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Rendiconti del Circolo Matematico diPalermo ISSN 0009-725XVolume 61Number 3 Rend. Circ. Mat. Palermo (2012)61:361-383DOI 10.1007/s12215-012-0096-0

Common coupled fixed point theorems for$$w^*$$-compatible mappings withoutmixed monotone property

Wutiphol Sintunavarat, Adrian Petruşel& Poom Kumam

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Rend. Circ. Mat. Palermo (2012) 61:361–383DOI 10.1007/s12215-012-0096-0

Common coupled fixed point theorems for w∗-compatiblemappings without mixed monotone property

Wutiphol Sintunavarat · Adrian Petrusel ·Poom Kumam

Received: 9 May 2012 / Accepted: 14 August 2012 / Published online: 8 September 2012© Springer-Verlag 2012

Abstract In this paper, we show that the mixed g-monotone property in coupled coinci-dence point theorems can be replaced by generalized property. Hence, these results can beapplied in a much wider class of problems. We also study the condition for the uniquenessof a common coupled fixed point and give some example of nonlinear contraction mappingswhere the existence of the common coupled fixed point cannot be obtained by the mixedmonotone property, but it follows by our results. At the end of this paper, we give an openproblems for further investigation.

Keywords Cone metric space · Coupled common fixed point · Coupled coincidencepoint · w∗-compatible maps · Mixed g-monotone property · (F, g)-invariant set

Mathematics Subject Classification (2010) 47H10 · 54H25

1 Introduction

The Banach’s contraction principle states that, if (X, d) is a complete metric space andT : X → X is a contraction mapping (i.e., d(T x, T y) ≤ kd(x, y) for all x, y ∈ X , where kis a non-negative number such that k < 1), then T has a unique fixed point. This principle

W. Sintunavarat · P. Kumam (B)Department of Mathematics, Faculty of Science,King Mongkut’s University of Technology Thonburi (KMUTT),Bangkok 10140, Thailande-mail: poom.kum@kmutt.ac.th

W. Sintunavarate-mail: poom_teun@hotmail.com

A. PetruselDepartment of Mathematics, Babes-Bolyai University Cluj-Napoca,Kogalniceanu 1, 400084 Cluj-Napoca, Romaniae-mail: petrusel@math.ubbcluj.ro

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play a major role for solving many problems in applied mathematics and sciences. So, it hasbeen generalized in many ways over the years (see [8,24,36,39–41,43,45,50,51]).

In 2007, Huang and Zhang [16], using the concept of cone metric space (in cone metricspaces the set of real numbers is replaced by an ordered Banach space, see e.g., [52]) gavesome fixed point theorems for contraction mappings. Later, many authors considered fixedpoint results in such spaces (see some reviews of these results in [12,13,17,18,38,49]).

On the other hand, Ran and Reurings [31, Theorem 2.1] extend the Banach’s contractionprinciple to metric spaces endowed with a partial ordering and they also gave applicationsof their results to matrix equations. Some extensions of this result were given in [30] byPetrusel and Rus. Then, Nieto and López [28] extended Ran and Reurings’s theorems fornondecreasing mappings and obtained a unique solution for a first order ordinary differentialequation with periodic boundary conditions. Then, order contractions in ordered cone metricspaces were considered in [3–5,19,29].

One of the interesting and crucial notion, coupled fixed point theorem was introduced byGuo and Lakshmikantham [14]. In 2006, Bhaskar and Lakshmikantham [9] introduced theconcept of mixed monotone property (see Definitions 2.4 in Sect. 2). They proved coupledfixed point theorems for mappings satisfying a mixed monotone property and gave someapplications concerning the existence and uniqueness of a solution for a periodic boundaryvalue problem

u′(t) = f (t, u(t)), t ∈ [0, T ]u(0) = u(T )

(1.1)

where the function f satisfies certain conditions. Following this result, many authors studiedthe existence and uniqueness of solutions of a nonlinear integral equation as an applicationof coupled fixed points. For example, Caballero et al. [10] have investigated the existenceand uniqueness of positive solutions for the following singular fractional boundary valueproblem

Dα0+u(t)+ f (t, u(t)) = 0, 0 < t < 1

u(0) = u(1) = 0,(1.2)

where α ∈ R such that 1 < α ≤ 2, Dα0+ is the standard Riemann–Liouville differentiation

and f : (0, 1] × [0,∞) → [0,∞) with limt→0+ f (t, ·) = ∞ ( f is singular at t = 0)for all t ∈ (0, 1]. Since their important role in the study of nonlinear differential equations,nonlinear integral equations and differential inclusions, a wide discussion on coupled fixedpoint theorems aimed the interest of many scientists. In 2009, Lakshmikantham and Ciric [21]extended the concept of a mixed monotone property to a mixed g-monotone mapping andproved coupled coincidence and coupled common fixed point theorems that extend theoremsfrom [9]. These results have a lot of applications, such as in proving existence of solutions ofperiodic boundary value problems (e.g., [7,9]) and integral equations of particular type (e.g.,[5,15,22,23]). A number of articles on coupled coincidence and coupled fixed point theoremsin ordered metric spaces have been dedicated to this topic; For several improvements andgeneralizations see [2,6,11,25,27,33,35,42,44,46,47] and reference therein.

Recently, Karapinar [20] proved coupled fixed point theorems for nonlinear contractions inordered cone metric spaces over normal cones without regularity. He considered the continuityof commuting mappings on the whole complete cone metric space. Shatanawi [37] provedcoupled coincidence fixed point theorems in cone metric spaces over cones which were notnecessarily normal. More recently, Nashine et al. [26] established coupled common fixedpoint theorems for mixed g-monotone andw∗-compatible mappings satisfying more general

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contractive conditions in ordered cone metric spaces over a cone that is only solid (i.e., has anonempty interior) which improve the works of Karapinar [20], Shatanawi [37] and Abbaset al. [1].

In this work, we introduce the new concept so-called (F, g)-invariant set which is moregeneral than the concept of the mixed g-monotone property in partially ordered sets. Fur-thermore, we will show that the mixed g-monotone property in coupled coincidence pointtheorems in ordered cone metric spaces can be replaced by such concept. Hence, our resultsgeneralize, extend and unify several well-known comparable results in the literature andthese results can be applied in a much wider class of problems. We also study the necessarycondition for the uniqueness of a common coupled fixed point. An illustrative example isalso presented, in the sense that our results can be used in proving the existence of a commoncoupled fixed point, while the results of Nashine et al. [26] and Abbas et al. [1] cannot. Atthe end of this paper, we give an open problems for further investigation.

2 Preliminaries

Let E be a real Banach space with respect to a given norm ‖ ·‖E and let 0E be the zero vectorof E . A nonempty subset P of E is called a cone if the following conditions hold:

(1) P is closed and P = {0E };(2) a, b ∈ R, a, b ≥ 0, x, y ∈ P �⇒ ax + by ∈ P;(3) x ∈ P, −x ∈ P �⇒ x = 0E .

Given a cone P ⊆ E , a partial ordering ≤P with respect to P is naturally defined byx ≤P y if and only if y − x ∈ P , for x, y ∈ E . We shall write x <P y to indicate thatx ≤P y but x = y, while x � y will stand for y − x ∈ int (P), where int (P) denotes theinterior of P .

The cone P is said to be normal if there exists a real number K > 0 such that for allx, y ∈ E ,

0E ≤P x ≤P y �⇒ ‖x‖E ≤ K‖y‖E .

The least positive number K satisfying the above statement is called the normal constantof P .

In what follows we always suppose that E is a real Banach space with cone P satisfyingint (P) = ∅ (such cones are called solid).

Definition 2.1 ([16]) Let X be a nonempty set and d : X × X → E satisfies

(1) 0E ≤P d(x, y) for all x, y ∈ X and(2) d(x, y) = d(y, x) for all x, y ∈ X ;(3) d(x, y) ≤P d(x, z)+ d(z, y) for all x, y, z ∈ X .

Then d is called a cone metric on X and (X, d) is called a cone metric space.

We obtain that the concept of a cone metric space is more general than a well-know metricspace.

Definition 2.2 ([16]) Let (X, d) be a cone metric space, {xn} be a sequence in X and x ∈ X .

(1) If for every c ∈ E with 0E �P c, there is N ∈ N such that d(xn, x) �P c for alln ≥ N , then {xn} is said to converge to x . This limit is denoted by limn→∞ xn = x orxn → x as n → ∞.

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(2) If for every c ∈ E with 0E �P c, there is N ∈ N such that d(xn, xm) �P c for alln,m > N , then {xn} is called a Cauchy sequence in X .

(3) If every Cauchy sequence in X is convergent in X , then (X, d) is called a complete conemetric space.

Let (X, d) be a cone metric space. Then the following properties are often used (particu-larly when dealing with cone metric spaces in which the cone need not be normal):

(p1) If E is a real Banach space with a cone P and if a ≤P ha where a ∈ P and h ∈ [0, 1),then a = 0E ;

(p2) if 0E ≤P u � c for each 0E � c, then u = 0E ;(p3) if u, v, w ∈ E , u ≤P v and v � w, then u � w;(p4) if c ∈ int P , 0 ≤P an ∈ E and an → 0E , then there exists k ∈ N such that for all

n > k we have an � c.

Definition 2.3 Let X be a nonempty set. Then (X, d,�) is called an ordered cone metricspace if:

(i) (X, d) is a cone metric space,(ii) (X,�) is a partially ordered set.

Let (X,�) be a partially ordered set. Elements x, y ∈ X are called comparable if x � yor y � x holds. If F : X → X is such that for all x, y ∈ X , x � y implies F(x) � F(y),then the mapping F is said to be non-decreasing. Similarly, a non-increasing mapping isdefined.

The concept of a mixed monotone property was introduced by Bhaskar and Lakshmikan-tham in [9] and further extended by Lakshmikantham and Ciric in [21].

Definition 2.4 ([9,21]) Let (X,�) be a partially ordered set and let F : X × X → X andg : X → X . The mapping F is said to have mixed g -monotone property if F is monotone g-non-decreasing in its first argument and monotone g-non-increasing in its second argument,that is, for any x, y ∈ X ,

x1, x2 ∈ X, gx1 � gx2 ⇒ F(x1, y) � F(x2, y) (2.1)

and

y1, y2 ∈ X, gy1 � gy2 ⇒ F(x, y1) � F(x, y2). (2.2)

hold. If in the previous relations g is the identity mapping then it is said that F has a mixedmonotone property.

Definition 2.5 ([9,21]) Let X be a nonempty set and F : X × X → X , g : X → X . Anelement (x, y) ∈ X × X is called:

(C1) a coupled fixed point of F if x = F(x, y) and y = F(y, x);(C2) a coupled coincidence point of mappings F and g if

gx = F(x, y) and gy = F(y, x),

and in this case (gx, gy) is called a coupled point of coincidence;(C3) a common coupled fixed point of mappings F and g if

x = gx = F(x, y) and y = gy = F(y, x).

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Definition 2.6 ([1]) Let X be a nonempty set. Mappings F : X × X → X and g : X → Xare called

(W1) w-compatible if gF(x, y) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x);(W2) w∗-compatible if gF(x, x) = F(gx, gx) whenever gx = F(x, x).

The following example shows that F and g may bew∗-compatible but notw-compatible.

Example 2.7 Let X = [0,+∞) and F : X × X → X and g : X → X be defined by

F(x, y) =⎧⎨

⎩

1, (x, y) = (0, 2)3, (x, y) = (2, 0)5, otherwise

, gx =

⎧⎪⎪⎨

⎪⎪⎩

1, x = 03, x = 25, x ∈ {4, 5, 6}7, otherwise.

Then g0 = 1 = F(0, 2) and g2 = 3 = F(2, 0) but gF(0, 2) = 7 = 5 = F(g0, g2). Thus,F and g are not w-compatible. However, F(x, x) = gx is possible only if x ∈ {4, 5, 6} andin all cases gF(x, x) = 5 = F(gx, gx). Therefore, F and g are w∗-compatible.

3 Main results

In this section, we prove some coupled coincidence and common coupled fixed point theoremsby using the concept of (F, g)-invariant set in cone metric spaces and apply our results inpartially ordered cone metric spaces. First of all, we give the concept of (F, g)-invariant set.

Definition 3.1 Let X be a nonempty set and F : X × X → X and g : X → X be givenmappings. Let M be a nonempty subset of X4. We say that M is (F, g) -invariant subset ofX4 if and only if for all x, y, z, w ∈ X , we have the following statements hold:

(F1) : (gx, gy, gz, gw) ∈ M ⇐⇒ (gw, gz, gy, gx) ∈ M;(F2) : (gx, gy, gz, gw) ∈ M �⇒ (F(x, y), F(y, x), F(z, w), F(w, z)) ∈ M.

If g is the identity mapping, then it is said that M is F -invariant subset of X4 which is dueto Samet and Vetro [34]; see also the work of Sintunavarat et al. [44].

Remark 3.2 We can easily to check that the set M = X4 is trivially (F, g)-invariant.

Example 3.3 Let X = {0, 1, 2, 3} and F : X × X → X and g : X → X be defined by

F(x, y) ={

0, x, y ∈ {0, 2}3, otherwise

, gx ={

0, x = 0, 2

1, x = 1, 3.

It easy to see that M = {0}4 ⊆ X4 is (F, g)-invariant.

Example 3.4 Let X = R and F : X × X → X and g : X → X be defined by

F(x, y) ={

x, x, y ∈ {π, 2π, 3π}sin(x + y), otherwise

, gx =

⎧⎪⎨

⎪⎩

π, x = π, 3π

2π, x = 2π

min{−1,−x2}, otherwise.

It easy to see that M = {π, 2π, 3π}4 ⊆ X4 is (F, g)-invariant.

Next example plays a key role in the proof of our main results in partially ordered set.

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Example 3.5 Let (X,�) be a partially ordered set, F : X × X → X and g : X → X . Let Fsatisfying the mixed g-monotone property, that is, for all x, y ∈ X , we have

x1, x2 ∈ X, gx1 � gx2 �⇒ F(x1, y) � F(x2, y) (3.1)

and

y1, y2 ∈ X, gy1 � gy2 �⇒ F(x, y2) � F(x, y1). (3.2)

If we define the subset M ⊆ X4 by

M = {(a, b, c, d) ∈ X4 : a � c and b � d},then M is (F, g)-invariant subset of X4.

Proof We claim that (F1) and (F2) hold.

Case (F1): Since for x, y, z, w ∈ X , we have

(gx, gy, gz, gw) ∈ M ⇐⇒ gx � gz and gy � gw

⇐⇒ gw � gy and gz � gx

⇐⇒ (gw, gz, gy, gx) ∈ M,

we get M satisfies condition (F1).

Case (F2): Since F satisfying the mixed g-monotone property, for all x, y, z, w ∈ X , wehave

(gx, gy, gz, gw) ∈ M ⇐⇒ gx � gz and gy � gw

�⇒ F(x, y) � F(z, y) � F(z, w) and

F(y, x) � F(w, x) � F(w, z)

⇐⇒ (F(x, y), F(y, x), F(z, w), F(w, z)) ∈ M,

we get M satisfies condition (F2). Therefore, M satisfies the condition (F1) and (F2) andthus M is (F, g)-invariant subset of X4.

Throughout this section we mean

(a, b, c, d) � M if and only if (a, b, c, d) ∈ M or (c, d, a, b) ∈ M.

Here we state that the coupled coincidence point theorem under the (F, g)-invariant set.

Theorem 3.6 Let (X, d) be a cone metric space over a solid cone P and M be subset of X4.Let F : X × X → X and g : X → X be mappings and there exist two elements x0, y0 ∈ Xwith

(F(x0, y0), F(y0, x0), gx0, gy0) ∈ M.

Suppose further that F, g satisfy

d(F(x, y), F(u, v)) ≤P a1d(gx, gu)+ a2d(F(x, y), gx)+ a3d(gy, gv)

+ a4d(F(u, v), gu)+ a5d(F(x, y), gu)+ a6d(F(u, v), gx),

(3.3)

for all (x, y), (u, v) ∈ X × X with (gx, gy, gu, gv) � M, where ai ≥ 0, for i = 1, 2, . . . , 6and

∑6i=1 ai < 1.

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Further suppose

(a): F(X × X) ⊆ g(X);(b): g(X) is a complete subspace of X;(c): M is (F, g)-invariant subset of X4;(d): X has the following properties:

if for two sequences {xn}, {yn} with (xn+1, yn+1, xn, yn) ∈ M and

{xn} → x, {yn} → y,

for all n ∈ N, then (x, y, xn, yn) ∈ M for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = gx and F(y, x) = gy,

that is, F and g have a coupled coincidence point (x, y) ∈ X × X.

Proof Let x0, y0 ∈ X be such that (F(x0, y0), F(y0, x0), gx0, gy0) ∈ M.Since F(X ×X) ⊆g(X), we can choose x1, y1 ∈ X such that

gx1 = F(x0, y0) and gy1 = F(y0, x0).

In the same way we construct

gx2 = F(x1, y1) and gy2 = F(y1, x1).

Continuing in this way we construct two sequences {xn} and {yn} in X such that

gxn+1 = F(xn, yn) and gyn+1 = F(yn, xn) ∀n ≥ 0. (3.4)

Since (gx1, gy1, gx0, gy0) = (F(x0, y0), F(y0, x0), gx0, gy0) ∈ M and M is (F, g)-invariant set, we get

(gx2, gy2, gx1, gy1) = (F(x1, y1), F(y1, x1), F(x0, y0), F(y0, x0)) ∈ M.

Again, using the fact that M is (F, g)-invariant set, we have

(gx3, gy3, gx2, gy2) = (F(x2, y2), F(y2, x2), F(x1, y1), F(y1, x1)) ∈ M.

By repeating argument to the above, we get

(gxn, gyn, gxn−1, gyn−1) ∈ M

for all n ∈ N.Since (gxn, gyn, gxn−1, gyn−1) ∈ M , we can apply condition (3.3) to obtain

d(gxn, gxn+1) = d(F(xn−1, yn−1), F(xn, yn))

≤P a1d(gxn−1, gxn)+ a2d(F(xn−1, yn−1), gxn−1)

+a3d(gyn−1, gyn)+ a4d(F(xn, yn), gxn)

+a5d(F(xn−1, yn−1), gxn)+ a6d(F(xn, yn), gxn−1)

= a1d(gxn−1, gxn)+ a2d(gxn, gxn−1)+ a3d(gyn−1, gyn)

+a4d(gxn+1, gxn)+ a5d(gxn, gxn)+ a6d(gxn+1, gxn−1)

≤P a1d(gxn−1, gxn)+ a2d(gxn, gxn−1)+ a3d(gyn−1, gyn)

+a4d(gxn+1, gxn)+ a6[d(gxn−1, gxn)+ d(gxn, gxn+1)]≤P (a1 + a2 + a6)d(gxn−1, gxn)+ a3d(gyn−1, gyn)

+(a4 + a6)d(gxn, gxn+1)

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which implies that

(1 − a4 − a6)d(gxn, gxn+1) ≤P (a1 + a2 + a6)d(gxn−1, gxn)+ a3d(gyn−1, gyn). (3.5)

Since M is (F, g)-invariant set and (gxn, gyn, gxn−1, gyn−1) ∈ M for all n ∈ N, we get(gyn−1, gxn−1, gyn, gxn) ∈ M for all n ∈ N.

Similarly, starting with d(gyn, gyn+1) = d(F(yn, xn), F(yn−1, xn−1)) and using that(gyn−1, gxn−1, gyn, gxn) ∈ M for all n ∈ N, we obtain

(1 − a4 − a6)d(gyn, gyn+1) ≤P (a1 + a2 + a6)d(gyn−1, gyn)+ a3d(gxn−1, gxn). (3.6)

Adding up (3.5) and (3.6) we obtain that

(1 − a4 − a6)[d(gxn, gxn+1)+ d(gyn, gyn+1)]≤P (a1 + a2 + a3 + a6)[d(gxn−1, gxn)+ d(gyn−1, gyn)]. (3.7)

Now, starting from d(gxn+1, gxn) = d(F(xn, yn), F(xn−1, yn−1) and using that(gxn, gyn, gxn−1, gyn−1) ∈ M for all n ∈ N, we get that

(1 − a2 − a5)d(gxn, gxn+1) ≤P (a1 + a4 + a5)d(gxn−1, gxn)+ a3d(gyn−1, gyn).

Similarly, starting from d(gyn+1, gyn) = d(F(yn, xn), F(yn−1, xn−1)) and using that(gyn−1, gxn−1, gyn, gxn) ∈ M for all n ∈ N, we get that

(1 − a2 − a5)d(gyn, gyn+1) ≤P (a1 + a4 + a5)d(gyn−1, gyn)+ a3d(gxn−1, gxn).

Again adding up, we obtain that

(1 − a2 − a5)[d(gxn, gxn+1)+ d(gyn, gyn+1)]≤P (a1 + a3 + a4 + a5)[d(gxn−1, gxn)+ d(gyn−1, gyn)]. (3.8)

Finally, adding up (3.7) and (3.8), it follows that

d(gxn, gxn+1)+ d(gyn, gyn+1) ≤P λ[d(gxn−1, gxn)+ d(gyn−1, gyn)], (3.9)

where

0 ≤ λ = 2a1 + a2 + 2a3 + a4 + a5 + a6

2 − a2 − a4 − a5 − a6< 1, (3.10)

since∑n

i=1 ai < 1.We repeat process (3.9) n times, we have

d(gxn, gxn+1)+ d(gyn, gyn+1) ≤P λ[d(gxn−1, gxn)+ d(gyn−1, gyn)]≤P λ

2[d(gxn−2, gxn−1)+ d(gyn−2, gyn−1)]...

≤P λn[d(gx0, gx1)+ d(gy0, gy1)].

If d(gx0, gx1)+ d(gy0, gy1) = 0E then (x0, y0) is a coupled coincidence point of F andg. So let 0E <P d(gx0, gx1)+ d(gy0, gy1).

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For any m > n ≥ 1, repeated use of triangle inequality gives

d(gxn, gxm)s + d(gyn, gym)

≤P d(gxn, gxn+1)+ d(gxn+1, gxn+2)+ · · · + d(gxm−1, gxm)

+ d(gyn, gyn+1)+ d(gyn+1, gyn+2)+ · · · + d(gym−1, gym)

≤P [λn + λn+1 + · · · + λm−1][d(gx0, gx1)+ d(gy0, gy1)]≤P

λn

1 − λ[d(gx0, gx1)+ d(gy0, gy1)].

Taking n → ∞ in above inequality, we get

λn

1 − λ[d(gx0, gx1)+ d(gy0, gy1)] → 0E .

From (p4), we have for 0E � c and large n:

λn

1 − λ[d(gx0, gx1)+ d(gy0, gy1)] � c,

thus, according to (p3), we get

d(gxn, gxm)+ d(gyn, gym) � c.

Since

d(gxn, gxm) ≤P d(gxn, gxm)+ d(gyn, gym)

and

d(gyn, gym) ≤P d(gxn, gxm)+ d(gyn, gym)

then by (p3), we have

d(gxn, gxm) � c and d(gyn, gym) � c

for n large enough and hence {gxn} and {gyn} are Cauchy sequences in g(X). By completenessof g(X), there exists gx, gy ∈ g(X) such that gxn → gx and gyn → gy as n → ∞.

Since (gxn, gyn, gxn−1, gyn−1) ∈ M for all n ∈ N and gxn → gx and gyn → gy asn → ∞, using the conditions (d), we have (gx, gy, gxn, gyn) ∈ M for all n ∈ N. Next, weprove that F(x, y) = gx and F(y, x) = gy. It follows from (gx, gy, gxn, gyn) ∈ M for alln ∈ N, we get

d(F(x, y), gx) ≤ P d(F(x, y), gxn+1)+ d(gxn+1, gx)

= d(F(x, y), F(xn, yn))+ d(gxn+1, gx)

≤P a1d(gx, gxn)+ a2d(F(x, y), gx)+ a3d(gy, gyn)

+a4d(F(xn, yn), gxn)+ a5d(F(x, y), gxn)

+a6d(F(xn, yn), gx)+ d(gxn+1, gx)

≤P a1d(gx, gxn)+ a2d(F(x, y), gx)+ a3d(gy, gyn)

+a4d(gxn+1, gx)+ a4d(gx, gxn)+a5d(F(x, y), gx)+ a5d(gx, gxn)

+a6d(gxn+1, gx)+ d(gxn+1, gx),

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which further implies that

d(F(x, y), gx) ≤Pa1 + a4 + a5

1 − a2 − a5d(gx, gxn)+ 1 + a4 + a6

1 − a2 − a5d(gxn+1, gx)

+ a3

1 − a2 − a5d(gy, gyn).

Since gxn → gx and gyn → gy then for 0E � c there exists N ∈ N such that

d(gxn, gx) � (1 − a2 − a5)c

3(a1 + a4 + a5),

d(gxn+1, gx) � (1 − a2 − a5)c

3(1 + a4 + a6),

and

d(gyn, gy) � (1 − a2 − a5)c

3a3,

for all n ≥ N . Thus,

d(F(x, y), gx) � c

3+ c

3+ c

3= c.

From (p2), we have d(F(x, y), gx) = 0E and so F(x, y) = gx . Similarly, we can provethat F(y, x) = gy. Therefore, (x, y) is a coupled coincidence point of mappings F and g.

��Corollary 3.7 Let (X, d) be a cone metric space over a solid cone P and M be subset of X4.Let F : X × X → X and g : X → X be mappings and there exist two elements x0, y0 ∈ Xwith

F(x0, y0), F(y0, x0), gx0, gy0) ∈ M.

Suppose further that F, g satisfy

d(F(x, y), F(u, v)) ≤P αd(gx, gu)+ βd(gy, gv)+ γ d(F(x, y), gu),

for all (x, y), (u, v) ∈ X × X with (gx, gy, gu, gv) � M, where α, β, γ ≥ 0 andα + β + γ < 1.

Further suppose

(a): F(X × X) ⊆ g(X);(b): g(X) is a complete subspace of X;(c): M is (F, g)-invariant subset of X4;(d): X has the following properties:

if for two sequences {xn}, {yn} with (xn+1, yn+1, xn, yn) ∈ M and

{xn} → x, {yn} → y,

for all n ∈ N, then (x, y, xn, yn) ∈ M for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = gx and F(y, x) = gy,

that is, F and g have a coupled coincidence point (x, y) ∈ X × X.

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Proof For the proof of this result is straightforward, following the same lines of the proof ofTheorem 3.6 with

a1 = α, a2 = 0, a3 = β, a4 = 0 and a5 = γ.

Then, in order to avoid repetition, the details are omitted. ��If we take g = IX , where IX is an identity mapping on X , in Theorem 3.6, then we get

the following corollary.

Corollary 3.8 Let (X, d) be a complete cone metric space over a solid cone P and M besubset of X4. Let F : X × X → X be a mapping and there exist two elements x0, y0 ∈ Xwith

(F(x0, y0), F(y0, x0), x0, y0) ∈ M.

Suppose further that F satisfy

d(F(x, y), F(u, v)) ≤P a1d(x, u)+ a2d(F(x, y), x)+ a3d(y, v)

+a4d(F(u, v), u)+ a5d(F(x, y), u)+ a6d(F(u, v), x), (3.11)

for all (x, y), (u, v) ∈ X × X with (x, y, u, v) � M, where ai ≥ 0, for i = 1, 2, . . . , 6 and∑6

i=1 ai < 1.Further suppose M is F-invariant subset of X4 and X has the following properties: if for

two sequences {xn}, {yn} with (xn+1, yn+1, xn, yn) ∈ M and

{xn} → x, {yn} → y,

for all n ∈ N, then (x, y, xn, yn) ∈ M for all n ∈ N.Then there exist x, y ∈ X such that

F(x, y) = x and F(y, x) = y,

that is, F has a coupled fixed point (x, y) ∈ X × X.

Next, we give a simple application of our results to coupled coincidence points theoremsin partially ordered metric spaces which are the main results of Nashine et al. [26].

Corollary 3.9 (Theorem 3.1 in [26]) Let (X, d,�) be an ordered cone metric space over asolid cone P. Let F : X × X → X and g : X → X be mappings such that F has the mixedg-monotone property on X and there exist two elements x0, y0 ∈ X with gx0 � F(x0, y0)

and gy0 � F(y0, x0). Suppose further that F, g satisfy

d(F(x, y), F(u, v)) ≤P a1d(gx, gu)+ a2d(F(x, y), gx)+ a3d(gy, gv)

+a4d(F(u, v), gu)+ a5d(F(x, y), gu)+ a6d(F(u, v), gx),

(3.12)

for all (x, y), (u, v) ∈ X × X with (gu � gx and gv � gy) or (gx � gu and gy � gv),where ai ≥ 0, for i = 1, 2, . . . , 6 and

∑6i=1 ai < 1. Further suppose

(1) F(X × X) ⊆ g(X);(2) g(X) is a complete subspace of X.

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Also, suppose that X has the following properties:

(i) if a non-decreasing sequence {xn} in X is such that xn → x, then xn � x for all n ∈ N,(ii) if a non-increasing sequence {yn} in X is such that yn → y, then yn � y for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = gx and F(y, x) = gy,

that is, F and g have a coupled coincidence point (x, y) ∈ X × X.

Proof First, we setting the subset M ⊆ X4 by

M = {(a, b, c, d) ∈ X4 : a � c and b � d}.From Example 3.5, we can conclude that M is (F, g)-invariant set. By (3.12), we have

d(F(x, y), F(u, v)) ≤P a1d(gx, gu)+ a2d(F(x, y), gx)+ a3d(gy, gv)

+a4d(F(u, v), gu)+ a5d(F(x, y), gu)+ a6d(F(u, v), gx),

for all (x, y), (u, v) ∈ X × X with (gx, gy, gu, gv) � M .Since x0, y0 ∈ X such that

gx0 � F(x0, y0) and gy0 � F(y0, x0),

we get

(F(x0, y0), F(y0, x0), gx0, gy0) ∈ M.

From the assumptions (i) and (ii), we get the condition (d) in Theorem 3.6 holds. Now, allhypothesis of Theorem 3.6 hold. Thus F and g have a coupled coincidence point. ��

Similarly, from Corollaries 3.7 and 3.8, we can get the following corollaries:

Corollary 3.10 (Corollary 3.3 in [26]) Let (X, d,�) be an ordered cone metric space overa solid cone P. Let F : X × X → X and g : X → X be mappings such that F has the mixedg-monotone property on X and there exist two elements x0, y0 ∈ X with gx0 � F(x0, y0)

and gy0 � F(y0, x0). Suppose that F, g satisfy that

d(F(x, y), F(u, v)) ≤P αd(gx, gu)+ βd(gy, gv)+ γ d(F(x, y), gu),

for all (x, y), (u, v) ∈ X × X with (gu � gx and gv � gy) or (gx � gu and gy � gv),where α, β, γ ≥ 0 and α + β + γ < 1. Further suppose

(1) F(X × X) ⊆ g(X);(2) g(X) is a complete subspace of X.

Also, suppose that X has the following properties:

(i) if a non-decreasing sequence {xn} in X is such that xn → x, then xn � x for all n ∈ N,(ii) if a non-increasing sequence {yn} in X is such that yn → y, then yn � y for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = gx and F(y, x) = gy,

that is, F and g have a coupled coincidence point (x, y) ∈ X × X.

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Corollary 3.11 (Corollary 3.4 in [26]) Let (X, d,�) be a complete ordered cone metricspace over a solid cone P. Let F : X × X → X be a mapping having the mixed monotoneproperty on X and there exist two elements x0, y0 ∈ X with x0 � F(x0, y0) and y0 �F(y0, x0). Suppose further that F satisfies

d(F(x, y), F(u, v)) ≤P a1d(x, u)+ a2d(F(x, y), x)+ a3d(y, v)

+a4d(F(u, v), u)+ a5d(F(x, y), u)+ a6d(F(u, v), x),

for all (x, y), (u, v) ∈ X × X with (u � x and v � y) or (x � u and y � v), where ai ≥ 0,for i = 1, 2, . . . , 6 and

∑6i=1 ai < 1. Also, suppose that X has the following properties:

(i) if a non-decreasing sequence {xn} in X is such that xn → x, then xn � x for all n ∈ N,(ii) if a non-increasing sequence {yn} in X is such that yn → y, then yn � y for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = x and F(y, x) = y,

that is, F has a coupled fixed point (x, y) ∈ X × X.

Next, we give our second main result is the following:

Theorem 3.12 Let (X, d) be a cone metric space over a solid cone P and M be subset of X4.Let F : X × X → X and g : X → X be mappings and there exist two elements x0, y0 ∈ Xwith

(F(x0, y0), F(y0, x0), gx0, gy0) ∈ M.

Suppose that there is some h ∈ [0, 1/2) such that for all (x, y), (u, v) ∈ X × X with(gx, gy, gu, gv) � M, there exists

�x,y,u,v ∈ {d(gx, gu), d(gy, gv), d(F(x, y), gu)}satisfying

d(F(x, y), F(u, v)) ≤P h�x,y,u,v. (3.13)

Further suppose

(a): F(X × X) ⊆ g(X);(b): g(X) is a complete subspace of X;(c): M is (F, g)-invariant subset of X4;(d): X has the following properties:

if for two sequences {xn}, {yn} with (xn+1, yn+1, xn, yn) ∈ M and

{xn} → x, {yn} → y,

for all n ∈ N, then (x, y, xn, yn) ∈ M for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = gx and F(y, x) = gy,

that is, F and g have a coupled coincidence point (x, y) ∈ X × X.

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Proof Let x0, y0 ∈ X be such that (F(x0, y0), F(y0, x0), gx0, gy0) ∈ M.Since F(X ×X) ⊆g(X), we can choose x1, y1 ∈ X such that

gx1 = F(x0, y0) and gy1 = F(y0, x0).

In the same way we construct

gx2 = F(x1, y1) and gy2 = F(y1, x1).

Continuing in this way we construct two sequences {xn} and {yn} in X such that

gxn+1 = F(xn, yn) and gyn+1 = F(yn, xn) ∀n ≥ 0. (3.14)

Repeating the steps from the proof of Theorem 3.6 we get

(gxn, gyn, gxn−1, gyn−1) ∈ M

for all n ∈ N.Now, we can apply condition (3.13) for (gxn, gyn, gxn−1, gyn−1) ∈ M for all n ∈ N to

obtain there exists

� 1 ∈ {d(gxn−1, gxn), d(gyn−1, gyn), d(F(xn−1, yn−1), gxn)}= {d(gxn−1, gxn), d(gyn−1, gyn), 0E }

such that

d(gxn, gxn+1) = d(F(xn−1, yn−1), F(xn, yn)) ≤P h�1.

Similarly, one can show that there exists

�2 ∈ {d(gxn−1, gxn), d(gyn−1, gyn), 0E }such that

d(gyn, gyn+1) = d(F(yn−1, xn−1), F(yn, xn)) ≤P h�2.

Now, denote δn = d(gxn, gxn+1)+ d(gyn, gyn+1). Since the cases �1 = 0E and �2 = 0E

are trivial, we have to consider the following four possibilities.

Case 1 d(gxn, gxn+1) ≤P hd(gxn−1, gxn) and d(gyn, gyn+1) ≤P hd(gyn−1, gyn). Addingup, we get that

δn ≤P hδn−1.

Case 2 d(gxn, gxn+1) ≤P hd(gxn−1, gxn) and d(gyn, gyn+1) ≤P hd(gxn−1, gxn). Then

δn ≤P 2hd(gxn−1, gxn) ≤P 2hd(gxn−1, gxn)+ 2hd(gyn−1, gyn) = 2hδn−1.

Case 3 d(gxn, gxn+1) ≤P hd(gyn−1, gyn) and d(gyn, gyn+1) ≤P hd(gxn−1, gxn). Thiscase is treated analogously to Case 1.

Case 4 d(gxn, gxn+1) ≤P hd(gyn−1, gyn) and d(gyn, gyn+1) ≤P hd(gyn−1, gyn). Thiscase is treated analogously to Case 2.

Thus, in any case, for each n ∈ N, δn ≤P 2hδn−1 holds, where 0 ≤ 2h < 1. Therefore,

δn ≤P 2hδn−1 ≤P (2h)2δn−2 ≤P · · · ≤P (2h)nδ0,

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and by the same argument as in Theorem 3.9 it is proved that {gxn} and {gyn} are Cauchysequences in g(X). By the completeness of g(X), there exists gx, gy ∈ g(X) such thatgxn → gx and gyn → gy.

Since (gxn, gyn, gxn−1, gyn−1) ∈ M for all n ∈ N and gxn → gx and gyn → gy asn → ∞, using the conditions (d), we have (gx, gy, gxn, gyn) ∈ M for all n ∈ N.

Next, we prove that F(x, y) = gx and F(y, x) = gy. As (gx, gy, gxn, gyn) ∈ M for alln ∈ N that

d(F(x, y), gx) ≤P d(F(x, y), gxn+1)+ d(gxn+1, gx)

= d(F(x, y), F(xn, yn))+ d(gxn+1, gx)

≤P h�x,y,xn ,yn + d(gxn+1, gx),

where �x,y,xn ,yn ∈ {d(gx, gxn), d(gy, gyn), d(F(x, y), gxn)}. Let c ∈ int(P) be fixed. If�x,y,xn ,yn = d(gx, gxn) or �x,y,xn ,yn = d(gy, gyn), then for n sufficiently large we havethat

d(F(x, y), gx) � h · c

2h+ c

2= c.

By property (p2), it follows that F(x, y) = gx . If �x,y,xn ,yn = d(F(x, y), gxn), then weget that

d(F(x, y), gx) ≤P hd(F(x, y), gxn)+ d(gxn+1, gx)

≤P hd(F(x, y), gx)+ hd(gx, gxn)+ d(gxn+1, gx).

Now it follows that for n sufficiently large,

d(F(x, y), gx) ≤Ph

1 − hd(gx, gxn)+ 1

1 − hd(gxn+1, gx)

� h

1 − h· 1 − h

h· c

2+ 1

1 − h(1 − h)

c

2= c.

Again using property (p2), we get F(x, y) = gx .Similarly, F(y, x) = gy is obtained. Therefore (x, y) is a coupled point of coincidence

of the mappings F and g. This complete the proof.

The following corollaries can be proved in the same way as Corollary 3.9.

Corollary 3.13 (Theorem 3.7 in [26] Let (X, d,�) be an ordered cone metric space overa solid cone P. Let F : X × X → X and g : X → X be mappings such that F hasthe mixed g-monotone property on X and there exist two elements x0, y0 ∈ X with gx0 �F(x0, y0) and gy0 � F(y0, x0). Suppose that there is some h ∈ [0, 1/2) such that for all(x, y), (u, v) ∈ X × X with (gx � gu and gy � gv) or (gu � gx and gv � gy), there exists

�x,y,u,v ∈ {d(gx, gu), d(gy, gv), d(F(x, y), gu)}satisfying

d(F(x, y), F(u, v)) ≤P h�x,y,u,v. (3.15)

Further suppose

(1) F(X × X) ⊆ g(X);(2) g(X) is a complete subspace of X.

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Also, suppose that X has the following properties:

(i) if a non-decreasing sequence {xn} in X is such that xn → x, then xn � x for all n ∈ N,(ii) if a non-increasing sequence {yn} in X is such that yn → y, then yn � y for all n ∈ N.

Then there exist x, y ∈ X such that

F(x, y) = gx and F(y, x) = gy,

that is, F and g have a coupled coincidence point (x, y) ∈ X × X.

Now, reasoning on Theorems 3.6 and 3.12, some questions arise naturally. To be precise,one can ask himself

Question Is possible to find the necessary condition for the existence and uniqueness of thecommon coupled fixed point of F and g?

Motivated by the interest in this research, we give positive answers to these questionsadding to Theorems 3.6 and 3.12 some hypotheses.

Theorem 3.14 In addition to the hypotheses of Theorem 3.6, suppose that for every(x, y), (y∗, x∗) ∈ X × X there exists (u, v) ∈ X × X such that

(F(x, y), F(y, x), F(u, v), F(v, u)) � M

and

(F(x∗, y∗), F(y∗, x∗), F(u, v), F(v, u)) � M.

Then F and g have a unique common coupled fixed point, that is, there exists a unique(u, v) ∈ X × X such that

u = gu = F(u, v) and v = gv = F(v, u),

provided F and g are w∗-compatible.

Proof From Theorem 3.6, we get F and g have a coupled coincidence point. Suppose (x, y)and (x∗, y∗) are coupled coincidence points of F and g, that is,

gx = F(x, y), gy = F(y, x), gx∗ = F(x∗, y∗) and gy∗ = F(y∗, x∗).

Next, we will provide the proof into 3 steps.

Step 1 We will prove that

gx = gx∗ and gy = gy∗. (3.16)

By assumption, there exists (u, v) ∈ X × X such that

(F(x, y), F(y, x), F(u, v), F(v, u)) � M (3.17)

and

(F(x∗, y∗), F(y∗, x∗), F(u, v), F(v, u)) � M. (3.18)

From (3.17), we get

(F(x, y), F(y, x), F(u, v), F(v, u)) ∈ M (3.19)

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or

(F(u, v), F(v, u), F(x, y), F(y, x)) ∈ M. (3.20)

From (3.18), we get

(F(x∗, y∗), F(y∗, x∗), F(u, v), F(v, u)) ∈ M (3.21)

or

(F(u, v), F(v, u), F(x∗, y∗), F(y∗, x∗)) ∈ M. (3.22)

Put u0 = u, v0 = v, and choose u1, v1 ∈ X so that gu1 = F(u0, v0) and gv1 = F(v0, u0).Then, similarly as in the proof of Theorem 3.6, we can inductively define sequences {gun},{gvn} with

gun+1 = F(un, vn) and gvn+1 = F(vn, un) ∀n.

Further, set x0 = x , y0 = y, x∗0 = x∗, y∗

0 = y∗ and, in a similar way, define the sequences{gxn}, {gyn} and {gx∗

n }, {gy∗n }. Then it is easy to show that

gxn → F(x, y), gyn → F(y, x)

and

gx∗n → F(x∗, y∗), gy∗

n → F(y∗, x∗)

as n → ∞.Since

(gu1, gv1, gx, gy) = (gu1, gv1, gx1, gy1) = (F(u, v), F(v, u), F(x, y), F(y, x)) ∈ M

or

(gx, gy, gu1, gv1) = (gx1, gy1, gu1, gv1) = (F(x, y), F(y, x), F(u, v), F(v, u)) ∈ M.

It is easy to show that, similarly

(gun, gvn, gx, gy) ∈ M

or

(gx, gy, gun, gvn) ∈ M

for all n ∈ N. Hence from (3.3), we have

d(gun+1, gx) = d(F(un, vn), F(x, y))

≤P a1d(gun, gx)+ a2d(F(un, vn), gun)+ a3d(gvn, gy)

+a4d(F(x, y), gx)+ a5d(F(un, vn), gx)+ a6d(F(x, y), gun)

= a1d(gun, gx)+ a2d(gun+1, gun)+ a3d(gvn, gy)

+a4d(gx, gx)+ a5d(gun+1, gx)+ a6d(gx, gun)

≤P a1d(gun, gx)+ a2[d(gun+1, gx)+ d(gx, gun)] + a3d(gvn, gy)

+a5d(gun+1, gx)+ a6d(gx, gun),

that is,

(1 − a2 − a5)d(gun+1, gx) ≤P (a1 + a2 + a6)d(gun, gx)+ a3d(gvn, gy).

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In the same way, starting from d(gvn+1, gy), we can show that

(1 − a2 − a5)d(gvn+1, gy) ≤P (a1 + a2 + a6)d(gvn, gy)+ a3d(gun, gx).

Adding up two above inequalities, we get

(1 − a2 − a5)[d(gun+1, gx)+ d(gvn+1, gy)]≤P (a1 + a2 + a3 + a6)[d(gun, gx)+ d(gvn, gy)] (3.23)

In a similar way, starting from d(gx, gun+1), resp., d(gy, gvn+1), and combine the obtainedinequalities, we get

(1 − a4 − a6)[d(gx, gun+1)+ d(gy, gvn+1)]≤P (a1 + a3 + a4 + a5)[d(gx, gun)+ d(gy, gvn)]. (3.24)

From (3.23) and (3.24), we get

[d(gun+1, gx)+ d(gvn+1, gy)] ≤P λ[d(gun, gx)+ d(gvn, gy)], (3.25)

where

0 ≤ λ = 2a1 + a2 + 2a3 + a4 + a5 + a6

2 − a2 − a4 − a5 − a6< 1,

since∑n

i=1 ai < 1.By repeating inequality (3.25) n times, we get

d(gun, gx)+ d(gvn, gy) ≤P λn[d(gu0, gx)+ d(gv0, gy)]

Since 0 ≤ λ < 1, if we take n → ∞, we have

d(gun, gx)+ d(gvn, gy) → 0E .

Hence d(gun, gx)+ d(gvn, gy) � c for each c ∈ int(P) and large n. Since

0E ≤P d(gun, gx) ≤P d(gun, gx)+ d(gvn, gy),

it follows from (p3) that d(gun, gx) � c for large n and then gun → gx as n → ∞.Similarly, we can prove that gvn → gy as n → ∞. By the same procedure one can showthat gun → gx∗ and gvn → gy∗ as n → ∞. By the uniqueness of the limit, we havegx = gx∗ and gy = gy∗. This complete the proof of Step 1. Therefore, (gx, gy) is theunique coupled point of coincidence of F and g.

Step 2 We will show that F and g have a common coupled fixed point.

Note that if (gx, gy) is a coupled point of coincidence of F and g, then (gy, gx) isalso a coupled point of coincidence of F and g. By the uniqueness of the coupled point ofcoincidence of F and g, we get gx = gy. Therefore, (gx, gx) is the unique coupled point ofcoincidence of F and g.

Now we let gx = u and so u = gx = F(x, x). Since F and g are w∗-compatibility, weget

gu = ggx = gF(x, x) = F(gx, gx) = F(u, u).

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Hence (gu, gu) is a coupled point of coincidence of F and g. By the uniqueness of a coupledpoint of coincidence of F and g, we have gu = gx . Therefore u = gu = F(u, u). Hence(u, u) is a common coupled fixed point of F and g.

Step 3 We prove the uniqueness of a common coupled fixed point of F and g. We may assumethat (u∗, u∗) ∈ X × X is another common coupled fixed point of F and g that is

u∗ = gu∗ = F(u∗, u∗).

Then (gu, gu) and (gu∗, gu∗) are two coupled points of coincidence of F and g. By unique-ness of a coupled point of coincidence of F and g, we must get gu = gu∗ and so

u = gu = gu∗ = u∗,

which is a contradiction. Therefore, we have (u, u) ∈ X × X is a unique common coupledfixed point of F and g. This finish the proof of this theorem.

Theorem 3.15 In addition to the hypotheses of Theorem 3.12, suppose that for every(x, y), (y∗, x∗) ∈ X × X there exists (u, v) ∈ X × X such that

(F(x, y), F(y, x), F(u, v), F(v, u)) � M

and

(F(x∗, y∗), F(y∗, x∗), F(u, v), F(v, u)) � M.

Then F and g have a unique common coupled fixed point, that is, there exists a unique(u, v) ∈ X × X such that

u = gu = F(u, v) and v = gv = F(v, u),

provided F and g are w∗-compatible.

Proof This result can be proved in the same way as Theorem 3.14.

Next, we apply our results to unique common coupled fixed points theorems in partiallyordered metric spaces. If (X,�) is a partially ordered set, then we endow the product spaceX × X with the following partial order:

for (x, y), (u, v) ∈ X × X, (u, v) � (x, y) ⇔ x � u, y � v.

Corollary 3.16 (Theorem 3.2 in [26]) In addition to the hypotheses of Corollary 3.9, sup-pose that for every (x, y), (y∗, x∗) ∈ X × X there exists (u, v) ∈ X × X such that(F(u, v), F(v, u)) is comparable both to (F(x, y), F(y, x)) and (F(x∗, y∗), F(y∗, x∗)).Then F and g have a unique common coupled fixed point, that is, there exists a unique(u, v) ∈ X × X such that

u = gu = F(u, v) and v = gv = F(v, u),

provided F and g are w∗-compatible.

Proof First, we setting the subset M ⊆ X4 by

M = {(a, b, c, d) ∈ X4 : a � c and b � d}.From Example 3.5, we can conclude that M is (F, g)-invariant set. From the hypothe-sis, we have for every (x, y), (y∗, x∗) ∈ X × X there exists (u, v) ∈ X × X such that(F(u, v), F(v, u)) is comparable both to (F(x, y), F(y, x)) and (F(x∗, y∗), F(y∗, x∗)).

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Since (F(u, v), F(v, u)) is comparable (F(x, y), F(y, x)), we have

(F(u, v), F(v, u)) � (F(x, y), F(y, x)) (3.26)

or

(F(x, y), F(y, x)) � (F(u, v), F(v, u)). (3.27)

From (3.26), we get F(x, y) � F(u, v) and F(y, x) � F(v, u) that is

(F(x, y), F(y, x), F(u, v), F(v, u)) ∈ M

or from (3.27), we have F(u, v) � F(x, y) and F(v, u) � F(y, x) that is

(F(u, v), F(v, u), F(x, y), F(y, x)) ∈ M.

Therefore, we have

(F(x, y), F(y, x), F(u, v), F(v, u)) � M.

Similarly, we can conclude that

(F(x∗, y∗), F(y∗, x∗), F(u, v), F(v, u)) � M.

Now, we have for every (x, y), (y∗, x∗) ∈ X × X there exists (u, v) ∈ X × X such that

(F(x, y), F(y, x), F(u, v), F(v, u)) � M

and

(F(x∗, y∗), F(y∗, x∗), F(u, v), F(v, u)) � M.

Therefore, all hypothesis of Theorem 3.14 satisfy and so F and g have a unique commoncoupled fixed point. ��Corollary 3.17 (Theorem 3.8 in [26]) In addition to the hypotheses of Corollary 3.13,suppose that for every (x, y), (y∗, x∗) ∈ X × X there exists (u, v) ∈ X × X such that(F(u, v), F(v, u)) is comparable to both (F(x, y), F(y, x)) and (F(x∗, y∗), F(y∗, x∗)).Then F and g have a unique common coupled fixed point, that is, there exists a unique(x, y) ∈ X × X such that

x = gx = F(x, y) and y = gy = F(y, x),

provided F and g are w∗-compatible.

4 Example

In this section, we give some example which shows that our Theorems are proper gener-alizations of the results from the paper of Nashine et al. [26] and for other works, such as[20,37,32].

Example 4.1 Let X = R be ordered by the following relation

x � y ⇐⇒ (x = y) or (x ≤ y for x, y ∈ [0, 1]).Let E = C1

R[0, 1] with ‖ f ‖ = ‖ f ‖∞ + ‖ f ′‖∞, f ∈ E and

P = { f ∈ E : f (t) ≥ 0 for t ∈ [0, 1] }.

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It is well known (see, e.g., [48]) that the cone P is not normal. Let d(x, y) = |x − y|ϕ,x, y ∈ X , for a fixed ϕ ∈ P (e.g., ϕ(t) = et for t ∈ [0, 1]). Then (X, d) is a completeordered cone metric space over a nonnormal solid cone.

Let g : X → X and F : X × X → X be defined by

gx = 1

10x and F(x, y) = x + y

30.

Consider y1 = 13 and y2 = 1

2 , we have for x = 1, we get gy1 = 130 � 1

20 = gy2 but

F(x, y1) = 4

90� 1

20= F(x, y2).

So the mapping F does not satisfy the mixed g-monotone property. Therefore, Theorem 3.1and 3.2 of Nashine et al. [26] and the works of Karapinar [20], Shatanawi [37] and Abbas etal. [1] cannot be used to reach this conclusion.

Next, we will check that condition (3.3) of Theorem 3.6 is fulfilled for M = X4. Takea1 = a3 = 1

3 and a2 = a4 = a5 = a6 = 0.For x, y, u, v ∈ X , we obtain that

d(F(x, y), F(u, v)) =∣∣∣∣x + y

30− u + v

30

∣∣∣∣ϕ

≤P1

30|x − u|ϕ + 1

30|y − v|ϕ

= 1

3d(x, u)+ 1

3d(y, v)

= a1d(gx, gu)+ a2d(F(x, y), gx)+ a3d(gy, gv)

+a4d(F(u, v), gu)+ a5d(F(x, y), gu)+ a6d(F(u, v), gx).

Now, we conclude that all the conditions of Theorems 3.14 is satisfied with trivially (F, g)-invariant set M = X4. Therefore, F and g have a unique coupled fixed point (0, 0).

Open problem In Theorems 3.6, 3.12, 3.14 and 3.15, can the condition of (F, g)-invariantset be replaced by other condition?

Acknowledgments The first author would like to thank the Research Professional Development Projectunder the Science Achievement Scholarship of Thailand (SAST) and the second author this work was supportedby a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0094. The third author this work was supported by the Commission on Higher Education,the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (KMUTT) (GrantNo. MRG5580213).

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