# Coupled Fixed Point Theorems for Weak Contraction Mappings underF-Invariant Set

Authors: Wutiphol Sintunavarat; Yeol Je Cho; Poom Kumam
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We extend the recent results of the coupled fixed point theorems of Cho et al. (2012) by weakening the concept of the mixed monotone property. We also give some examples of a nonlinear contraction mapping, which is not applied to the existence of the coupled fixed point by the results of Cho et al. but can be applied to our results. The main results extend and unify the results of Cho et al. and many results of the coupled fixed point theorems.


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## Body

## 1. Introduction

Since Banach's fixed point theorem in 1922, because of its simplicity and usefullnes, it has become a very popular tool in solving the existence problems in many branches of nonlinear analysis. For some more results of the generalization of this principle, refer to [1–9] and references mentioned therein. Ran and Reurings [10] extended the Banach contraction principle to metric spaces endowed with a partial ordering, and they gave applications of their results to matrix equations. Afterward, Nieto and Rodriguez-López [11] extended Ran and Reurings’s theorems in [10] for nondecreasing mappings and obtained a unique solution for a first order-ordinary differential equation with periodic boundary conditions.In 2006, Gnana Bhaskar and Lakshmikantham [12] introduced the concept of the mixed monotone property and a coupled fixed point. They also established some coupled fixed point theorems for mappings that satisfy the mixed monotone property and gave some applications in the existence and uniqueness of a solution for a periodic boundary value problem. Because of their important role in the study of nonlinear differential equations, nonlinear integral equations and differential inclusions, a wide discussion on coupled fixed point theorems has been studied by many authors. A number of articles in this topic have been dedicated to the improvement and generalization; see [13–22] and references therein.on the other hand, Huang and Zhang [23] introduced the notion of cone metric spaces and established the fixed point theorems for mappings on this space. In 2011, Cho et al. [24] introduced a new concept of the c-distance in cone metric spaces, which is a cone version of the w-distance of Kada et al. [25] and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the c-distance. Afterward, Sintunavarat et al. [26] studied and established the fixed point theorems for the generalized contraction mappings by using this concept. For some more results on the fixed point theory and applications in cone metric spaces by using the cone metric, w-cone-distance, and c-distance, we refer the readers to [27–35].Recently, Cho et al. [36] established new coupled fixed point theorems under weak contraction mappings by using the concept of mixed monotone property and c-distance in partially cone metric spaces as follows.Theorem 1.1 (see [36]).
Let(X,⊑) be a partially ordered set, and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X and F:X×X→X a continuous function having the mixed monotone property such that
(1.1)q(F(x,y),F(x*,y*))⪯k2(q(x,x*)+q(y,y*))
for some k∈[0,1) and all x,y,x*,y*∈X with
(1.2)(x⊑x*)∧(y⊒y*)or(x⊒x*)∧(y⊑y*).
If there exist x0,y0∈X such that
(1.3)x0⊑F(x0,y0),F(y0,x0)⊑y0,
then F has a coupled fixed point (u,v). Moreover, one has q(v,v)=θ and q(u,u)=θ.Theorem 1.2 (see [36]).
In addition to the hypotheses of Theorem1.1, suppose that any two elements x and y in X are comparable. Then, the coupled fixed point has the form (u,u), where u∈X.Theorem 1.3 (see [36]).
Let(X,⊑) be a partially ordered set, and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X and F:X×X→X a function having the mixed monotone property such that
(1.4)q(F(x,y),F(x*,y*))⪯k4(q(x,x*)+q(y,y*))
for some k∈(0,1) and all x,y,x*,y*∈X with
(1.5)(x⊑x*)∧(y⊒y*)or(x⊒x*)∧(y⊑y*).
Also, suppose that X has the following properties:(a)
if(xn) is a nondecreasing sequence in X with xn→x, then xn⊑x for all n∈ℕ;(b)
if(xn) is a nonincreasing sequence in X with xn→x, then x⊑xn for all n∈ℕ.Assume there exist x0,y0∈X such that
(1.6)x0⊑F(x0,y0),F(y0,x0)⊑y0.
If y0⊑x0, then F has a coupled fixed point.In this paper, we weaken the condition of the mixed monotone property in results of Cho et al. [36] by using the concept of the F-invariant set due to Samet and Vetro [37] in the cone version. We also give the example of a nonlinear contraction mapping, which is not applied by the main results of Cho et al. but can be applied to our results. The presented results extend and complement some recent results of Cho et al. [36] and some classical coupled fixed point theorems and several results in the literature.


## 2. Preliminaries

Throughout this paper(X,⊑) denotes a partially ordered set with the partial order ⊑. By x⊏y, we mean x⊑y but x≠y. A mapping f:X→X is said to be nondecreasing (non-increasing) if, for all x,y∈X, x⊑y implies f(x)⊑f(y)(f(y)⊑f(x), resp.).Definition 2.1 (see [12]).
Let(X,⊑) be a partial ordered set. A mapping F:X×X→X is said to have a mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x,y∈X(2.1)x1,x2∈X,x1⊑x2⟹F(x1,y)⊑F(x2,y),y1,y2∈X,y1⊑y2⟹F(x,y1)⊒F(x,y2).Definition 2.2 (see [12]).
LetX be a nonempty set. An element (x,y)∈X×X is called a coupled fixed point of mapping F:X×X→X if
(2.2)x=F(x,y),y=F(y,x).
Next, we give some terminology of cone metric spaces and the concept ofc-distance in cone metric spaces due to Cho et al. [24], which is a generalization of the w-distance of Kada et al. [25].
Let(E,∥·∥) be a real Banach space, θ a zero element in E, and P a subset of E with int(P)≠∅. Then, P is called a cone if the following conditions are satisfied:(1)
P is closed and P≠{θ};(2)
a,b∈ℝ+, x,y∈P implies ax+by∈P;(3)
x∈P∩-P implies x=θ.
For a coneP, define the partial ordering⪯ with respect to P by x⪯y if and only if y-x∈P. We write x≺y to indicate that x⪯y but x≠y, while x≪y stands for y-x∈int(P).
It can be easily shown thatλint(P)⊆int(P) for all positive scalars λ.
The coneP is called normal if there is a number K>0 such that, for all x,y∈E,
(2.3)0≤x≤y⟹‖x‖≤K‖y‖.
The least positive number satisfying the above is called the normal constant of P.Definition 2.3 (see [23]).
LetX be a nonempty set. Suppose that the mapping d:X×X→E satisfies the following conditions:(1)
θ⪯d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y;(2)
d(x,y)=d(y,x) for all x,y∈X;(3)
d(x,y)⪯d(x,z)+d(y,z) for all x,y,z∈X. Then, d is called a cone metric on X and (X,d) is called a cone metric space.Definition 2.4 (see [23]).
Let(X,d) be a cone metric space. Let {xn} be a sequence in X and x∈X.(1)
If, for anyc∈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}. One denotes this by limn→∞xn=x or xn→x as n→∞.(2)
If, for anyc∈E with θ≪c, there exists N∈ℕ such that d(xn,xm)≪c for all n,m≥N, then {xn} is called a Cauchy sequence in X.(3)
The space(X,d) is called a complete cone metric space if every Cauchy sequence is convergent.Definition 2.5 (see [24]).
Let(X,d) be a cone metric space. Then a function q:X×X→E is called a c-distance on X if the following are satisfied:(q1)
θ⪯q(x,y) for all x,y∈X;(q2)
q(x,z)⪯q(x,y)+q(y,z) for all x,y,z∈X;(q3)
for anyx∈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;(q4)
for anyc∈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 2.6.
Thec-distance q is a w-distance on X if we take (X,d) is a metric space, E=ℝ+, P=[0,∞), and (q3) is replaced by the following condition:
for anyx∈X, q(x,·):X→ℝ+ is lower semicontinuous.
Moreover, (q3) holds wheneverq(x,·) is lower semi-continuous. Thus, if (X,d) is a metric space, E=ℝ+, and P=[0,∞), then every w-distance is a c-distance. But the converse is not true in the general case. Therefore, the c-distance is a generalization of the w-distance.Example 2.7.
Let(X,d) be a cone metric space and P a normal cone. Define a mapping q:X×X→E by q(x,y)=d(x,y) for all x,y∈X. Then, q is c-distance.Example 2.8.
LetE=Cℝ1[0,1] with ∥x∥=∥x∥∞+∥x′∥∞ and
(2.4)P={x∈E:x(t)≥0on⁡[0,1]}
(this cone is not normal). Let X=[0,∞), and define a mapping d:X×X→E by
(2.5)d(x,y)=|x-y|φ
for all x,y∈X, where φ:[0,1]→ℝ such that φ(t)=et. Then, (X,d) is a cone metric space. Define a mapping q:X×X→E by
(2.6)q(x,y)=(x+y)φ
for all x,y∈X. Then, q is a c-distance.Example 2.9.
Let(X,d) be a cone metric space and P a normal cone. Define a mapping q:X×X→E by
(2.7)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.Example 2.10.
LetE=ℝ and P={x∈E:x≥0}. Let X=[0,∞), and define a mapping d:X×X→E by
(2.8)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
(2.9)q(x,y)=y
for all x,y∈X. Then, q is a c-distance.Remark 2.11.
From Examples2.9 and 2.10, we have two important results. For the c-distance, q(x,y)=q(y,x) does not necessarily hold and q(x,y)=θ is not necessarily equivalent to x=y for all x,y∈X.The following lemma is crucial in proving our results.Lemma 2.12 (see [24]).
Let(X,d) be a cone metric space and q 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 converging to θ. Then, the following hold.(1)
Ifq(xn,y)⪯un and q(xn,z)⪯un, then y=z.(2)
Ifq(xn,yn)⪯un and q(xn,z)⪯un, then {yn} converges to a point z∈X.(3)
Ifq(xn,xm)⪯un for each m>n, then {xn} is a Cauchy sequence in X.(4)
Ifq(y,xn)⪯un, then {xn} is a Cauchy sequence in X.


## 3. Coupled Fixed Point underF-Invariant Set

In this section, we prove some coupled fixed point theorems by using thec-distance under the concept of the F-invariant in cone metric spaces and apply our results in partially ordered cone metric spaces. First of all, we give the concept of the F-invariant set in the cone version.Definition 3.1.
Let(X,d) be a cone metric space and F:X×X→X a given mapping. Let M be a nonempty subset of X4. One says that M is the F-invariant subset of X4 if and only if, for all x,y,z,w∈X, one has(a)
(x,y,z,w)∈M⇔(w,z,y,x)∈M;(b)
(x,y,z,w)∈M⇒(F(x,y),F(y,x),F(z,w),F(w,z))∈M.
We obtain that the setM=X4 is trivially F-invariant.
The next example plays a key role in the proof of our main results in partially ordered set.Example 3.2.
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 all x,y∈X, we have
(3.1)x1,x2∈X,x1⊑x2⟹F(x1,y)⊑F(x2,y),y1,y2∈X,y1⊑y2⟹F(x,y1)⊒F(x,y2).
Define the subset M⊆X4 by
(3.2)M={(a,b,c,d):c⊑a,b⊑d}.
Then, M is the F-invariant of X4.
Next, we prove the main results of this work.Theorem 3.3.
Let(X,d) be a complete cone metric space. Let q be a c-distance on X, M a nonempty subset of X4, and F:X×X→X a continuous function such that
(3.3)q(F(x,y),F(x*,y*))⪯k2(q(x,x*)+q(y,y*))
for some k∈[0,1) and all x,y,x*,y*∈X with
(3.4)(x,y,x*,y*)∈Mor(x*,y*,x,y)∈M.
If M is an F-invariant and there exist x0,y0∈X such that
(3.5)(F(x0,y0),F(y0,x0),x0,y0)∈M,
then F has a coupled fixed point (u,v). Moreover, if (u,v,u,v)∈M, then q(v,v)=θ and q(u,u)=θ.Proof.
AsF(X×X)⊆X, we can construct sequences {xn} and {yn} in X such that
(3.6)xn=F(xn-1,yn-1),yn=F(yn-1,xn-1)∀n∈N.
Since (F(x0,y0),F(y0,x0),x0,y0)=(x1,y1,x0,y0)∈M and M is an F-invariant set, we get
(3.7)(F(x1,y1),F(y1,x1),F(x0,y0),F(y0,x0))=(x2,y2,x1,y1)∈M.
Again, using the fact that M is an F-invariant set, we have
(3.8)(F(x2,y2),F(y2,x2),F(x1,y1),F(y1,x1))=(x3,y3,x2,y2)∈M.
By repeating the argument to the above, we get
(3.9)(F(xn-1,yn-1),F(yn-1,xn-1),xn-1,yn-1)=(xn,yn,xn-1,yn-1)∈M
for all n∈ℕ. From (3.3), we have
(3.10)q(xn,xn+1)=q(F(xn-1,yn-1),F(xn,yn))⪯k2(q(xn-1,xn)+q(yn-1,yn)),(3.11)q(xn+1,xn)=q(F(xn,yn),F(xn-1,yn-1))⪯k2(q(xn,xn-1)+q(yn,yn-1)).
Combining (3.10) and (3.11), we get
(3.12)q(xn,xn+1)+q(xn+1,xn)⪯k2(q(xn-1,xn)+q(yn-1,yn)+q(xn,xn-1)+q(yn,yn-1)).
Since (xn,yn,xn-1,yn-1)∈M for all n∈ℕ and M is an F-invariant set, we get
(3.13)(yn-1,xn-1,yn,xn)∈M
for all n∈ℕ. From (3.3), we have
(3.14)q(yn,yn+1)=q(F(yn-1,xn-1),F(yn,xn))⪯k2(q(yn-1,yn)+q(xn-1,xn)),q(yn+1,yn)=q(F(yn,xn),F(yn-1,xn-1))⪯k2(q(yn,yn-1)+q(xn,xn-1)).
From (3.14), we have
(3.15)q(yn,yn+1)+q(yn+1,yn)⪯k2(q(xn-1,xn)+q(yn-1,yn)+q(xn,xn-1)+q(yn,yn-1)).
Adding (3.12) and (3.15), we get
(3.16)q(xn,xn+1)+q(xn+1,xn)+q(yn,yn+1)+q(yn+1,yn)⪯k(q(xn-1,xn)+q(yn-1,yn)+q(xn,xn-1)+q(yn,yn-1)).
We repeat the above process for n-times, and we get
(3.17)q(xn,xn+1)+q(xn+1,xn)+q(yn,yn+1)+d(yn+1,yn)⪯kn(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
From (3.17), we can conclude that
(3.18)q(xn,xn+1)⪯kn(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,yn+1)⪯kn(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
Let m,n∈ℕ with m>n. Since
(3.19)q(xn,xm)⪯∑i=nm-1q(xi,xi+1),q(yn,ym)⪯∑i=nm-1q(yi,yi+1),
and 0≤k<1, we have
(3.20)q(xn,xm)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,ym)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
Using Lemma 2.12(3), we have {xn} and {yn} are Cauchy sequences in (X,d). By the completeness of X, we get xn→u and yn→v for some u,v∈X.
SinceF is continuous, taking n→∞ in (3.6), we get
(3.21)lim⁡n→∞xn+1=lim⁡n→∞F(xn,yn)=F(lim⁡n→∞xn,lim⁡n→∞yn)=F(u,v),lim⁡n→∞yn+1=lim⁡n→∞F(yn,xn)=F(lim⁡n→∞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 a coupled fixed point of F.
Finally, we assume that(u,v,u,v)∈M and so (v,u,v,u)∈M. By (3.3), we have
(3.22)q(u,u)=q(F(u,v),F(u,v))⪯k2(q(u,u)+q(v,v)),q(v,v)=q(F(v,u),F(v,u))⪯k2(q(v,v)+q(u,u)).
Therefore, we get
(3.23)q(u,u)+q(v,v)⪯k(q(v,v)+q(u,u)).
Since 0≤k<1, 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.Theorem 3.4.
In addition to the hypotheses of Theorem3.3, suppose that any two elements x and y in X, we have
(3.24)(y,x,x,y)∈Mor(x,y,y,x)∈M.
Then, the coupled fixed point has the form (u,u), where u∈X.Proof.
As in the proof of Theorem3.3, there exists a coupled fixed point (u,v)∈X×X. Hence,
(3.25)u=F(u,v),v=F(v,u).
From the additional hypothesis and (3.3), we get
(3.26)q(u,v)=q(F(u,v),F(v,u))⪯k2(q(u,v)+q(v,u)),q(v,u)=q(F(v,u),F(u,v))⪯k2(q(v,u)+q(u,v)).
Therefore, we have
(3.27)q(u,v)+q(v,u)⪯k(q(v,u)+q(u,v)).
Since 0≤k<1, we get q(u,v)+q(v,u)=θ. Therefore, q(u,v)=θ and q(v,u)=θ.
Letun=θ and xn=u. Then,
(3.28)q(xn,u)⪯un,q(xn,v)⪯un.
From Lemma 2.12(1), we have u=v. Therefore, the coupled fixed point of F has the form (u,u). This completes the proof.Next, we apply Theorems3.3 and 3.4 in partially ordered cone metric spaces. If we set M as Example 3.2, then we get to Theorems 3.1 and 3.2 of Cho et al. [36].Corollary 3.5 (see [36, Theorem 3.1]).
Let(X,⊑) be a partially ordered set, and suppose that (X,d) is a complete cone metric space. Let q a c-distance on X and F:X×X→X be a continuous function having the mixed monotone property such that
(3.29)q(F(x,y),F(x*,y*))⪯k2(q(x,x*)+q(y,y*))
for some k∈[0,1) and all x,y,x*,y*∈X with
(3.30)(x⊑x*)∧(y⊒y*)or(x⊒x*)∧(y⊑y*).
If there exist x0,y0∈X such that
(3.31)x0⊑F(x0,y0),F(y0,x0)⊑y0,
then F has a coupled fixed point (u,v). Moreover, one has q(v,v)=θ and q(u,u)=θ.Corollary 3.6 (see [36, Theorem 3.2]).
In addition to the hypotheses of Corollary3.5, suppose that any two elements x and y in X are comparable. Then the coupled fixed point has the form (u,u), where u∈X.Theorem 3.7.
Let(X,d) be a complete cone metric space. Let q be a c-distance on X, M a subset of X4, and F:X×X→X a function such that
(3.32)q(F(x,y),F(x*,y*))⪯k4(q(x,x*)+q(y,y*))
for some k∈[0,1) and all x,y,x*,y*∈X with
(3.33)(x,y,x*,y*)∈Mor(x*,y*,x,y)∈M.
Also, suppose that(i)
there existx0,y0∈X such that (F(x0,y0),F(y0,x0),x0,y0)∈M,(ii)
if two sequences{xn}, {yn} are such that  (xn+1,yn+1,xn,yn)∈M for all n∈ℕ and {xn}→x,{yn}→y, then (x,y,xn,yn)∈M for all n∈ℕ.If M is an F-invariant set, then F has a coupled fixed point.Proof.
As in the proof of Theorem3.3, we can construct two Cauchy sequences {xn} and {yn} in X such that
(3.34)(xn,yn,xn-1,yn-1)∈M
for all n∈ℕ. Moreover, we have that {xn} converges to a point u∈X and {yn} converges to v∈X:
(3.35)q(xn,xm)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,ym)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1))
for each m>n≥1. By (q3), we have
(3.36)q(xn,u)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1))q(yn,v)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1))
and so
(3.37)q(xn,u)+q(yn,v)⪯2kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
By assumption (ii), we have (u,v,xn-1,yn-1)∈M and (yn-1,xn-1,v,u)∈M. From (3.32), we have
(3.38)q(xn,F(u,v))=q(F(xn-1,yn-1),F(u,v))⪯k4(q(xn-1,u)+q(yn-1,v)),q(yn,F(v,u))=q(F(yn-1,xn-1),F(v,u))⪯k4(q(yn-1,v)+q(xn-1,u)).
Thus, we have
(3.39)q(xn,F(u,v))+q(yn,F(v,u))⪯k2(q(xn-1,u)+q(yn-1,v)).
By (3.37), we get
(3.40)q(xn,F(u,v))+q(yn,F(v,u))⪯k2⋅2kn-11-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1))=kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
Therefore, we have
(3.41)q(xn,F(u,v))⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,F(v,u))⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
By using (3.36) and (3.41), Lemma 2.12(1) shows that u=F(u,v) and v=F(v,u). Therefore, (u,v) is a coupled fixed point of F. This completes the proof.Corollary 3.8 (see [36, Theorem 3.3]).
Let(X,⊑) be a partially ordered set, and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X and F:X×X→X a function having the mixed monotone property such that
(3.42)q(F(x,y),F(x*,y*))⪯k4(q(x,x*)+q(y,y*))
for some k∈(0,1) and all x,y,x*,y*∈X with
(3.43)(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. Assume there exist x0,y0∈X such that
(3.44)x0⊑F(x0,y0),F(y0,x0)⊑y0.
If y0⊑x0, then F has a coupled fixed point.Proof.
LetM={(a,b,c,d):c⊑a,b⊑d}. We obtain that M is an F-invariant set. By (3.42), we have
(3.45)q(F(x,y),F(x*,y*))⪯k4(q(x,x*)+q(y,y*))
for some k∈[0,1) and all x,y,x*,y*∈X with (x,y,x*,y*)∈M or (x*,y*,x,y)∈M. From assumptions (a) and (b), we know that, for two sequences {xn}, {yn} such that {xn} is a non-decreasing sequence in X with xn→x and {yn} is a non-increasing sequence in X with yn→y,
(3.46)x0⊑x1⊑⋯⊑xn⊑⋯⊑x,y0⊒y1⊒⋯yn⊒⋯⊒y
for all n∈ℕ. Therefore, we have (x,y,xn,yn)∈M for all n∈ℕ. Since y0⊑x0, we have
(3.47)y⊑yn⊑y0⊑x0⊑xn⊑x
for all n∈ℕ. Therefore, we can conclude that (yn,xn,y,x)∈M for all n∈ℕ. Now, all the hypotheses of Theorem 3.7 hold. Thus, F has a coupled fixed point.Example 3.9.
LetE=Cℝ1[0,1] with ∥x∥=∥x∥∞+∥x′∥∞ and
(3.48)P={x∈E:x(t)≥0,t∈[0,1]}.
Let X=[0,∞) (with usual order), and let d:X×X→E be defined by
(3.49)d(x,y)(t)=|x-y|et.
Then, (X,d) is an ordered cone metric space (see [24, Example 2.9]). Further, let q:X×X→E be defined by
(3.50)q(x,y)(t)=yet.
It is easy to check that q is a c-distance. Consider now the function F:X×X→X defined by
(3.51)F(x,y)={17(x+y),x≥y,0,x<y.
For y1=2 and y2=3, we have y1⊑y2 but F(x,y1)⊑F(x,y2) for all x>3. So the mapping F does not satisfy the mixed monotone property. Hence, the main results of Cho et al. [36] cannot be applied to this example. But it is easy to see that
(3.52)q(F(x,y),F(u,v))⪯15(q(x,u)+q(y,v))
for all x,y,u,v∈X4. Note that there exists 0,1∈X such that
(3.53)(F(0,1),F(1,0),0,1)∈X4.
Now, we can apply Theorem 3.3 with M=X4. Therefore, F has a unique coupled fixed point, that is, a point (0,0).Remark 3.10.
Although the main results of Cho et al. [36] are an essential tool in the partially ordered cone metric spaces to claim the existence of the coupled fixed points the mappings do not have the mixed monotone property in general case such as the mapping in the above example. Therefore, it is very interesting to consider our theorems as another auxiliary tool to claim the existence of a coupled fixed point.



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*Source: 324874-2012-05-09.xml*