# Common Coupled Fixed Point Theorems of Single-Valued Mapping forc-Distance in Cone Metric Spaces

Authors: Zaid Mohammed Fadail; Abd Ghafur Bin Ahmad
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The existence and uniqueness of the common coupled fixed point in cone metric spaces have been studied by considering different types of contractive conditions. A new concept of thec-distance in cone metric space has been recently introduced in 2011. Then, coupled fixed point results for contraction-type mappings in ordered cone metric spaces and cone metric spaces have been considered. In this paper, some common coupled fixed point results on c-distance in cone metric spaces are obtained. Some supporting examples are given.


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## Body

## 1. Introduction

In 2007, Huang and Zhang [1] introduced the concept of cone metric space where each pair of points is assigned to a member of a real Banach space with a cone. Subsequently, several authors have studied the existence and uniqueness of the fixed point and common fixed point for self-map f by considering different types of contractive conditions. Some of these works are noted in [2–12].In [13], Bhaskar and Lakshmikantham introduced the concept of coupled fixed point for a given partially ordered set X. Lakshmikantham and Ćirić [14] proved some more coupled fixed point theorems in partially ordered set.In [15], Sabetghadam et al. considered the corresponding definition of coupled fixed point for the mapping in complete cone metric space and proved some coupled fixed point theorems. Subsequently, several authors have studied the existence and uniqueness of the coupled fixed point and common coupled fixed point by considering different types of contractive conditions. Some of these works are noted in [16–23].Recently, Cho et al. [23] introduced a new concept of the c-distance in cone metric spaces (also see [24]) and proved some fixed point theorems in ordered cone metric spaces. This is more general than the classical Banach contraction mapping principle. Sintunavarat et al. [25] extended and developed the Banach contraction theorem on c-distance of Cho et al. [23]. Wang and Guo [24] proved some common fixed point theorems for this new distance. Subsequently, several authors have studied on the generalized distance in cone metric space. Some of this works are noted in [26–31].In [30], Fadail and Ahmad proved some coupled fixed point theorems in cone metric spaces by using the concept of c-distance.Recall that an element(x,y)∈X×X is said to be a coupled fixed point of the mapping F:X×X→X if F(x,y)=x and F(y,x)=y.Definition 1.1.
An element(x,y)∈X×X is called (1)
a coupled coincidence point of mappingsF:X×X→X and g:X→X if gx=F(x,y) and gy=F(y,x), and (gx,gy) is called coupled point of coincidence.(2)
a common coupled fixed point of mappingsF:X×X→X and g:X→X if x=gx=F(x,y),y=gy=F(y,x).Abbas et al. [20] introduced the following definition.Definition 1.2.
The mappingsF:X×X→X and g:X→X are called w-compatible if g(F(x,y))=F(gx,gy), whenever gx=F(x,y) and gy=F(y,x).The aim of this paper is to continue the study of common coupled fixed points of mappings but now forc-distance in cone metric space. Our theorems extend and develop some theorems in literature to c-distance in cone metric spaces. In this paper, we do not impose the normality condition for the cones, the only assumption is that cone P has nonempty interior.


## 2. Preliminaries

LetE be a real Banach space and θ denote the zero element in E. A cone P is a subset of E such that (1)
P is nonempty set closed and P≠{θ},(2)
ifa,b are nonnegative real numbers and x,y∈P, then ax+by∈P,(3)
x∈P and -x∈P implies x=θ.For any coneP⊂E, the partial ordering ⪯ with respect to P is defined by x⪯y if and only if y-x∈P. The notation of ≺ stand for x⪯y but x≠y. Also, we used x≪y to indicate that y-x∈intP, where intP denotes the interior of P. A cone P is called normal if there exists a number K such that
(2.1)θ⪯x⪯y⇒∥x∥≤K∥y∥,
for all x,y∈E. The least positive number K satisfying the above condition is called the normal constant of P.Definition 2.1 (see [1]).
LetX be a nonempty set and E a real Banach space equipped with the partial ordering ⪯ with respect to the cone P. Suppose that the mapping d:X×X→E satisfies the following condition: (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,y)+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 and {xn} a sequence in X and x∈X. (1)
For allc∈E with θ≪c, if there exists a positive integer N such that d(xn,x)≪c for all n>N, then xn is said to be convergent and x is the limit of {xn}. We denote this by xn→x.(2)
For allc∈E with θ≪c, if there exists a positive integer N such that d(xn,xm)≪c for all n,m>N, then {xn} is called a Cauchy sequence in X.(3)
A cone metric space(X,d) is called complete if every Cauchy sequence in X is convergent.Lemma 2.3 (see [8]).
(1)
IfE be a real Banach space with a cone P and a⪯λa, where a∈P and 0≤λ<1, then a=θ.
(2)
Ifc∈intP, θ⪯an and an→θ, then there exists a positive integer N such that an≪c for all n≥N.Next, we give the notation ofc-distance on a cone metric space which is a generalization of ω-distance of Kada et al. [32] with some properties.Definition 2.4 (see [23]).
Let(X,d) be a cone metric space. A function q:X×X→E is called a c-distance on X if the following conditions hold: (q1)
θ⪯q(x,y) for all x,y∈X,(q2)
q(x,y)⪯q(x,y)+q(y,z) for all x,y,z∈X,(q3)
for eachx∈X and n≥1, if q(x,yn)⪯u for some u=ux∈P, then q(x,y)⪯u whenever {yn} is a sequence in X converging to a point y∈X,(q4)
for allc∈E with θ≪c, there exists e∈E with θ≪e such that q(z,x)≪e and q(z,y)≪e imply d(x,y)≪c.Example 2.5 (see [23]).
LetE=ℝ and P={x∈E:x≥0}. Let X=[0,∞) and define a mapping d: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 on X.Lemma 2.6 (see [23]).
Let(X,d) be a cone metric space and q is a c-distance on X. Let {xn} and {yn} be sequences in X and x,y,z∈X. Suppose that un is a sequences 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 z.(3)
Ifq(xn,xm)⪯un for m>n, then {xn}is a Cauchy sequence in X.(4)
Ifq(y,xn)⪯un, then {xn} is a Cauchy sequence in X.Remark 2.7 (see [23]).
(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.


## 3. Main Results

In this section, we prove some common coupled fixed point results usingc-distance in cone metric space. Also, we generalize the contractive conditions in literature by replacing the constants with functions.Theorem 3.1.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Let F:X×X→X and g:X→X be two mappings and suppose that there exists mappings k,l:X×X→[0,1) such that the following hold: (a)
k(F(x,y),F(u,v))≤k(gx,gy) and l(F(x,y),F(u,v))≤l(gx,gy) for all x,y,u,v∈X,(b)
k(x,y)=k(y,x) and l(x,y)=l(y,x) for all x,y∈X,(c)
(k+l)(x,y)<1 for all x,y∈X,(d)
q(F(x,y),F(u,v))⪯k(gx,gy)q(gx,gu)+l(gx,gy)q(gy,gv) for all x,y,u,v∈X. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Proof.
Choosex0,y0∈X. Set gx1=F(x0,y0),gy1=F(y0,x0), this can be done because F(X×X)⊆g(X). Continuing this process, we obtain two sequences {xn} and {yn} such that gxn+1=F(xn,yn),gyn+1=F(yn,xn). Then we have
(3.1)q(gxn,gxn+1)=q(F(xn-1,yn-1),F(xn,yn))⪯k(gxn-1,gyn-1)q(gxn-1,gxn)+l(gxn-1,gyn-1)q(gyn-1,gyn)=k(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,gxn)+l(F(xn-2,yn-2),F(yn-2,xn-2))q(gyn-1,gyn)⪯k(gxn-2,gyn-2)q(gxn-1,gxn)+l(gxn-2,gyn-2)q(gyn-1,gyn)⋮⪯k(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gyn-1,gyn).
Similarly, we have
(3.2)q(gyn,gyn+1)=q(F(yn-1,xn-1),F(yn,xn))⪯k(gyn-1,gxn-1)q(gyn-1,gyn)+l(gyn-1,gxn-1)q(gxn-1,gxn)=k(F(yn-2,xn-2),F(xn-2,yn-2))q(gyn-1,gyn)+l(F(yn-2,xn-2),F(xn-2,yn-2))q(gxn-1,gxn)⪯k(gyn-2,gxn-2)q(gyn-1,gyn)+l(gyn-2,gxn-2)q(gxn-1,gxn)⋮⪯k(gy1,gx1)q(gyn-1,gyn)+l(gy1,gx1)q(gxn-1,gxn)=k(gx1,gy1)q(gyn-1,gyn)+l(gx1,gy1)q(gxn-1,gxn).
Put qn=q(gxn,gxn+1)+q(gyn,gyn+1). Then we have
(3.3)qn=q(gxn,gxn+1)+q(gyn,gyn+1)⪯k(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gyn-1,gyn)+k(gx1,gy1)q(gyn-1,gyn)+l(gx1,gy1)q(gxn-1,gxn)=[k(gx1,gy1)+l(gx1,gy1)][q(gxn-1,gxn)+q(gyn-1,gyn)]=[k(gx1,gy1)+l(gx1,gy1)]qn-1=hqn-1⋮⪯hn-1q1,
where h=k(gx1,gy1)+l(gx1,gy1)<1.
Letm>n≥1. It follows that
(3.4)q(gxn,gxm)⪯q(gxn,gxn+1)+q(gxn+1,gxn+2)+⋯+q(gxm-1,gxm),q(gyn,gym)⪯q(gyn,gyn+1)+q(gyn+1,gxn+2)+⋯+q(gym-1,gym).
Then we have
(3.5)q(gxn,gxm)+q(gyn,gym)⪯qn+qn+1+⋯+qm-1⪯hnq1+hn+1q1+⋯+hm-1q1=(hn-1+hn+⋯+hm-2)q1⪯hn-11-hq1.
Consequently,
(3.6)q(gxn,gxm)⪯hn-11-hq1,q(gyn,gym)⪯hn-11-hq1.
Thus, Lemma 2.6 (3) shows that {gxn} and {gyn} are Cauchy sequences in g(X). Since g(X) is complete, there exists x* and y* in X such that gxn→gx* and gyn→gy* as n→∞. Using q3, we have
(3.7)q(gxn,gx*)⪯hn-11-hq1,q(gyn,gy*)⪯hn-11-hq1.
On the other hand,(3.8)q(gxn,F(x*,y*))=q(F(xn-1,yn-1),F(x*,y*))⪯k(gxn-1,gyn-1)q(gxn-1,gx*)+l(gxn-1,gyn-1)q(gyn-1,gy*)=k(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,gx*)+l(F(xn-2,yn-2),F(yn-2,xn-2))q(gyn-1,gy*)⪯k(gxn-2,gyn-2)q(gxn-1,gx*)+l(gxn-2,gyn-2)q(gyn-1,gy*)⋮⪯k(gx1,gy1)q(gxn-1,gx*)+l(gx1,gy1)q(gyn-1,gy*)⪯k(gx1,gy1)hn-21-hq1+l(x1,y1)hn-21-hq1=[k(gx1,gy1)+l(gx1,gy1)]hn-21-hq1=hhn-21-hq1=hn-11-hq1.
Thus, Lemma 2.6 (1), (3.5), and (3.8) show that gx*=F(x*,y*). By similar way, we can prove that gy*=F(y*,x*). Therefore, (x*,y*) is a coupled coincidence point of F and g.
Suppose thatu=gx*=F(x*,y*) and v=gy*=F(y*,x*). Then we have
(3.9)q(u,u)=q(gx*,gx*)=q(F(x*,y*),F(x*,y*))⪯k(gx*,gy*)q(gx*,gx*)+l(gx*,gy*)q(gy*,gy*)=k(u,v)q(u,u)+l(u,v)q(v,v),(3.10)q(v,v)=q(gy*,gy*)=q(F(y*,x*),F(y*,x*))⪯k(gy*,gx*)q(gy*,gy*)+l(gy*,gx*)q(gx*,gx*)=k(v,u)q(v,v)+l(v,u)q(u,u)=k(u,v)q(v,v)+l(u,v)q(u,u).
This implies that
(3.11)q(u,u)+q(v,v)⪯k(u,v)q(u,u)+l(u,v)q(v,v)+k(u,v)q(v,v)+l(u,v)q(u,u)=[k(u,v)+l(u,v)][q(u,u)+q(v,v)]=[(k+l)(u,v)][q(u,u)+q(v,v)].
Since (k+l)(u,v)<1, Lemma 2.3 (1) shows that q(u,u)+q(v,v)=θ. But q(u,u)≽θ and q(v,v)≽θ. Consequently,  q(u,u)=θ and q(v,v)=θ.
Finally, suppose there is another coupled point of coincidence(u1,v1) of F and g such that u1=gx′=F(x′,y′) and v1=gy′=F(y′,x′) for some (x′,y′) in X×X. Then we have
(3.12)q(u,u1)=q(gx*,gx′)=q(F(x*,y*),F(x′,y′))⪯k(gx*,gy*)q(gx*,gx′)+l(gx*,gy*)q(gy*,gy′)=k(v,u)q(v,v1)+l(v,u)q(u,u1)=k(u,v)q(u,u1)+l(u,v)q(v,v1),
and also,(3.13)q(v,v1)=q(gy*,gy′)=q(F(y*,x*),F(y′,x′))⪯k(gy*,gx*)q(gy*,gy′)+l(gy*,gx*)q(gx*,gx′)=k(v,u)q(v,v1)+l(v,u)q(u,u1)=k(u,v)q(v,v1)+l(u,v)q(u,u1).
This implies that
(3.14)q(u,u1)+q(v,v1)⪯k(u,v)q(u,u1)+l(u,v)q(v,v1)+k(u,v)q(v,v1)+l(u,v)q(u,u1)=[k(u,v)+l(u,v)][q(u,u1)+q(v,v1)]=[(k+l)(u,v)][q(u,u1)+q(v,v1)].
Since (k+l)(u,v)<1, Lemma 2.3 (1) shows that q(u,u1)+q(v,v1)=θ. But q(u,u1)≽θ and q(v,v1)≽θ. Hence q(u,u1)=θ and q(v,v1)=θ. Also we have q(u,u)=θ and q(v,v)=θ. Thus, Lemma 2.6 (1) shows that u=u1 and v=v1, which implies that (u,v)=(u1,v1). Similarly, we can prove that u=v1 and v=u1. Thus, u=v. Therefore, (u,u) is the unique coupled point of coincidence. Now, let u=gx*=F(x*,y*). Since F and g are w-compatible, then we have
(3.15)gu=g(gx*)=gF(x*,y*)=F(gx*,gy*)=F(gx*,gx*)=F(u,u).
Thus (gu,gu) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that gu=u. Therefore, u=gu=F(u,u). Hence (u,u) is the unique common coupled fixed point of F and g.The following corollaries can be obtained as consequences of this theorem.Corollary 3.2.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.16)q(F(x,y),F(u,v))⪯kq(gx,gu)+lq(gy,gv),
for all x,y,u,v∈X, where k,l are nonnegative constants with k+l<1. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a coupled coincidence point in X. Further, if gx1=F(x1,y1) and gy1=F(y1,x1), then q(gx1,gx1)=θ and q(gy1,gy1)=θ. Moreover, if F and g are w -compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Corollary 3.3.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.17)q(F(x,y),F(u,v))⪯k[q(gx,gu)+q(gy,gv)],
for all x,y,u,v∈X, where k∈[0,1/2) is a constants. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w -compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Theorem 3.4.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Let F:X×X→X and g:X→X be two mappings and suppose that there exists mappings k,l:X×X→[0,1) such that the following hold: (a)
k(F(x,y),F(u,v))≤k(gx,gy) and l(F(x,y),F(u,v))≤l(gx,gy) for all x,y,u,v∈X,(b)
(k+l)(x,y)<1 for all x,y∈X,(c)
q(F(x,y),F(u,v))⪯k(gx,gy)q(gx,F(x,y))+l(gx,gy)q(gu,F(u,v)) for all x,y,u,v∈X.
IfF(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w -compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Proof.
Choosex0,y0∈X. Set gx1=F(x0,y0),gy1=F(y0,x0). This can be done because F(X×X)⊆g(X). Continuing this process, we obtain two sequences {xn} and {yn} such that gxn+1=F(xn,yn), gyn+1=F(yn,xn).
Then we have(3.18)q(gxn,gxn+1)=q(F(xn-1,yn-1),F(xn,yn))⪯k(gxn-1,gyn-1)q(gxn-1,F(xn-1,yn-1))+l(gxn-1,gyn-1)q(gxn,F(xn,yn))=k(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,gxn)+l(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn,gxn+1)⪯k(gxn-2,gyn-2)q(gxn-1,gxn)+l(gxn-2,gyn-2)q(gxn,gxn+1)⋮⪯k(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gxn,gxn+1).
Hence(3.19)q(gxn,gxn+1)⪯k(gx1,gy1)1-l(gx1,gy1)q(gxn-1,gxn)=hq(gxn-1,gxn)⪯h2q(gxn-2,gxn-1)⋮⪯hn-1q(gx1,gx2),
where h=k(gx1,gy1)/(1-l(gx1,gy1))<1. It follows that
(3.20)q(F(xn-1,yn-1),F(xn,yn))=q(gxn,gxn+1)⪯hq(gxn-1,gxn).
Similarly, we have(3.21)q(gyn,gyn+1)⪯k(gy1,gx1)1-l(gy1,gx1)q(gyn-1,gyn)=dq(gyn-1,gyn)⪯d2q(gyn-2,gyn-1)⋮⪯dn-1q(gy1,gy2),
where d=k(gy2,gx2)/(1-l(gy0,gx0))<1. It follows that
(3.22)q(F(yn-1,xn-1),F(yn,xn))=q(gyn,gyn+1)⪯dq(gyn-1,gyn).
Letm>n≥1. Then, it follows that
(3.23)q(gxn,gxm)⪯q(gxn,gxn+1)+q(gxn+1,gxn+2)+⋯+q(gxm-1,gxm)⪯(hn-1+hn+⋯+hm-2)q(gx1,gx2)⪯hn-11-hq(gx1,gx2),
and also,
(3.24)q(gyn,gym)⪯q(gyn,gyn+1)+q(gyn+1,gyn+2)+⋯+q(gym-1,gym)⪯(dn-1+dn+⋯+dm-2)q(gy1,gy2)⪯dn1-dq(gy1,gy2).
Thus, Lemma 2.6 (3) shows that {gxn} and {gyn} are Cauchy sequences in g(X). Since g(X) is complete, there exists x* and y* in X such that gxn→gx* and gyn→gy* as n→∞. Using (q3), we have
(3.25)q(gxn,gx*)⪯hn-11-hq(gx1,gx2),(3.26)q(gyn,gy*)⪯dn-11-dq(gy1,gy2).
On the other hand and by using (3.20), we have
(3.27)q(gxn,F(x*,y*))=q(F(xn-1,yn-1),F(x*,y*))⪯hq(gxn-1,gx*)⪯hhn-21-hq(gx1,gx2)=hn-11-hq(gx1,gx2),
also by using (3.22), we have
(3.28)q(gyn,F(y*,x*))=q(F(yn-1,xn-1),F(y*,x*))⪯dq(gyn-1,gy*)⪯ddn-21-dq(gy1,gy2)=dn-11-dq(gy1,gy2).
Thus, Lemma 2.6 (1), (3.25), and (3.27) show that gx*=F(x*,y*). Again, Lemma 2.6 (1), (3.26), and (3.28) show that gy*=F(y*,x*). Therefore, (x*,y*) is a coupled coincidence point of F and g.
Suppose thatu=gx*=F(x*,y*) and v=gy*=F(y*,x*). Then we have
(3.29)q(u,u)=q(gx*,gx*)=q(F(x*,y*),F(x*,y*))⪯k(gx*,gy*)q(gx*,F(x*,y*))+l(gx*,gy*)q(gx*,F(x*,y*))=k(u,v)q(u,u)+l(u,v)q(u,u)=[k(u,v)+l(u,v)]q(u,u)=[(k+l)(u,v)]q(u,u).
Since (k+l)(u,v)<1, Lemma 2.3 (1) shows that q(u,u)=θ. By similar way, we have q(v,v)=θ.
Finally, suppose there is another coupled point of coincidence(u1,v1) of F and g such that u1=gx′=F(x′,y′) and v1=gy′=F(y′,x′) for some (x′,y′) in X×X. Then we have
(3.30)q(u,u1)=q(gx*,gx′)=q(F(x*,y*),F(x′,y′))⪯k(gx*,gy*)q(gx*,F(x*,y*))+l(gx*,gy*)q(gx′,F(x′,y′))=k(u,v)q(u,u)+l(u,v)q(u1,u1)=θ,
and also,
(3.31)q(v,v1)=q(gy*,gy′)=q(F(y*,x*),F(y′,x′))⪯k(gy*,gx*)q(gy*,F(y*,x*))+l(gy*,gx*)q(gy′,F(y′,x′))=k(v,u)q(v,v)+l(v,u)q(v1,v1)=θ.
Also, we have q(u,u)=θ and q(v,v)=θ. Thus, Lemma 2.6 (1) shows that u=u1 and v=v1, which implies that (u,v)=(u1,v1). Similarly, we can prove that u=v1 and v=u1. Thus, u=v. Therefore, (u,u) is the unique coupled point of coincidence. Now, let u=gx*=F(x*,y*). Since F and g are w-compatible, then we have
(3.32)gu=g(gx*)=gF(x*,y*)=F(gx*,gy*)=F(gx*,gx*)=F(u,u).
Then, (gu,gu) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that gu=u. Therefore, u=gu=F(u,u). Hence, (u,u) is the unique common coupled fixed point of F and g.The following corollaries can be obtained as consequences of Theorem3.4.Corollary 3.5.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.33)q(F(x,y),F(u,v))⪯kq(gx,F(x,y))+lq(gu,F(u,v)),
for all x,y,u,v∈X, where k,l are nonnegative constants with k+l<1. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1), and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w -compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Corollary 3.6.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.34)q(F(x,y),F(u,v))⪯k[q(gx,F(x,y))+q(gu,F(u,v))],
for all x,y,u,v∈X, where k∈[0,1/2) is constants. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Finally, we provide another result with another contractive type.Theorem 3.7.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Let F:X×X→X and g:X→X be two mappings and suppose that there exists mappings k,l,r:X×X→[0,1) such that the following hold: (a)
k(F(x,y),F(u,v))≤k(gx,gy), l(F(x,y),F(u,v))≤l(gx,gy) and  r(F(x,y),F(u,v))≤r(gx,gy) for all x,y,u,v∈X,(b)
(k+2l+r)(x,y)<1 for all x,y∈X,(c)
(1-r(gx,gy))q(F(x,y),F(u,v))⪯k(gx,gy)q(gx,F(x,y))+l(gx,gy)q(gx,F(u,v)) for all x,y,u,v∈X.
IfF(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Proof.
Choosex0,y0∈X. Set gx1=F(x0,y0),gy1=F(y0,x0) this can be done because F(X×X)⊆g(X). Continuing this process, we obtain to sequences {xn} and {yn} such that gxn+1=F(xn,yn),gyn+1=F(yn,xn). Observe that
(3.35)(1-r(gx,gy))q(F(x,y),F(u,v))⪯k(gx,gy)q(gx,F(x,y))+l(gx,gy)q(gx,F(u,v)),
equivalently
(3.36)q(F(x,y),F(u,v))⪯k(gx,gy)q(gx,F(x,y))+l(gx,gy)q(gx,F(u,v))+r(gx,gy)q(F(x,y),F(u,v)).
Then, we have
(3.37)q(gxn,gxn+1)=q(F(xn-1,yn-1),F(xn,yn))⪯k(gxn-1,gyn-1)q(gxn-1,F(xn-1,yn-1))+l(gxn-1,gyn-1)q(gxn-1,F(xn,yn))+r(gxn-1,gyn-1)q(F(xn-1,yn-1),F(xn,yn))=k(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,gxn)+l(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,gxn+1)+r(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn,gxn+1)⪯k(gxn-2,gyn-2)q(gxn-1,gxn)+l(gxn-2,gyn-2)q(gxn-1,gxn+1)+r(gxn-2,gyn-2)q(gxn,gxn+1)⋮⪯k(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gxn-1,gxn+1)+r(gx1,gy1)q(gxn,gxn+1)⪯k(gx1,gy1)q(xn-1,xn)+l(gx1,gy1)q(xn-1,xn)+l(gx1,gy1)q(gxn,gxn+1)+r(gx1,gy1)q(gxn,gxn+1).
Hence
(3.38)q(gxn,gxn+1)⪯k(gx1,gy1)+l(gx1,gy1)1-l(gx1,gy1)-r(gx1,gy1)q(gxn-1,gxn)=hq(gxn-1,gxn)⪯h2q(gxn-2,gxn-1)⋮⪯hn-1q(gx1,gx2),
where h=(k(gx1,gy1)+l(gx1,gy1))/(1-l(gx1,gy1)-r(gx1,gy1))<1.
Similarly, we have(3.39)q(gyn,gyn+1)⪯k(gy1,gx1)+l(gy1,gx1)1-l(gy1,gx1)-r(gy1,gx1)q(gyn-1,gyn)=dq(gyn-1,gyn)⪯d2q(gyn-2,gyn-1)⋮⪯dn-1q(gy1,gy2),
where d=(k(gy1,gx1)+l(gy1,gx1))/(1-l(gy1,gx1)-r(gy1,gx1))<1.
Letm>n≥1. Then, it follows that
(3.40)q(gxn,gxm)⪯q(gxn,gxn+1)+q(gxn+1,gxn+2)+⋯+q(gxm-1,gxm)⪯(hn-1+hn+⋯+hm-2)q(gx1,gx2)⪯hn1-hq(gx1,gx2),q(gyn,gym)⪯q(gyn,gyn+1)+q(gyn+1,gyn+2)+⋯+q(gym-1,gym)⪯(dn-1+dn+⋯+dm-2)q(gy1,gy2)⪯dn1-dq(gy1,gy2).
Thus, Lemma 2.6 (3) shows that {gxn} and {gyn} are Cauchy sequences in g(X). Since g(X) is complete, there exists x*,y*∈X such that gxn→gx* and gyn→gy* as n→∞. Using (q3), we have
(3.41)q(gxn,gx*)⪯hn1-hq(gx1,gx2),(3.42)q(gyn,gy*)⪯dn1-dq(gy1,gy2).
On the other hand,(3.43)q(gxn,F(x*,y*))=q(F(xn-1,yn-1),F(x*,y*))⪯k(gxn-1,gyn-1)q(gxn-1,F(xn-1,yn-1))+l(gxn-1,gyn-1)q(gxn-1,F(x*,y*))+r(gxn-1,gyn-1)q(F(xn-1,yn-1),F(x*,y*))=k(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,xn)+l(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn-1,F(x*,y*))+r(F(xn-2,yn-2),F(yn-2,xn-2))q(gxn,F(x*,y*))⪯k(gxn-2,gyn-2)q(gxn-1,gxn)+l(gxn-2,gyn-2)q(gxn-1,F(x*,y*))+r(gxn-2,gyn-2)q(gxn,F(x*,y*))⋮⪯k(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gxn-1,F(x*,y*))+r(gx1,gy1)q(gxn,F(x*,y*))⪯k(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gxn-1,gxn)+l(gx1,gy1)q(gxn,F(x*,y*))+r(gx1,gy1)q(gxn,F(x*,y*)).
Then,
(3.44)q(gxn,F(x*,y*))⪯k(gx1,gy1)+l(gx1,gy1)1-l(gx1,gy1)-r(gx1,gy1)q(gxn-1,gxn)=hq(gxn-1,gxn)⪯hhn-2q(gx1,gx2)=hn-1q(gx1,gx2)⪯hn-11-hq(gx1,gx2).
Thus, Lemma 2.6 (1), (3.41), and (3.44) show that gx*=F(x*,y*). Similarly, we can prove that gy*=F(y*,x*). Therefore, (x*,y*) is a coupled coincidence point of F and g. Hence, (gx*,gy*) is the coupled point of coincidence.
Suppose thatu=gx*=F(x*,y*) and v=gy*=F(y*,x*). Then, we have
(3.45)q(u,u)=q(gx*,gx*)=q(F(x*,y*),F(x*,y*))⪯k(gx*,gy*)q(gx*,F(x*,y*))+l(gx*,gy*)q(gx*,F(x*,y*))+r(gx*,gy*)q(F(x*,y*),F(x*,y*))=k(u,v)q(u,u)+l(u,v)q(u,u)+r(u,v)q(u,u)=k(u,v)q(u,u)+l(u,v)q(u,u)+l(u,v)q(u,u)+r(u,v)q(u,u)=[k(u,v)+l(u,v)+l(u,v)+r(u,v)]q(u,u)=[(k+2l+r)(u,v)]q(u,u).
Since (k+2l+r)(u,v)<1, Lemma 2.3(1) shows that q(u,u)=θ. Similarly, we have q(v,v)=θ.
Finally, suppose there is another coupled point of coincidence(u1,v1) of F and g such that u1=gx′=F(x′,y′) and v1=gy′=F(y′,x′) for some (x′,y′) in X×X. Then, we have
(3.46)q(u,u1)=q(gx*,gx′)=q(F(x*,y*),F(x′,y′))⪯k(gx*,gy*)q(gx*,F(x*,y*))+l(gx*,gy*)q(gx*,F(x′,y′))+r(gx*,gy*)q(F(x*,y*),F(x′,y′))=k(u,v)q(u,u)+l(u,v)q(u,u1)+r(u,v)q(u,u1)=l(u,v)q(u,u1)+r(u,v)q(u,u1)⪯k(uv)q(u,u1)+l(u,v)q(u,u1)+l(u,v)q(u,u1)+r(u,v)q(u,u1)=[k(u,v))+l(u,v)+l(u,v)+r(u,v)]q(u,u1)=[(k+2l+r)(u,v)]q(u,u1).
Since (k+2l+r)(u,v)<1, Lemma 2.3 (1) shows that q(u,u1)=θ. By similar way, q(v,v1)=θ. Also, we have q(u,u)=θ and q(v,v)=θ. Thus, Lemma 2.6 (1) shows that u=u1 and v=v1, which implies that (u,v)=(u1,v1). Similarly, we can prove that u=v1 and v=u1. Thus, u=v. Therefore, (u,u) is the unique coupled point of coincidence. Now, let u=gx*=F(x*,y*). Since F and g are w-compatible, then we have
(3.47)gu=g(gx*)=gF(x*,y*)=F(gx*,gy*)=F(gx*,gx*)=F(u,u).
Then, (gu,gu) is a coupled point of coincidence. The uniqueness of the coupled point of coincidence implies that gu=u. Therefore, u=gu=F(u,u). Hence, (u,u) is the unique common coupled fixed point of F and g.The following corollaries can be obtained as consequences of Theorem3.7.Corollary 3.8.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Let F:X×X→X and g:X→X be two mappings and suppose that there exists mappings k,l:X×X→[0,1) such that the following hold: (a)
k(F(x,y),F(u,v))≤k(gx,gy) and l(F(x,y),F(u,v))≤l(gx,gy),(b)
(k+2l)(x,y)<1,(c)
q(F(x,y),F(u,v))⪯k(gx,gy)q(gx,F(x,y))+l(gx,gy)q(gx,F(u,v)). for all x,y,u,v∈X. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Corollary 3.9.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.48)(1-r)q(F(x,y),F(u,v))⪯kq(gx,F(x,y))+lq(gx,F(u,v)),
for all x,y,u,v∈X, where k,l,r are nonnegative constants with k+2l+r<1. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1) then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Corollary 3.10.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.49)q(F(x,y),F(u,v))⪯kq(gx,F(x,y))+lq(gx,F(u,v)),
for all x,y,u,v∈X, where k,l are nonnegative constants with k+2l<1. If F(X×X)⊆g(X) and g(X) is a complete subspace of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.Corollary 3.11.
Let(X,d) be a cone metric space with a cone P having nonempty interior and q is a c-distance on X. Suppose the mappings F:X×X→X and g:X→X satisfy the following contractive condition:
(3.50)q(F(x,y),F(u,v))⪯k[q(gx,F(x,y))+q(gx,F(u,v))],
for all x,y,u,v∈X, where k∈[0,1/3) is a constants. If F(X×X)⊆g(X) and g(X) is a complete subset of X, then F and g have a unique coupled point of coincidence (u,v) in X×X. Further, if u=gx1=F(x1,y1) and v=gy1=F(y1,x1), then q(u,u)=θ and q(v,v)=θ. Moreover, if F and g are w-compatible, then F and g have a unique common coupled fixed point and the common coupled fixed point of F and g is of the form (u,u) for some u∈X.


## 4. Some Examples

Example 4.1.
LetE=ℝ and P={x∈E:x≥0}. Let X=[0,1] and define a mapping d:X×X→E by d(x,y)=|x-y| for all x,y∈X. Clearly, (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 on X. Define the mappings F:X×X→X by F(x,y)=(xy)2/16 and g:X→X by gx=x/2 for all x,y∈X. Obviously that F(X×X)⊆g(X) and g(X) is a complete subspace of X. Take k(x,y)=l(x,y)=(xy+1)/8, where x,y∈X. Observe the following: (a)
k(F(x,y),F(u,v))=l(F(x,y),F(u,v)) =(((xy)2(uv)2/162)+1)/8 =(1/8)(((xy)2(uv)2/162)+1) ≤(1/8)((xy)2/16+1) ≤(1/8)(xy/16+1) ≤(1/8)(xy/4+1) =k(gx,gy) for all x,y,u,v∈X.(b)
k(x,y)=k(y,x) and l(x,y)=l(y,x) for all x,y∈X.(c)
k(x,y)+l(x,y)=(xy+1)/4<1 for all x,y∈X.(d)
For allx,y,u,v∈X, we have
(4.1)q(F(x,y),F(u,v))=F(u,v)=(uv)216⪯uv16⪯u+v16=u16+v16=18u2+18v2⪯(xy32+18)u2+(xy32+18)v2=(xy/48+18)u2+(xy/48+18)v2=(xy/4)+18u2+(xy/4)+18v2=k(gx,gy)q(gx,gu)+l(gx,gy)q(gy,gv).Clearly that F and g are w-compatible. Therefore, F and g satisfy all the conditions of Theorem 3.1. Hence, F and g have a unique common coupled fixed point (u,u)=(0,0) and g(0)=F(0,0)=0 with q(0,0)=0.Example 4.2.
Consider Example2.5. Define the mappings F:X×X→X by F(x,y)=(x+y)/4 and g:X→X by gx=x for all x,y∈X. Obviously that F(X×X)⊆g(X) and g(X) is a complete subspace of X. Then, we have q(F(x,y),F(u,v))=F(u,v)=(u+v)/4⪯(3/7)(u+v)=k[q(gx,gu)+q(gy,gv)] with k=3/7∈[0,1/2). Clearly that F and g are w-compatible. Therefore, F and g satisfy all the conditions of Corollary 3.3. Hence, F and g have a unique common coupled fixed point (u,u)=(0,0) and g(0)=F(0,0)=0 with q(0,0)=0.Example 4.3.
LetE=ℝ2 and P={(x,y)∈E:x,y≥0}. Let X=[0,1) and define a mapping d:X×X→E by d(x,y)=(|x-y|,|x-y|) for all x,y∈X. Then (X,d) is a complete cone metric space (see [15]). Define a mapping q:X×X→E by q(x,y)=(y,y) for all x,y∈X. Then q is a c-distance on X. In fact (q1)–(q3) are immediate. Let c∈E with θ≪c and put e=c/2. If q(z,x)≪e and q(z,y)≪e, then we have
(4.2)d(x,y)=(|x-y|,|x-y|)⪯(x+y,x+y)=(x,x)+(y,y)=q(z,x)+q(z,y)≪e+e=c.
This show, that (q4) holds. Therefore, q is a c-distance on X. Define the mappings F:X×X→X by F(x,y)=(x+y)/16 and g:X→X by gx=x/2 for all x,y∈X. Obviously that F(X×X)⊆g(X) and g(X) is a complete subspace of X. Then we have
(4.3)q(F(x,y),F(u,v))=(F(u,v),F(u,v))=(u+v16,u+v16)=(u16,u16)+(v16,v16)=(u/28,u/28)+(v/28,v/28)=18(u2,u2)+18(v2,v2)⪯14(u2,u2)+14(v2,v2)=14[(u2,u2)+(v2,v2)]=k[q(gx,gu)+q(gy,gv)],
with k=1/4∈[0,1/2). Clearly that F and g are w-compatible. Therefore, F and g satisfy all the conditions of Corollary 3.3. Hence, F and g have a unique common coupled fixed point (u,u)=(0,0) and g(0)=F(0,0)=0 with q(0,0)=0.



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*Source: 901792-2012-08-21.xml*