# Coupled Fixed Point Theorems under Weak Contractions

Authors: Y. J. Cho; Z. Kadelburg; R. Saadati; W. Shatanawi
Journal: Discrete Dynamics in Nature and Society
 (2012)

---

## Abstract

Cho et al. [Comput. Math. Appl. 61(2011), 1254–1260] studied common fixed point
theorems on cone metric spaces by using the concept ofc-distance. In this paper, we prove
some coupled fixed point theorems in ordered cone metric spaces by using the concept of
c-distance in cone metric spaces.


---

## Body

## 1. Introduction

Many fixed point theorems have been proved for mappings on cone metric spaces in the sense of Huang and Zhang [1]. For some more results on fixed point theory and applications in cone metric spaces, we refer the readers to [2–15]. Recently, Bhaskar and Lakshmikantham [16] introduced the concept of a coupled coincidence point of a mapping F from X×X into X and a mapping g from X into X and studied fixed point theorems in partially ordered metric spaces. For some more results on couple fixed point theorems, refer to [17–23].Recently, Cho et al. [7] introduced a new concept of c-distance in cone metric spaces, which is a cone version of w-distance of Kada et al. [24] (see also [25]) and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the c-distance.In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept ofc-distance.


## 2. Preliminaries

In this paper, assume thatE is a real Banach space. Let P be 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 stand for y-x∈intP.It can be easily shown thatλint(P)⊆int(P) for all positive scalars λ.Definition 2.1 (see [1]).
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 2.2 (see [1]).
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). We denote 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 2.3 (see [7]).
Let(X,⊑) be a partially ordered set, and let F:X×X→X be a function. Then the mapping F is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y; that is,
(2.1)x1⊑x2impliesF(x1,y)⊑F(x2,y)
for all y∈X and
(2.2)y1⊑y2impliesF(x,y2)⊑F(x,y1)
for all x∈X.Definition 2.4 (see [7]).
An element(x,y)∈X×X is called a coupled fixed point of a mapping F:X×X→X if F(x,y)=x and F(y,x)=y.Recently, Cho et al. [7] introduced the concept of c-distance on cone metric space (X,d) which is a generalization of w-distance of Kada et al. [24].Definition 2.5 (see [7]).
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≥1, 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 0≤e such that q(z,x)≪e and q(z,y)≪c imply d(x,y)≪c.Cho et al. [7] noticed the following important remark in the concept of c-distance on cone metric spaces.Remark 2.6 (see [7]).
Letq be a c-distance on a cone metric space (X,d). Then(1)
q(x,y)=q(y,x) does not necessarily hold for all x,y∈X,(2)
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 2.7 (see [7]).
Let(X,d) be a cone metric space, and let 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 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. Main Results

In this section, we prove some coupled fixed point theorems by usingc-distance in partially ordered cone metric spaces.Theorem 3.1.
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 let F:X×X→X be a continuous function having the mixed monotone property such that
(3.1)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 (x⊑x*)∧(y⊒y*) or (x⊒x*)∧(y⊑y*). If there exist x0,y0∈X such that x0⊑F(x0,y0) and 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.Proof.
Letx0,y0∈X be such that x0⊑F(x0,y0) and F(y0,x0)⊑y0. Let x1=F(x0,y0) and y1=F(y0,x0). Since F has the mixed monotone property, we have x0⊑x1 and y1⊑y0. Continuing this process, we can construct two sequences (xn) and (yn) in X such that
(3.2)xn=F(xn-1,yn-1)⊑xn+1=F(xn,yn),yn+1=F(yn,xn)⊑yn=F(yn-1,xn-1).
Let n∈N. Now, by (3.1), we have
(3.3)q(xn,xn+1)=q(F(xn-1,yn-1),F(xn,yn))⪯k2(q(xn-1,xn)+q(yn-1,yn)),q(xn+1,xn)=q(F(xn,yn),F(xn-1,yn-1))⪯k2(q(xn,xn-1)+q(yn,yn-1)).
From (3.3), it follows that
(3.4)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)).
Similarly, we have(3.5)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)).
Thus it follows from (3.4) and (3.5) that
(3.6)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)).
Repeating (3.6) n-times, we get
(3.7)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)).
Thus we have
(3.8)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
(3.9)q(xn,xm)⪯∑i=nm-1q(xi,xi+1),q(yn,ym)⪯∑i=nm-1q(yi,yi+1),
and k<1, we have
(3.10)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)).
From Lemma 2.7 (3), it follows that (xn) and (yn) are Cauchy sequences in (X,d). Since X is complete, there exist u,v∈X such that xn→u and yn→v. Since F is continuous, we have
(3.11)xn+1=F(xn,yn)⟶F(u,v),yn+1=F(yn,xn)⟶F(v,u).
By the uniqueness of the limits, we get u=f(u,v) and v=F(v,u). Thus (u,v) is a coupled fixed point of F.
Moreover, by (3.1), we have
(3.12)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
(3.13)q(u,u)+q(v,v)⪯k(q(v,v)+q(u,u)).
Since 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 3.2.
In addition to the hypotheses of Theorem3.1, 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.Proof.
As in the proof of Theorem3.1, there exists a coupled fixed point (u,v)∈X×X. Here u=F(u,v) and v=F(v,u). By the additional assumption and (3.1), we have
(3.14)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)).
Thus we have
(3.15)q(u,v)+q(v,u)⪯k(q(v,u)+q(u,v)).
Since k<1, we get q(u,v)+q(v,u)=θ. Hence q(u,v)=θ and q(v,u)=θ. Let un=θ and xn=u. Then
(3.16)q(xn,u)⪯un,q(xn,v)⪯un.
From Lemma 2.7 (1), we have u=v. Hence the coupled fixed point of F has the form (u,u). This completes the proof.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 let F:X×X→X be a function having the mixed monotone property such that
(3.17)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⊑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≥1;(b)
if(xn) is a nonincreasing sequence in X with xn→x, then x⊑xn for all n≥1.Assume there exist x0,y0∈X such that x0⊑F(x0,y0) and F(y0,x0)⊑y0. If y0⊑x0, then F has a coupled fixed point.Proof.
As in the proof of Theorem3.1, we can construct two Cauchy sequences (xn) and (yn) in X such that
(3.18)x0⊑x1⊑⋯⊑xn⊑⋯,y0⊒y1⊒⋯yn⊒⋯.
Moreover, we have that(xn) converges to a point u∈X and (yn) converges to v∈X,
(3.19)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 n>m≥1. By (q3), we have
(3.20)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
(3.21)q(xn,u)+q(yn,v)⪯2kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
By the properties (a) and (b), we have
(3.22)v⊑yn⊑y0⊑x0⊑xn⊑u.
By (3.17), we have
(3.23)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
(3.24)q(xn,F(u,v))+q(yn,F(v,u))⪯k2(q(xn-1,u)+q(yn-1,v)).
By (3.21), we get
(3.25)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
(3.26)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 (3.20) and (3.26), Lemma 2.7 (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.Example 3.4.
LetE=CR1[0,1] with ∥x∥=∥x∥∞+∥x′∥∞ and P={x∈E:x(t)≥0,t∈[0,1]}. Let X=[0,+∞) (with usual order), and let d:X×X→E be defined by d(x,y)(t)=|x-y|et. Then (X,d) is an ordered cone metric space (see [7, Example 2.9]). Further, let q:X×X→E be defined by 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
(3.27)F(x,y)={18(x-y),x≥y,0,x<y.
Then it is easy to see that
(3.28)q(F(x,y),F(u,v))⪯16(q(x,u)+q(y,v))
for all x,y,u,v∈X with (x≤u)∧(y≥v) or (x≥u)∧(y≤v). Note that 0≤F(0,1) and 1≥F(1,0). Thus, by Theorem 3.1, it follows that F has a coupled fixed point in E. Here (0,0) is a coupled fixed point of F.



---

*Source: 184534-2012-03-14.xml*