119910) and therefore we obtain the same results in [9] Soour theorem is more general than Theorem 31 in [9] for apair of weakly commuting mappings

Using the concept of commuting mappings (see eg[10]) we can give the following result

Theorem 19 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof The proof is very similar to the proof of Theorem 17with some simple modifications so it will be omitted

Corollary 20 Let (119883 119889C) be a complete complex valuedmetric space119870lowast a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (14) and

120593 [1003816100381610038161003816119889C (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889C (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889C (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889C (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889C (119879119910 119865119909)

1003816100381610038161003816]

(53)

for all 119909 119910 isin 119870lowast with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1Let 119865 119879 119870

lowastrarr 119883 be such that 119865 is generalized 119879-

contractive satisfying the conditions

(i) 120597119870lowast sube 119879119870lowast 119865119870lowast sube 119879119870lowast(ii) 119879119909 isin 120597119870lowast rArr 119865119909 isin 119870

lowast

(iii) 119865 and 119879 are weakly mappings(iv) 119879 is continuous at 119870lowast

then there exists a unique common fixed point 119911 in 119870lowast suchthat 119911 = 119879119911 = 119865119911

Proof Since each element 119909 isin H can be written in the form119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are the

basis elements of H and 119899 = 1 2 3 Putting 1199092= 1199093= 0 we

obtain an element in C So the proof can be obtained fromTheorem 17 directly

4 Fixed Points in Normal Cone Metric Spaces

In this section we prove a fixed point theorem in normalcone metric spaces including results which generalize aresult due to Huang and Zhang in [11] as well as a resultdue to Abbas and Rhoades [12] The obtained result gives afixed point theorem for four mappings without appealing tocommutativity conditions defined on a cone metric space

Let 119864 be a real Banach space A subset 119875 of 119864 is called acone if and only if

(a) 119875 is closed and nonempty and 119875 = 0(b) 119886 119887 isin R 119886 119887 ge 0 119886 119887 isin 119875 implies that 119886119909 + 119887119910 isin 119875(c) 119875⋂(minus119875) = 0

Given cone 119875 sub 119864 we define a partial ordering lewith respectto 119875 by 119909 le 119910 if and only if 119910minus119909 isin 119875 Cone 119875 is called normalif there is a number 119870

1gt 0 such that for all 119909 119910 isin 119864

0 le 119909 le 119910 implies 119909 le 1198701

10038171003817100381710038171199101003817100381710038171003817 (54)

The least positive number satisfying the above inequality iscalled the normal constant of 119875 while 119909 ≪ 119910 stands for 119910 minus119909 isin int119875 (interior of 119875) We will write 119909 lt 119910 to indicate that119909 ≪ 119910 but 119909 = 119910

Definition 21 (see [11]) Let119883 be a nonempty set Suppose thatthe mapping 119889 119883 times 119883 rarr 119864 satisfies

(d1) 0 le 119889(119909 119910) for all 119909 119910 isin 119883 and 119889(119909 119910) = 0 if and onlyif 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

(d3) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889 is called a conemetric on119883 and (119883 119889) is called a conemetric space

Definition 22 (see [11]) Let (119883 119889) be a conemetric space 119909119899

a sequence in 119883 and 119909 isin 119883 For every 119888 isin 119864 with 0 ≪ 119888 wesay that 119909

119899 is

(1) a Cauchy sequence if there is an 119873 such that for all119899119898 gt 119873 119889(119909

119899 119909119898) ≪ 119888

(2) a convergent sequence if there is an 119873 such that forall 119899 gt 119873 119889(119909

119899 119909) ≪ 119888 for some 119909 isin 119883 119889(119909 119910) le

119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 6: Fixed Point Theorems in Quaternion-Valued Metric Spaces

6 Abstract and Applied Analysis

a contradiction thereby giving 120593(|119889H(119911 119879119911)|) = 0 whichimplies |119889H(119911 119879119911)| = 0 so that 119889H(119911 119879119911) = 0 and hence119911 = 119879119911 Thus we have shown that 119911 = 119879119911 = 119865119911 so 119911 is acommon fixed point of 119865 and 119879 To show that 119911 is unique let119908 be another fixed point of 119865 and 119879 then

120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

= 120593 (1003816100381610038161003816119889H (119865119908 119865119911)

1003816100381610038161003816)

le 119887 120593 [1003816100381610038161003816119889H (119879119908 119865119908)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889H (119879119911 119865119911)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889H (119879119908 119865119911)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889H (119879119911 119865119908)

1003816100381610038161003816]

= 119888120593 (1003816100381610038161003816119889H (119908 119911)

1003816100381610038161003816)

(52)

a contradiction therefore giving 120593(|119889H(119908 119911)|) = 0 whichimplies that |119889H(119908 119911)| = 0 so that 119889H(119908 119911) = 0 thus 119908 = 119911This completes the proof

Remark 18 If Im119904119889H(119909 119910) = 0 119904 = 119894 119895 119896 then 119889H(119909 119910) =

119889(119909 119910) and therefore we obtain the same results in [9] Soour theorem is more general than Theorem 31 in [9] for apair of weakly commuting mappings

Using the concept of commuting mappings (see eg[10]) we can give the following result

Theorem 19 Let (119883 119889H) be a complete quaternion valuedmetric space119870 a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (13) and (14)Let 119865 119879 119870 rarr 119883 be such that 119865 is generalized 119879-contractivesatisfying the conditions

(i) 120597119870 sube 119879119870 119865119870 sube 119879119870(ii) 119879119909 isin 120597119870 rArr 119865119909 isin 119870(iii) 119865 and 119879 are commuting mappings(iv) 119879 is continuous at 119870

then there exists a unique common fixed point 119911 in119870 such that119911 = 119879119911 = 119865119911

Proof The proof is very similar to the proof of Theorem 17with some simple modifications so it will be omitted

Corollary 20 Let (119883 119889C) be a complete complex valuedmetric space119870lowast a nonempty closed subset of119883 and 120593 R+ rarrR+ an increasing continuous function satisfying (14) and

120593 [1003816100381610038161003816119889C (119865119909 119865119910)

1003816100381610038161003816]

le 119887 120593 [1003816100381610038161003816119889C (119879119909 119865119909)

1003816100381610038161003816] + 120593 [

1003816100381610038161003816119889C (119879119910 119865119910)

1003816100381610038161003816]

+ 119888min 120593 [1003816100381610038161003816119889C (119879119909 119865119910)

1003816100381610038161003816] 120593 [

1003816100381610038161003816119889C (119879119910 119865119909)

1003816100381610038161003816]

(53)

for all 119909 119910 isin 119870lowast with 119909 = 119910 119887 119888 ge 0 2119887 + 119888 lt 1Let 119865 119879 119870

lowastrarr 119883 be such that 119865 is generalized 119879-

contractive satisfying the conditions

(i) 120597119870lowast sube 119879119870lowast 119865119870lowast sube 119879119870lowast(ii) 119879119909 isin 120597119870lowast rArr 119865119909 isin 119870

lowast

(iii) 119865 and 119879 are weakly mappings(iv) 119879 is continuous at 119870lowast

then there exists a unique common fixed point 119911 in 119870lowast suchthat 119911 = 119879119911 = 119865119911

Proof Since each element 119909 isin H can be written in the form119902 = 119909

0+ 1199091119894 + 1199092119895 + 1199093119896 119909119899isin R where 1 119894 119895 119896 are the

basis elements of H and 119899 = 1 2 3 Putting 1199092= 1199093= 0 we

obtain an element in C So the proof can be obtained fromTheorem 17 directly

4 Fixed Points in Normal Cone Metric Spaces

In this section we prove a fixed point theorem in normalcone metric spaces including results which generalize aresult due to Huang and Zhang in [11] as well as a resultdue to Abbas and Rhoades [12] The obtained result gives afixed point theorem for four mappings without appealing tocommutativity conditions defined on a cone metric space

Let 119864 be a real Banach space A subset 119875 of 119864 is called acone if and only if

(a) 119875 is closed and nonempty and 119875 = 0(b) 119886 119887 isin R 119886 119887 ge 0 119886 119887 isin 119875 implies that 119886119909 + 119887119910 isin 119875(c) 119875⋂(minus119875) = 0

Given cone 119875 sub 119864 we define a partial ordering lewith respectto 119875 by 119909 le 119910 if and only if 119910minus119909 isin 119875 Cone 119875 is called normalif there is a number 119870

1gt 0 such that for all 119909 119910 isin 119864

0 le 119909 le 119910 implies 119909 le 1198701

10038171003817100381710038171199101003817100381710038171003817 (54)

The least positive number satisfying the above inequality iscalled the normal constant of 119875 while 119909 ≪ 119910 stands for 119910 minus119909 isin int119875 (interior of 119875) We will write 119909 lt 119910 to indicate that119909 ≪ 119910 but 119909 = 119910

Definition 21 (see [11]) Let119883 be a nonempty set Suppose thatthe mapping 119889 119883 times 119883 rarr 119864 satisfies

(d1) 0 le 119889(119909 119910) for all 119909 119910 isin 119883 and 119889(119909 119910) = 0 if and onlyif 119909 = 119910

(d2) 119889(119909 119910) = 119889(119910 119909) for all 119909 119910 isin 119883

(d3) 119889(119909 119910) le 119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Then 119889 is called a conemetric on119883 and (119883 119889) is called a conemetric space

Definition 22 (see [11]) Let (119883 119889) be a conemetric space 119909119899

a sequence in 119883 and 119909 isin 119883 For every 119888 isin 119864 with 0 ≪ 119888 wesay that 119909

119899 is

(1) a Cauchy sequence if there is an 119873 such that for all119899119898 gt 119873 119889(119909

119899 119909119898) ≪ 119888

(2) a convergent sequence if there is an 119873 such that forall 119899 gt 119873 119889(119909

119899 119909) ≪ 119888 for some 119909 isin 119883 119889(119909 119910) le

119889(119909 119911) + 119889(119911 119910) for all 119909 119910 119911 isin 119883

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 7: Fixed Point Theorems in Quaternion-Valued Metric Spaces

Abstract and Applied Analysis 7

A cone metric space119883 is said to be complete if every Cauchysequence in119883 is convergent in119883

Now we give the following result

Theorem 23 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119909 119892119910) + (1 minus 120582) 119889 (1198911119909 1198921119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119909) + 119889 (119910 119892119910))

+ (1 minus 120582) (119889 (119909 1198911119909) + 119889 (119910 119892

1119910))]

+ 120574 [120582 (119889 (119909 119892119910) + 119889 (119910 119891119909))

+ (1 minus 120582) (119889 (119909 1198921119909) + 119889 (119910 119891

1119909))]

(55)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in 119883

Proof If 120582 = 0 or 120582 = 1 the proof is already known from [12]So we consider the case when 0 lt 120582 lt 1 Suppose 119909

0is an

arbitrary point of119883 and define 119909119899 by 119909

2119899+1= 1198911199092119899= 11989111199092119899

and 1199092119899+2

= 1198921199092119899+1

= 11989211199092119899+1

119899 = 0 1 2 Then we have

119889 (1199092119899+1

1199092119899+2

)

= 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (1198911199092119899 1198921199092119899+1

)

= 120582119889 (1198911199092119899 1198921199092119899+1

) + (1 minus 120582) 119889 (11989111199092119899 11989211199092119899+1

)

le 120572119889 (1199092119899 1199092119899+1

)

+ 120573 [120582 (119889 (1199092119899 1198911199092119899) + 119889 (119909

2119899+1 1198921199092119899+1

))

+ (1 minus 120582) (119889 (1199092119899 11989111199092119899) + 119889 (119909

2119899+1 11989211199092119899+1

))]

+ 120574 [120582 (119889 (1199092119899 1198921199092119899+1

) + 119889 (1199092119899+1

1198911199092119899))

+ (1 minus 120582) (119889 (1199092119899 11989211199092119899) + 119889 (119909

2119899+1 11989111199092119899))]

= (120572 + 120573 + 120574) 119889 (1199092119899 1199092119899+1

) + (120573 + 120574) 119889 (1199092119899+1

1199092119899+2

)

(56)

this yields that

119889(1199092119899+1

1199092119899+2

)le120575119889 (1199092119899 1199092119899+1

) where 120575 =120572 + 120573 + 120574

1minus 120573 minus 120574

lt1

(57)

Therefore for all 119899 we deduce that

119889 (119909119899+1

119909119899+2

) le 120575119889 (119909119899 119909119899+1

) le sdot sdot sdot le 120575119899+1

119889 (1199090 1199091) (58)

Now for119898 gt 119899 we obtain that

119889 (119909119898 119909119899) le

120575119899

1 minus 120575

119889 (1199090 1199091) (59)

From (54) we have

1003817100381710038171003817119889 (119909119898 119909119899)1003817100381710038171003817le

1198701120575119899

1 minus 120575

1003817100381710038171003817119889 (1199090 1199091)1003817100381710038171003817 (60)

which implies that 119889(119909119898 119909119899) rarr 0 as 119899119898 rarr infin Hence

119909119899 is a Cauchy sequence Since 119883 is complete there exists a

point 119901 isin 119883 such that 119909119899rarr 119901 as 119899 rarr infin Now using (55)

we obtain that

120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)

le 120582 [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

119892119901)]

+ (1 minus 120582) [119889 (119901 1199092119899+1

) + 119889 (1199092119899+1

1198921119901)]

le 119889 (119901 1199092119899+1

) + 120572119889 (1199092119899 119901)

+ 120573 [119889 (1199092119899 1199092119899+1

) + 120582 119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921119901)]

+ 120574 [119889 (1199092119899 119901) + 120582 119889 (119901 119892119901)

+ (1 minus 120582) 119889 (119901 1198921119901) + 119889 (119901 119909

2119899+1)]

(61)

Now using (54) and (61) we obtain that

|120582|1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 1198921

119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(62)

Therefore1003817100381710038171003817119889 (119901 119892119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(63)

Since the right hand side of the above inequality approacheszero as 119899 rarr 0 hence 119889(119901 119892119901) = 0 and then 119901 = 119892119901 Alsowe have

|1 minus 120582|1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le1003817100381710038171003817120582119889 (119901 119892119901) + (1 minus 120582) 119889 (119901 119892

1119901)1003817100381710038171003817

le

1198701

1 minus 120573 minus 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817

+ 1205731003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817

+1205741003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817

(64)

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 8: Fixed Point Theorems in Quaternion-Valued Metric Spaces

8 Abstract and Applied Analysis

which implies that

1003817100381710038171003817119889 (119901 119892

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+ 1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(65)

Letting 119899 rarr 0 we deduce that 119889(119901 1198921119901) = 0 and then

119901 = 1198921119901 Similarly by replacing 119892 by 119891 and 119892

1by 1198911in (61)

we deduce that1003817100381710038171003817119889 (119901 119891119901)

1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|120582| (1 minus 120573 minus 120574))minus1

(66)

where 119889(119901 119891119901) = 0 as 119899 rarr 0 Hence 119889(119901 119891119901) = 0 andthen 119901 = 119891119901 Also

1003817100381710038171003817119889 (119901 119891

1119901)1003817100381710038171003817

le (11987011003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817+ 120572

1003817100381710038171003817119889 (119909119899+1

119901)1003817100381710038171003817+ 120573

1003817100381710038171003817119889 (119909119899 119909119899+1

)1003817100381710038171003817

+1205741003817100381710038171003817119889 (119909119899 119901)

1003817100381710038171003817+ 120574

1003817100381710038171003817119889 (119901 119909

119899+1)1003817100381710038171003817)

times (|1 minus 120582| (1 minus 120573 minus 120574))minus1

(67)

Letting 119899 rarr 0 we deduce that 119889(119901 1198911119901) = 0 and then

119901 = 1198911119901

To prove uniqueness suppose that 119902 is another fixed pointof 119891 119892 119892

1 and 119891

1 and then

119889 (119901 119902) = 120582119889 (119901 119902) + (1 minus 120582) 119889 (119901 119902)

= 120582119889 (119891119901 119892119902) + (1 minus 120582) 119889 (1198911119901 1198921119902)

le 120572119889 (119901 119902)

+ 120573 [120582119889 (119901 119891119901) + (1 minus 120582) 119889 (119901 1198911119901)

+ 120582119889 (119902 119892119902) + (1 minus 120582) 119889 (119902 1198921119902)]

+ 120574 [120582119889 (119901 119892119902) + (1 minus 120582) 119889 (119901 1198921119902)

+ 120582119889 (119902 119891119901) + (1 minus 120582) 119889 (119902 1198911119901)]

= (120572 + 2120574) 119889 (119901 119902)

(68)

which gives 119889(119901 119902) = 0 and hence 119901 = 119902 This completes theproof

Now we give the following result

Corollary 24 Let (119883 119889) be a complete cone metric space and119875 a normal cone with normal constant 119870

1 Suppose that the

mappings 119891 119892 1198911 1198921are four self-maps of119883 satisfying

120582119889 (119891119898119909 119892119899119910) + (1 minus 120582) 119889 (119891

119898

1119909 119892119899

1119910)

le 120572119889 (119909 119910) + 120573 [120582 (119889 (119909 119891119898119909) + 119889 (119910 119892

119899119910)) + (1 minus 120582)

times (119889 (119909 119891119898

1119909) + 119889 (119910 119892

119899

1119910))]

+ 120574 [120582 (119889 (119909 119892119899119910) + 119889 (119910 119891

119898119909))

+ (1 minus 120582) (119889 (119909 119892119899

1119909) + 119889 (119910 119891

119898

1119909))]

(69)

for all 119909 119910 isin 119883 where 0 le 120582 le 1 and 120572 120573 120574 ge 0 with 120572+ 2120573+2120574 lt 1 Then 119891 119892 119891

1 and 119892

1have a unique common fixed

point in119883119898 119899 gt 1

Proof Inequality (69) is obtained from (55) by setting 119891 equiv

119891119898 1198911equiv 119891119898

1 and 119892 equiv 119892

119898 1198921equiv 119892119898

1 Then the result follows

fromTheorem 23

Remark 25 If we put 120582 = 0 or 120582 = 1 inTheorem 23 we obtainTheorem 21 in [12]

Remark 26 It should be remarked that the quaternionmetricspace is different from conemetric space which is introducedin [11] (see also [12ndash19]) The elements in R or C form analgebra while the same is not true in H

Open Problem It is still an open problem to extend our resultsin this paper for some kinds of mappings like biased weaklybiased weakly compatible mappings compatible mappingsof type (119879) and 119898-weaklowastlowast commuting mappings See [20ndash30] for more studies on these mentioned mappings

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Taif University Saudi Arabiafor its financial support of this research under Project no18364331 The first author would like to thank ProfessorRashwan Ahmed Rashwan from Assiut University Egyptfor introducing him to the subject of fixed point theory andProfessorKlausGurlebeck fromBauhausUniversityWeimarGermany for introducing him to the subject of Cliffordanalysis

References

[1] F Brackx R Delanghe and F SommenClifford Analysis vol 76of Research Notes in Mathematics Pitman Boston Mass USA1982

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 9: Fixed Point Theorems in Quaternion-Valued Metric Spaces

Abstract and Applied Analysis 9

[2] K Gurlebeck K Habetha and W Sproszligig Holomorphic Func-tions in the Plane and n-Dimensional Space Birkhauser BaselSwitzerland 2008

[3] K Gurlebeck andW Sproszligig Quaternionic and Clifford Calcu-lus for Engineers and Physicists John Wiley amp Sons ChichesterUK 1997

[4] A Sudbery ldquoQuaternionic analysisrdquo Mathematical Proceedingsof the Cambridge Philosophical Society vol 85 no 2 pp 199ndash224 1979

[5] A Azam B Fisher and M Khan ldquoCommon fixed point the-orems in complex valued metric spacesrdquo Numerical FunctionalAnalysis and Optimization vol 32 no 3 pp 243ndash253 2011

[6] N A Assad and W A Kirk ldquoFixed point theorems forset-valued mappings of contractive typerdquo Pacific Journal ofMathematics vol 43 pp 553ndash562 1972

[7] S Sessa ldquoOn a weak commutativity condition of mappings infixed point considerationsrdquo Institut Mathematique vol 32 pp149ndash153 1982

[8] O Hadzic and L Gajic ldquoCoincidence points for set-valuedmappings in convex metric spacesrdquo Univerzitet u Novom SaduZbornik Radova Prirodno-Matematickog Fakulteta vol 16 no 1pp 13ndash25 1986

[9] A J Asad and Z Ahmad ldquoCommon fixed point of a pairof mappings in Banach spacesrdquo Southeast Asian Bulletin ofMathematics vol 23 no 3 pp 349ndash355 1999

[10] G Jungck ldquoCommuting mappings and fixed pointsrdquo TheAmerican Mathematical Monthly vol 83 no 4 pp 261ndash2631976

[11] L-G Huang and X Zhang ldquoConemetric spaces and fixed pointtheorems of contractive mappingsrdquo Journal of MathematicalAnalysis and Applications vol 332 no 2 pp 1468ndash1476 2007

[12] M Abbas and B E Rhoades ldquoFixed and periodic point resultsin cone metric spacesrdquo Applied Mathematics Letters vol 22 no4 pp 511ndash515 2009

[13] M Abbas VC Rajic T Nazir and S Radenovic ldquoCommonfixed point of mappings satisfying rational inequalities inordered complex valued generalized metric spacesrdquo AfrikaMatematika 2013

[14] A P Farajzadeh A Amini-Harandi and D Baleanu ldquoFixedpoint theory for generalized contractions in cone metricspacesrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 2 pp 708ndash712 2012

[15] Z Kadelburg and S Radenovic ldquoA note on various types ofcones and fixed point results in cone metric spacesrdquo AsianJournal of Mathematics and Applications 2013

[16] N Petrot and J Balooee ldquoFixed point theorems for set-valued contractions in ordered cone metric spacesrdquo Journal ofComputational Analysis and Applications vol 15 no 1 pp 99ndash110 2013

[17] P D Proinov ldquoA unified theory of cone metric spaces and itsapplications to the fixed point theoryrdquo Fixed Point Theory andApplications vol 2013 article 103 2013

[18] S Radenovic ldquoA pair of non-self mappings in cone metricspacesrdquo Kragujevac Journal of Mathematics vol 36 no 2 pp189ndash198 2012

[19] H Rahimi S Radenovic G S Rad and P Kumam ldquoQuadru-pled fixed point results in abstract metric spacesrdquo Computa-tional and Applied Mathematics 2013

[20] A El-Sayed Ahmed ldquoCommon fixed point theorems for m-weaklowastlowast commuting mappings in 2-metric spacesrdquo AppliedMathematics amp Information Sciences vol 1 no 2 pp 157ndash1712007

[21] A El-Sayed and A Kamal ldquoFixed points and solutions ofnonlinear functional equations in Banach spacesrdquo Advances inFixed Point Theory vol 2 no 4 pp 442ndash451 2012

[22] A El-Sayed and A Kamal ldquoSome fixed point theorems usingcompatible-typemappings in Banach spacesrdquoAdvances in FixedPoint Theory vol 4 no 1 pp 1ndash11 2014

[23] A Aliouche ldquoCommon fixed point theorems of Gregus type forweakly compatible mappings satisfying generalized contractiveconditionsrdquo Journal of Mathematical Analysis and Applicationsvol 341 no 1 pp 707ndash719 2008

[24] J Ali andM Imdad ldquoCommon fixed points of nonlinear hybridmappings under strict contractions in semi-metric spacesrdquoNonlinear Analysis Hybrid Systems vol 4 no 4 pp 830ndash8372010

[25] G Jungck and H K Pathak ldquoFixed points via lsquobiased mapsrsquordquoProceedings of the American Mathematical Society vol 123 no7 pp 2049ndash2060 1995

[26] J K Kim and A Abbas ldquoCommon fixed point theoremsfor occasionally weakly compatible mappings satisfying theexpansive conditionrdquo Journal of Nonlinear and Convex Analysisvol 12 no 3 pp 535ndash540 2011

[27] R P Pant and R K Bisht ldquoOccasionally weakly compatiblemappings and fixed pointsrdquoBulletin of the BelgianMathematicalSociety vol 19 no 4 pp 655ndash661 2012

[28] H K Pathak S M Kang Y J Cho and J S Jung ldquoGregus typecommon fixed point theorems for compatible mappings of type(T) and variational inequalitiesrdquo Publicationes MathematicaeDebrecen vol 46 no 3-4 pp 285ndash299 1995

[29] V Popa M Imdad and J Ali ldquoUsing implicit relations to proveunified fixed point theorems in metric and 2-metric spacesrdquoBulletin of the Malaysian Mathematical Sciences Society vol 33no 1 pp 105ndash120 2010

[30] T Suzuki and H K Pathak ldquoAlmost biased mappings andalmost compatible mappings are equivalent under some con-ditionrdquo Journal of Mathematical Analysis and Applications vol368 no 1 pp 211ndash217 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Page 10: Fixed Point Theorems in Quaternion-Valued Metric Spaces

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Optimal Inverse Littlewood-Offord theorems

Optimal Inverse Littlewood-Offord theorems

Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces

Coupled fixed-point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces

Generalised interval-valued fuzzy soft set

Generalised interval-valued fuzzy soft set

Multiplicative valued difference fields

Multiplicative valued difference fields

New coincidence and common fixed point theorems

New coincidence and common fixed point theorems

Fixed point theorems and best approximation in convex metric spaces

Fixed point theorems and best approximation in convex metric spaces

metric-units-answers.pdf - Corbettmaths

metric-units-answers.pdf - Corbettmaths

Theory of vector-valued hyperfunctions

Theory of vector-valued hyperfunctions

Helmholtz Theorems, Gauge Transformations, General ...

Helmholtz Theorems, Gauge Transformations, General ...

Theorems on area - Praadis Education

Theorems on area - Praadis Education

Maximal infinite-valued constraint languages

Maximal infinite-valued constraint languages

The Extended Edit Distance Metric

The Extended Edit Distance Metric

Conical plurisubharmonic measure and new cross theorems

Conical plurisubharmonic measure and new cross theorems

Common fixed point theorems for non compatible mappings in fuzzy metric spaces

Common fixed point theorems for non compatible mappings in fuzzy metric spaces

Metric-Driven RoSy Fields Design

Metric-Driven RoSy Fields Design

Metric Semantics for Concurrency

Metric Semantics for Concurrency

Metric Conversions Ladder Method

Metric Conversions Ladder Method

Single-layered complex-valued neural network for real-valued classification problems

Single-layered complex-valued neural network for real-valued classification problems

Dear Valued Shareholders

Dear Valued Shareholders

Coupled fixed point theorems in cone metric spaces with a c-distance and applications

Coupled fixed point theorems in cone metric spaces with a c-distance and applications