Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194http://www.fixedpointtheoryandapplications.com/content/2012/1/194

RESEARCH Open Access

Coupled fixed point theorems in cone metricspaces with a c-distance and applicationsErdal Karapinar1, Poom Kumam2* and Wutiphol Sintunavarat2*

*Correspondence:poom.kum@kmutt.ac.th;poom_teun@hotmail.com2Department of Mathematics,Faculty of Science, King Mongkut’sUniversity of Technology Thonburi(KMUTT), Bangkok, 10140, ThailandFull list of author information isavailable at the end of the article

AbstractIn this paper, we extend the very recent result of Sintunavarat et al. in the paper‘Coupled fixed point theorems for weak contraction mapping under F-invariant set’(Abstr. Appl. Anal. 2012:324874, 2012). In particular, we give an example of a nonlinearcontraction mapping for which our result successfully detects a coupled fixed pointin contrast to the result of Sintunavarat et al., which is not applied to show theexistence of a coupled fixed point. As a consequence, the main results in this paperextend and unify many results in the topic of coupled fixed points including theresults of Sintunavarat et al. Also, some applications of the main results are given.MSC: 54H25; 47H10

Keywords: coupled fixed point; F-invariant set; partially ordered cone metric space;c-distance

1 IntroductionLet X be an arbitrary nonempty set and f : X → X be a mapping. A fixed point for f is apoint x ∈ X such that fx = x. Many useful mathematical facts can be expressed by asser-tions that say that certain mappings have fixed points. Fixed point theory, one of the veryactive research areas in mathematics as well as quantitative sciences, focuses onmaps andabstract spaces for which one can assure the existence and/or uniqueness of fixed pointswhen they are put together. For example, fixed point theorems are vital for the existenceand uniqueness of differential equations, matrix equations, and integral equations (see,e.g., [, ]). Moreover, it has applications in many fields such as chemistry, biology, statis-tics, economics, computer science and engineering (see, e.g., [–]). For example, fixedpoint results are incredibly useful when it comes to proving the existence of various typesof Nash equilibria (see, e.g., []) in economics.The classical contraction mapping principle of Banach is one of the most powerful the-

orems in fixed point theory because of its simplicity and usefulness. So, it has become avery popular tool for solving many problems in many branches of mathematical analysisand also in many other fields. Refer to [–] and references mentioned therein for cer-tain extensions of this principle. In , the Banach contraction principle was extendedtometric spaces endowed with a partial ordering by Ran and Reurings []. They also gavesome applications of their results to matrix equations. Afterwards, Nieto and Rodríguez-López [] extended the results of Ran and Reurings for nondecreasing mappings andstudied the existence and uniqueness of solutions for a first-order ordinary differentialequation with periodic boundary conditions.

© 2012 Karapinar et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative CommonsAttribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproductionin any medium, provided the original work is properly cited.

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In , Bhaskar and Lakshmikantham [] introduced the concept of a mixed mono-tone property for the first time and gave their classical coupled fixed point theorems formappings which satisfy the mixed monotone property. They also produced some applica-tions in the existence and uniqueness of solutions for the periodic boundary value prob-lem

u′(t) = f(t,u(t)

), t ∈ [,T],

u() = u(T),(.)

where the function f satisfies certain conditions. Afterwards, Harjani et al. [] and Luongand Thuan [, ] studied the existence and uniqueness of solutions of nonlinear integralequations as an application of coupled fixed points. Recently, Caballero et al. [] have in-vestigated the existence and uniqueness of positive solutions for the following singularfractional boundary value problem:

Dα+u(t) + f

(t,u(t)

)= , < t < ,

u() = u() = ,(.)

where α ∈ R such that < α ≤ . Here Dα+ is the standard Riemann-Liouville differen-

tiation. The function f : (, ] × [,∞) → [,∞) has the property limt→+ f (t, ·) = ∞ forall t ∈ (, ] (i.e., f is singular at t = ). Very recently, motivated by the work of Caballeroet al. [], Jleli and Samet [] discussed the existence and uniqueness of a positive so-lution for the singular nonlinear fractional differential equation boundary value prob-lem

Dα+u(t) = f

(t,u(t),u(t)

), < t < ,

u(a) = u′(b) = , a,b ∈ {, },(.)

where α ∈ R such that < α ≤ . Above Dα+ is also the Riemann-Liouville frac-

tional derivative. The function f : (, ] × [,∞) × [,∞) → [,∞) is continuous,limt→+ f (t, ·, ·) = ∞ for all t ∈ (, ], and nondecreasing with respect to the first com-ponent, decreasing with respect to its second and third components.Because of their important roles in the study of the existence and uniqueness of solutions

of the periodic boundary value problems, nonlinear integral equations and the existenceand uniqueness of positive solutions for the singular nonlinear fractional differential equa-tions with boundary value, discussions on coupled fixed point theorems are of interest tomany scientists. A considerable number of articles have been dedicated to the improve-ment and generalization of this topic (see [–] and references therein).On the other hand,Huang andZhang [] reintroduced the notion of conemetric spaces

and established fixed point theorems for mappings in such spaces. In , Cho et al. []introduced a new concept of a c-distance, a cone version of aw-distance of Kada et al. [],in cone metric spaces and proved some fixed point theorems in partially ordered conemetric spaces using the notion of a c-distance. Afterwards, Sintunavarat et al. [] (seealso []) established fixed point theorems for some type of generalization of contractionmappings by using this concept.

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Recently, Cho et al. [] established new coupled fixed point theorems under contrac-tion mappings by using the concept of the mixed monotone property and c-distance inpartially ordered cone metric spaces as follows.

Theorem . ([]) Let (X,�) be a partially ordered set and suppose that (X,d) is a com-plete cone metric space. Let q be a c-distance on X and F : X × X → X be a continuousfunction having the mixed monotone property such that

q(F(x, y),F

(x*, y*

)) � k(q(x,x*

)+ q

(y, y*

))

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x� x*

) ∧ (y y*

)or

(x x*

) ∧ (y� y*

).

If there exist x, y ∈ X such that

x � F(x, y) and F(y,x)� y,

then F has a coupled fixed point (u, v).Moreover, we have q(v, v) = θ and q(u,u) = θ .

Theorem . ([]) In addition to the hypotheses of Theorem ., suppose that any twoelements x and y in X are comparable. Then the coupled fixed point has the form (u,u),where u ∈ X.

Theorem . ([]) Let (X,�) be a partially ordered set and suppose that (X,d) is a com-plete cone metric space. Let q be a c-distance on X and F : X ×X → X be a function havingthe mixed monotone property such that

q(F(x, y),F

(x*, y*

)) � k

(q(x,x*

)+ q

(y, y*

))

for some k ∈ (, ) and all x, y,x*, y* ∈ X with

(x� x*

) ∧ (y y*

)or

(x x*

) ∧ (y� y*

).

Also, suppose that X has the following properties:(a) if (xn) is a non-decreasing sequence in X with xn → x, then xn � x for all n ∈N;(b) if (xn) is a non-increasing sequence in X with xn → x, then x� xn for all n ∈ N.

Assume there exist x, y ∈ X such that

x � F(x, y) and F(y,x)� y.

If y � x, then F has a coupled fixed point.

Very recently, Sintunavarat et al. [] have weakened the condition of the mixed mono-tone property in results of Cho et al. [] by using the concept of an F-invariant set (seethe notion of an F-invariant set in Section ).

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Theorem . ([]) Let (X,d) be a complete cone metric space. Let q be a c-distance on X,M be a nonempty subset of X and F : X ×X → X be a continuous function such that

q(F(x, y),F

(x*, y*

)) � k(q(x,x*

)+ q

(y, y*

))(.)

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x, y,x*, y*

) ∈M or(x*, y*,x, y

) ∈M.

If M is F-invariant and there exist x, y ∈ X such that

(F(x, y),F(y,x),x, y

) ∈M,

then F has a coupled fixed point (u, v). Moreover, if (u, v,u, v) ∈ M, then q(v, v) = θ andq(u,u) = θ .

Theorem . ([]) In addition to the hypotheses of Theorem ., suppose that for any twoelements x and y in X, we have

(y,x,x, y) ∈ M or (x, y, y,x) ∈M.

Then the coupled fixed point has the form (u,u), where u ∈ X.

Theorem . ([]) Let (X,d) be a complete cone metric space. Let q be a c-distance onX,M be a subset of X and F : X ×X → X be a function such that

q(F(x, y),F

(x*, y*

)) � k

(q(x,x*

)+ q

(y, y*

))(.)

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x, y,x*, y*

) ∈M or(x*, y*,x, y

) ∈M.

Also, suppose that(i) there exist x, y ∈ X such that (F(x, y),F(y,x),x, y) ∈M,(ii) two sequences {xn}, {yn} with (xn+, yn+,xn, yn) ∈ M for all n ∈ N and {xn} → x,

{yn} → y, then (x, y, yn,xn) ∈M and (yn,xn, y,x) ∈M for all n ∈N.If M is an F-invariant set, then F has a coupled fixed point.

Inspired by the results of Sintunavarat et al. [], we prove some coupled fixed pointtheorems in cone metric spaces by using the concept of an F-invariant set and give someexample which is not applied to the existence of a coupled fixed point by the results of Sin-tunavarat et al. [], but can be applied to our results. Moreover, we show that our resultscan be applied to the result in partially ordered cone metric spaces. We also consider anapplication to illustrate our result is useful (see Section ).

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2 PreliminariesThroughout this paper (X,�) denotes a partially ordered set. By x� y, we mean x � y butx �= y. A mapping f : X → X is said to be non-decreasing (non-increasing) if for all x, y ∈ X,x � y implies f (x)� f (y) (f (y)� f (x) respectively).

Definition . ([]) Let (X,�) be a partially ordered set. A mapping F : X × X → Xis said to have the mixed monotone property if F is monotone non-decreasing in its firstargument and is monotone non-increasing in its second argument; that is, for any x, y ∈ X,

x,x ∈ X, x � x =⇒ F(x, y)� F(x, y) (.)

and

y, y ∈ X, y � y =⇒ F(x, y) F(x, y). (.)

Definition . ([]) Let X be a nonempty set. An element (x, y) ∈ X ×X is called a cou-pled fixed point of mapping F : X ×X → X if

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

Next, we give some terminology of cone metric spaces and the concept of a c-distancein cone metric spaces due to Cho et al. [] which is a generalization of a w-distance ofKada et al. [].Let (E,‖·‖E) be a real Banach space, θ be a zero element in E and P be a proper nonempty

subset of E. Then P is called a cone if the following conditions are satisfied:() P is closed and P �= {θ};() a,b ∈R

+, x, y ∈ P implies ax + by ∈ P;() x ∈ P ∩ (–P) implies x = θ .We shall always assume that the cone P has a nonempty interior (such cones are called

solid). For the cone P, define the partial ordering �with respect to P by x � y if and only ify–x ∈ P.Wewrite x≺ y to indicate that x� y but x �= y, while x � y stand for y–x ∈ int(P),where int(P) is an interior of P.It can be easily shown that λ int(P)⊆ int(P) for all positive scalars λ.The cone P is called normal if there is a number K > such that for all x, y ∈ E,

≤ x≤ y =⇒ ‖x‖E ≤ K‖y‖E .

The least positive number satisfying the above is called the normal constant of P. It is clearthat K ≥ .In what follows, we always suppose that E is a real Banach space with the cone P.

Definition . ([]) Let X be a nonempty set. Suppose that the mapping d : X ×X → Esatisfies the following conditions:() θ � d(x, y) for all x, y ∈ X and d(x, y) = θ if and only if x = y;() d(x, y) = d(y,x) for all x, y ∈ X ;() d(x, y)� 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.

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Definition . ([]) Let (X,d) be a cone metric space. Let {xn} be a sequence in X andx ∈ X.() If for any c ∈ X with θ � c, there exists N ∈N such that d(xn,x)� c for all n≥ N ,

then {xn} is said to be convergent to a point x ∈ X and x is the limit of {xn}. Wedenote this by limn→∞ xn = x or xn → x as n → ∞.

() If for any c ∈ E with θ � c, there exists N ∈ N such that d(xn,xm)� c for alln,m ≥ N , then {xn} is called a Cauchy sequence in X .

() The space (X,d) is called a complete cone metric space if every Cauchy sequence isconvergent.

Definition . ([]) Let (X,d) be a cone metric space. Then a function q : X ×X → E iscalled a c-distance on X if the following are satisfied:(q) θ � q(x, y) for all x, y ∈ X ;(q) q(x, y)� q(x, z) + q(z, y) for all x, y, z ∈ X ;(q) for any x ∈ X , if there exists u = ux ∈ P such that q(x, yn)� u for each n ∈N, then

q(x, y)� u whenever {yn} is a sequence in X converging to a point y ∈ X ;(q) for any c ∈ E with θ � c, there exists e ∈ E with θ � e such that q(z,x)� e and

q(z, y)� c imply d(x, y)� c.

Remark . The c-distance q is a w-distance on X if we take (X,d) is a metric space,E =R

+, P = [,∞) and (q) is replaced by the following condition:For any x ∈ X, q(x, ·) : X →R

+ is lower semi-continuous.Therefore, everyw-distance is a c-distance. But the converse is not true in a general case.

Therefore, the c-distance is a generalization of the w-distance.

Example . Let (X,d) be a cone metric space and P be a normal cone. Define a mappingq : X ×X → E by q(x, y) = d(x, y) for all x, y ∈ X. Then q is a c-distance.

Example . Let E = CR[, ] with ‖x‖E = ‖x‖∞+‖x′‖∞ and P = {x ∈ E : x(t)≥ on [, ]}

(this cone is not normal). Let X = [,∞) and define a mapping d : X ×X → E by

d(x, y) = |x – y|ϕ

for all x, y ∈ X, where ϕ : [, ]→R such that ϕ(t) = et . Then (X,d) is a cone metric space.Define a mapping q : X ×X → E by

q(x, y) = (x + y)ϕ

for all x, y ∈ X. Then q is a c-distance.

Example . Let (X,d) be a cone metric space and P be a normal cone. Define a mappingq : X ×X → E by

q(x, y) = d(u, y)

for all x, y ∈ X, where u is a fixed point in X. Then q is a c-distance.

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Example . Let E = R and P = {x ∈ E : x ≥ }. Let X = [,∞) and define a mappingd : X ×X → E by

d(x, y) = |x – y|

for all x, y ∈ X. Then (X,d) is a cone metric space. Define a mapping q : X ×X → E by

q(x, y) = y

for all x, y ∈ X. Then q is a c-distance.

Remark . From Examples . and ., we have the following important results for ac-distance q:• q(x, y) = q(y,x) does not necessarily hold.• q(x, y) = θ is not necessarily equivalent to x = y for all x, y ∈ X .

The following lemma is crucial in proving main results.

Lemma . ([]) Let (X,d) be a cone metric space and q be a c-distance on X. Let {xn}and {yn} be sequences in X and x, y, z ∈ X. Suppose that {un} is a sequence in P convergingto θ . Then the following hold:() If q(xn, y)� un and q(xn, z)� un, then y = z.() If q(xn, yn)� un and q(xn, z)� un, then {yn} converges to a point z ∈ X .() If q(xn,xm)� un for each m > n, then {xn} is a Cauchy sequence in X .() If q(y,xn)� un, then {xn} is a Cauchy sequence in X .

Next, we give the concept of an F-invariant set in a cone version due to Sintunavarat etal. [].

Definition . ([]) Let (X,d) be a cone metric space and F : X × X → X be a givenmapping. Let M be a nonempty subset of X. We say that M is an F-invariant subset ofX if and only if for all x, y, z,w ∈ X, we have

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

We obtain that the setM = X is trivially F-invariant.Next example plays a key role in the proof of our main results in a partially ordered set.

Example . ([]) Let (X,d) be a cone metric space endowed with a partial order �.Let F : X × X → X be a mapping satisfying the mixed monotone property; that is, for allx, y ∈ X, we have

x,x ∈ X, x � x =⇒ F(x, y)� F(x, y)

and

y, y ∈ X, y � y =⇒ F(x, y) F(x, y).

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Define the subsetM ⊆ X by

M ={(a,b, c,d) : c� a,b � d

}.

Then,M is F-invariant of X.

3 Main resultsFirst, we show the weakness of Theorem . with the following example.

Example . Let E = CR[, ] with ‖x‖E = ‖x‖∞+‖x′‖∞ and P = {x ∈ E,x(t)≥ , t ∈ [, ]}.

Let X = [,∞) (with usual order �), and let d : X × X → E be defined by d(x, y)(t) = |x –y|t . Then (X,d) is a complete conemetric space. Let, further, q : X×X → E be defined byq(x, y)(t) = yt . It is easy to check that q is a c-distance. Consider the mapping F : X×X →X by

F(x, y) =

⎧⎨⎩

(x + y) if x ≥ y, if x < y.

LetM = X and soM is an F-invariant subset of X. Now, we show that there is no k ∈ [, )for which (.) holds. To prove this, suppose the contrary; that is, there is k ∈ [, ) suchthat

q(F(x, y),F

(x*, y*

)) � k(q(x,x*

)+ q

(y, y*

))

for all x, y,x*, y* ∈ X with

(x, y,x*, y*

) ∈M or(x*, y*,x, y

) ∈M.

Take x = , y = , x* = and y* = . Then

q(F(, ),F(, )

)(t)� k

(q(, ) + q(, )

)(t).

This implies

F(, )t = t � k

t .

Hence, k ≥ is a contradiction. Therefore, there is no k for which (.) holds.

Moreover, for y = and y =

, we have for x = , we get y = �

= y but

F(x, y) =

�

= F(x, y).

So, the mapping F does not satisfy the mixed monotone property. Therefore, main theo-rems of Cho et al. [] cannot be used to reach this conclusion.

The following theorem is the extension of Theorem .

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Theorem . Let (X,d) be a complete cone metric space. Let q be a c-distance on X,M bea nonempty subset of X and F : X ×X → X be a continuous function such that

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k(q(x,x*

)+ q

(y, y*

))(.)

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x, y,x*, y*

) ∈M or(x*, y*,x, y

) ∈M.

If M is F-invariant and there exist x, y ∈ X such that

(F(x, y),F(y,x),x, y

) ∈M,

then F has a coupled fixed point (u, v). Moreover, if (u, v,u, v) ∈ M, then q(u,u) = θ andq(v, v) = θ .

Proof Since F(X ×X)⊆ X, we can construct two sequences {xn} and {yn} in X such that

xn = F(xn–, yn–) and yn = F(yn–,xn–) for all n ∈N. (.)

Since (F(x, y),F(y,x),x, y) = (x, y,x, y) ∈M andM is an F-invariant set, we get

(F(x, y),F(y,x),F(x, y),F(y,x)

)= (x, y,x, y) ∈M.

Again, using the fact thatM is an F-invariant set, we have

(F(x, y),F(y,x),F(x, y),F(y,x)

)= (x, y,x, y) ∈ M.

By repeating the argument similar to the above, we get

(F(xn–, yn–),F(yn–,xn–),xn–, yn–

)= (xn, yn,xn–, yn–) ∈M

for all n ∈N. From (.), we have

q(xn,xn+) + q(yn, yn+) � q(F(xn–, yn–),F(xn, yn)

)+ q

(F(yn–,xn–),F(yn,xn)

)� k

(q(xn–,xn) + q(yn–, yn)

). (.)

We repeat the above process for n-times, we get

q(xn,xn+) + q(yn, yn+)� kn(q(x,x) + q(y, y)

). (.)

From (.), we can conclude that

q(xn,xn+)� kn(q(x,x) + q(y, y)

)(.)

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and

q(yn, yn+)� kn(q(x,x) + q(y, y)

). (.)

Let m,n ∈N with m > n. Since

q(xn,xm)�m–∑i=n

q(xi,xi+),

q(yn, ym)�m–∑i=n

q(yi, yi+)

and ≤ k < , we have

q(xn,xm)� kn

– k(q(x,x) + q(y, y)

)

and

q(yn, ym)� kn

– k(q(x,x) + q(y, y)

).

Using Lemma .(), we have {xn} and {yn} are Cauchy sequences in (X,d). By complete-ness of X, we get xn → u and yn → v as n→ ∞ for some u, v ∈ X.Since F is continuous, taking n→ ∞ in (.), we get

limn→∞xn+ = lim

n→∞F(xn, yn) = F(

limn→∞xn, lim

n→∞ yn)= F(u, v)

and

limn→∞ yn+ = lim

n→∞F(yn,xn) = F(

limn→∞ yn, lim

n→∞xn)= F(v,u).

By the uniqueness of the limits, we get u = F(u, v) and v = F(v,u). Therefore, (u, v) is acoupled fixed point of F .Finally, we assume that (u, v,u, v) ∈M. By (.), we have

q(u,u) + q(v, v) = q(F(u, v),F(u, v)

)+ q

(F(v,u),F(v,u)

) � k(q(u,u) + q(v, v)

).

Since ≤ k < , we conclude that q(u,u) + q(v, v) = θ and hence q(u,u) = θ and q(v, v) = θ .This completes the proof. �

Remark . We obtain that the mapping F in Example . has a coupled fixed point.Indeed, for all x, y,x*, y* ∈ X with

(x, y,x*, y*

) ∈M or(x*, y*,x, y

) ∈M

we have

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � (q(x,x*

)+ q

(y, y*

)).

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Also, we note that there exists points , ∈ X such that (F(, ),F(, ), , ) ∈M. Thus, byTheorem ., we have F has a coupled fixed point that is a point (, ).

Theorem . In addition to the hypotheses of Theorem ., suppose that for any two ele-ments x and y in X, we have

(y,x,x, y) ∈ M or (x, y, y,x) ∈M.

Then the coupled fixed point has the form (u,u), where u ∈ X.

Proof As in the proof of Theorem ., there exists a coupled fixed point (u, v) ∈ X × X.Hence,

u = F(u, v) and v = F(v,u).

Moreover, q(u,u) = θ and q(v, v) = θ if (u, v,u, v) ∈ M.From the additional hypothesis, we have (u, v, v,u) ∈ M or (v,u,u, v) ∈ M. By (.), we

get

q(u, v) + q(v,u) = q(F(u, v),F(v,u)

)+ q

(F(v,u),F(u, v)

) � k(q(u, v) + q(v,u)

).

Since ≤ k < , we get q(u, v) + q(v,u) = θ . Therefore, q(u, v) = θ and q(v,u) = θ .Let un = θ and xn = u. Then

q(xn,u)� un

and

q(xn, v)� un.

From Lemma .(), we have u = v. Therefore, the coupled fixed point of F has the form(u,u). This completes the proof. �

Theorem . Let (X,d) be a complete cone metric space. Let q be a c-distance on X,M bea subset of X and F : X ×X → X be a function such that

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k(q(x,x*

)+ q

(y, y*

))(.)

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x, y,x*, y*

) ∈M or(x*, y*,x, y

) ∈M.

Also, suppose that(i) there exist x, y ∈ X such that (F(x, y),F(y,x),x, y) ∈M,(ii) two sequences {xn}, {yn} with (xn+, yn+,xn, yn) ∈ M for all n ∈ N and {xn} → x,

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

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 12 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

If M is an F-invariant set, then F has a coupled fixed point. Moreover, if (u, v,u, v) ∈ M,then q(u,u) = θ and q(v, v) = θ .

Proof As in the proof of Theorem ., we can construct two Cauchy sequences {xn} and{yn} in X such that

(xn, yn,xn–, yn–) ∈M

for all n ∈ N. Moreover, we have that {xn} converges to a point u ∈ X and {yn} convergesto v ∈ X,

q(xn,xm)� kn

– k(q(x,x) + q(y, y)

)

and

q(yn, ym)� kn

– k(q(x,x) + q(y, y)

)

for eachm > n≥ . By (q), we have

q(xn,u)� kn

– k(q(x,x) + q(y, y)

)(.)

and

q(yn, v)� kn

– k(q(x,x) + q(y, y)

)(.)

and so

q(xn,u) + q(yn, v)� kn

– k(q(x,x) + q(y, y)

). (.)

By the assumption (ii), we have (u, v,xn, yn) ∈M. From (.) and (.), we have

q(xn,F(u, v)

)+ q

(yn,F(v,u)

)= q

(F(xn–, yn–),F(u, v)

)+ q

(F(yn–,xn–),F(v,u)

)� k

(q(xn–,u) + q(yn–, v)

)

� k · kn–

– k(q(x,x) + q(y, y)

)

=kn

– k(q(x,x) + q(y, y)

).

Therefore, we have

q(xn,F(u, v)

) � kn

– k(q(x,x) + q(y, y)

)(.)

and

q(yn,F(v,u)

) � kn

– k(q(x,x) + q(y, y)

). (.)

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 13 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

From (.), we get

q(xn,u)� kn

– k(q(x,x) + q(y, y)

)(.)

and

q(yn, v)� kn

– k(q(x,x) + q(y, y)

). (.)

Since (.), (.), (.) and (.) hold, by using Lemma .(), we get u = F(u, v) andv = F(v,u). Therefore, (u, v) is a coupled fixed point of F . The proof of q(u,u) = θ andq(v, v) = θ is the same as the proof in Theorem .. This completes the proof. �

The following theorem can be proved in the same way as Theorem ..

Theorem . In addition to the hypotheses of Theorem ., suppose that for any two ele-ments x and y in X, we have

(y,x,x, y) ∈ M or (x, y, y,x) ∈M.

Then the coupled fixed point has the form (u,u), where u ∈ X.

Remark . Since the relations (.) and (.) imply (.), we can apply Theorems ., .and . to the main result of Sintunavarat et al. [].

Next, we apply Theorems ., ., . and . to the results in a partially ordered conemetric spaces.

Corollary . Let (X,�) be a partially ordered set and suppose that (X,d) is a completecone metric space. Let q be a c-distance on X and F : X ×X → X be a continuous functionhaving the mixed monotone property such that

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k(q(x,x*

)+ q

(y, y*

))(.)

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x� x*

) ∧ (y y*

)or

(x x*

) ∧ (y� y*

).

If there exist x, y ∈ X such that

x � F(x, y) and F(y,x)� y,

then F has a coupled fixed point (u, v).Moreover, we have q(v, v) = θ and q(u,u) = θ .

Proof LetM = {(a,b, c,d) : c � a,b � d} ⊆ X. We obtain thatM is an F-invariant set. By(.), we have

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k(q(x,x*

)+ q

(y, y*

))

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 14 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

for some k ∈ [, ) and all x, y,x*, y* ∈ X with (x, y,x*, y*) ∈ M or (x*, y*,x, y) ∈ M. Now, allthe hypotheses of Theorem . hold. Thus, F has a coupled fixed point. �

Corollary . In addition to the hypotheses of Corollary ., suppose that for any twoelements x and y in X, we have x and y are comparable. Then the coupled fixed point hasthe form (u,u), where u ∈ X.

Proof This result is obtained by Theorem .. �

Corollary . Let (X,�) be a partially ordered set and suppose that (X,d) is a completecone metric space. Let q be a c-distance on X and F : X × X → X be a function having themixed monotone property such that

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k(q(x,x*

)+ q

(y, y*

))(.)

for some k ∈ [, ) and all x, y,x*, y* ∈ X with

(x� x*

) ∧ (y y*

)or

(x x*

) ∧ (y� y*

).

Also, suppose that X has the following properties:(a) if {xn} is a non-decreasing sequence in X with xn → x, then xn � x for all n ∈N;(b) if {yn} is a non-increasing sequence in X with yn → y, then y � yn for all n ∈N.

If there exist x, y ∈ X such that

x � F(x, y) and F(y,x)� y,

then F has a coupled fixed point (u, v).Moreover, we have q(v, v) = θ and q(u,u) = θ .

Proof LetM = {(a,b, c,d) : c � a,b � d} ⊆ X. We obtain thatM is an F-invariant set. By(.), we have

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k(q(x,x*

)+ q

(y, y*

))

for some k ∈ [, ) and all x, y,x*, y* ∈ X with (x, y,x*, y*) ∈ M or (x*, y*,x, y) ∈ M. From as-sumptions (a) and (b), we get two sequences {xn}, {yn}, with {xn} being a non-decreasingsequence in X with xn → x and {yn} being a non-increasing sequence in X with yn → y,

x � x � · · · � xn � · · · � x

and

y y · · · yn · · · y

for all n ∈ N. Therefore, we get the condition (ii) in Theorem . holds. Now, all the hy-potheses of Theorem . hold. Thus, F has a coupled fixed point. �

Corollary . In addition to the hypotheses of Corollary ., suppose that for any twoelements x and y in X, we have x and y are comparable. Then the coupled fixed point hasthe form (u,u), where u ∈ X.

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 15 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

4 ApplicationsIn this section, we apply our theorem to the existence theorem for a solution of the fol-lowing integral equations:

x(t) =∫ T

f(t,x(s), y(s)

)ds, t ∈ [,T];

y(t) =∫ T

f(t, y(s),x(s)

)ds, t ∈ [,T],

(.)

where T > and f : [,T]×R×R →R.In what follows, we always letC([,T],R) denote the class ofR-valued continuous func-

tions on the interval [,T], where T is a real number such that T > .

Definition . An element α,β ∈ C([,T],R)×C([,T],R) is called a coupled lower andupper solution of the integral equation (.) if α(t)≤ β(t) and

α(t)≤∫ T

f(t,α(s),β(s)

)ds

and

β(t)≥∫ T

f(t,β(s),α(s)

)ds

for all t ∈ [,T].

Now, we consider the following assumptions:(a) f : [,T]×R×R→R is continuous;(b) for all t ∈ [,T] and for all x, y,x*, y* ∈R for which x* ≤ x and y≤ y*, we have

≤ f (t,x, y) – f(t,x*, y*

) ≤ T

ϕ

(x – x* + y* – y

),

where ϕ : [,∞)→ [,∞) is continuous non-decreasing and satisfies: there exists≤ k < such that

ϕ(r)≤ kr ∀r ≥ .

Next, we give the existence theorem for a solution of the integral equations (.).

Theorem . Suppose that assumptions (a) and (b) hold. Then the existence of a coupledlower and upper solution for (.) provides the existence of a solution (u, v) ∈ C([,T],R)×C([,T],R) for the integral equations (.).

Proof Let E =R and P ⊆ E be the cone defined by

P ={(x, y) ∈R

|x≥ , y≥ }.

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 16 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

We endow X with the cone metric d : X ×X → E defined by

d(x, y) =(

supt∈[,T]

∣∣x(t) – y(t)∣∣, supt∈[,T]

∣∣x(t) – y(t)∣∣), ∀u, v ∈ X.

It is clear that (X,d) is a complete cone metric space. Let q(x, y) = d(x, y) for all x, y ∈ X.Then, q is a c-distance.Define the self-mapping F : C([,T],R)×C([,T],R)→ C([,T],R) by

F(x, y)(t) =∫ T

f(t,x(s), y(s)

)ds, x, y ∈ C

([,T],R

), t ∈ [,T].

LetM = {(x, y,x*, y*) ∈ X : x*(t)≤ x(t) and y(t)≤ y*(t) for all t ∈ [,T]}.Next, we prove that M is an F-invariant subset of X. It obvious that M satisfies (F).

Now, we show that (F) holds. Let (x, y,x*, y*) ∈M and then x*(t)≤ x(t) and y(t)≤ y*(t) forall t ∈ [,T]. By (b), for all t ∈ [,T], we have

≤ f(t,x(t), y(t)

)– f

(t,x*(t), y*(t)

)(.)

and

≤ f(t, y*(t),x*(t)

)– f

(t, y(t),x(t)

). (.)

From (.), we get

f(t,x*(t), y*(t)

) ≤ f(t,x(t), y(t)

)

and so∫ T

f(t,x*(s), y*(s)

)ds≤

∫ T

f(t,x(s), y(s)

)ds.

This implies that

F(x*, y*

)(t)≤ F(x, y)(t). (.)

Similarly, from (.), we get

F(y,x)(t)≤ F(y*,x*

)(t). (.)

From (.) and (.), we get (F(x, y),F(y,x),F(x*, y*),F(y*,x*)) ∈ M, and thus M satisfies(F). Therefore,M is an F-invariant subset of X.Now, let (x, y,x*, y*) ∈ M and so x*(t) ≤ x(t) and y(t) ≤ y*(t) for all t ∈ [,T]. Using (b),

for all t ∈ [,T], we have

∣∣F(x, y)(t) – F(x*, y*

)(t)

∣∣=

∫ T

[f(t,x(s), y(s)

)– f

(t,x*(s), y*(s)

)]ds

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 17 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

≤ T

∫ T

ϕ

(x(s) – x*(s) + y*(s) – y(s)

)ds

≤ T

∫ T

ϕ

(supz∈[,T] |x(z) – x*(z)| + supz∈[,T] |y(z) – y*(z)|

)ds

= ϕ

(supz∈[,T] |x(z) – x*(z)| + supz∈[,T] |y(z) – y*(z)|

)

≤ k[

supz∈[,T] |x(z) – x*(z)| + supz∈[,T] |y(z) – y*(z)|

],

which implies that

supt∈[,T]

∣∣F(x, y)(t) – F(x*, y*

)(t)

∣∣

≤ k[

supz∈[,T] |x(z) – x*(z)| + supz∈[,T] |y(z) – y*(z)|

]. (.)

Similarly, we can get

supt∈[,T]

∣∣F(y,x)(t) – F(y*,x*

)(t)

∣∣

≤ k[

supz∈[,T] |x(z) – x*(z)| + supz∈[,T] |y(z) – y*(z)|

]. (.)

Adding up (.) and (.), we get

supt∈[,T]

∣∣F(x, y)(t) – F(x*, y*

)(t)

∣∣ + supt∈[,T]

∣∣F(y,x)(t) – F(y*,x*

)(t)

∣∣

≤ k[

supz∈[,T]

∣∣x(z) – x*(z)∣∣ + sup

z∈[,T]

∣∣y(z) – y*(z)∣∣].

Thus,

q(F(x, y),F

(x*, y*

))+ q

(F(y,x),F

(y*,x*

)) � k[q(x,x*

)+ q

(y, y*

)]

for all (x, y,x*, y*) ∈M. This implies that the condition (.) of Theorem . is satisfied.Now, let (α,β) be a coupled lower and upper solution of the integral equations (.), then

we have α(t)≤ F(α,β)(t) and β(t)≤ F(β ,α)(t) for all t ∈ [,T], that is, α ≤ F(α,β) and β ≥F(β ,α). Therefore, (F(α,β),F(β ,α),α,β) ∈M, and then the condition (i) in Theorem . issatisfied. Moreover, it is easy to see that the condition (ii) in Theorem . is also satisfied.Finally, applying our Theorem ., we get the desired result. �

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.

Author details1Department of Mathematics, Atilim University, Incek, Ankara 06836, Turkey. 2Department of Mathematics, Faculty ofScience, King Mongkut’s University of Technology Thonburi (KMUTT), Bangkok, 10140, Thailand.

Karapinar et al. Fixed Point Theory and Applications 2012, 2012:194 Page 18 of 19http://www.fixedpointtheoryandapplications.com/content/2012/1/194

AcknowledgementsThis work was supported by the Higher Education Research Promotion and National Research University Project ofThailand, Office of the Higher Education Commission. The third author would like to thank the Research ProfessionalDevelopment Project under the Science Achievement Scholarship of Thailand (SAST).

Received: 19 July 2012 Accepted: 15 October 2012 Published: 29 October 2012

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doi:10.1186/1687-1812-2012-194Cite this article as: Karapinar et al.: Coupled fixed point theorems in cone metric spaces with a c-distance andapplications. Fixed Point Theory and Applications 2012 2012:194.