# Common Fixed Point and Coupled Coincidence Point Theorems for Geraghty’s Type Contraction Mapping with Two Metrics Endowed with a Directed Graph

Authors: P. Charoensawan; W. Atiponrat
Journal: Journal of Mathematics
 (2017)

---

## Abstract

The purpose of this paper is to present some existence and uniqueness results for common fixed point theorems forθ-ϕ contraction mappings with two metrics endowed with a directed graph. In addition, by using our main results, we obtain some results about coupled coincidence points endowed with a directed graph. Our results generalize those presented in previous papers.


---

## Body

## 1. Introduction and Preliminaries

Geraghty [1] introduced an interesting class Θ of functions θ:[0,∞)→0,1 satisfying that (1)θtn⟶1⟹tn⟶0,and they have shown some results which are a generalization of Banach’s contraction principle in 1973.Recently, Martínez-Moreno et al. [2] gave some new common fixed point theorems for Geraghty’s type contraction mappings employing the monotone property with two metrics by using d-compatibility and g-uniform continuity where the concept of d-compatibility is defined as follows.Definition 1 (see [3]).
Let(X,d) be a metric space, and let f,g:X→X be two mappings. The mappings g and f are said to be d-compatible if(2)limn→∞⁡dgfxn,fgxn=0whenever xn is a sequence in X such that limn→∞⁡fxn=limn→∞⁡gxn.In [2], there are many results on multidimensional common fixed point theorems which generalize the results in paper [4].The fixed point theorem employing the concept of metric spaces endowed with a graph was initiated by Jachymski [5], which generalizes the Banach contraction principle to mappings on metric spaces equipped with a graph. Also, the definitions of G-continuous and property A were given in [5].Definition 2 (see [5]).
A functionf:X→X is said to be G-continuous if and only if, for any x∈X which has a sequence {xn} in X, xn→x, and (xn,xn+1)∈E(G) for n∈N, we have that f(xn)→f(x).Definition 3 (see [5]).
Suppose that(X,d) is a metric space and G is a directed graph. The triple (X,d,G) is said to have property A if and only if for any sequence {xn} in X such that xn→x and (xn,xn+1)∈E(G), where n∈N, we have that (xn,x)∈E(G).After the discovery above, many researchers have studied the topic of existence of a fixed point for single-valued mappings and multivalued mappings in various spaces equipped with a graph; see [6–11].In the case of coupled fixed point with a graph, which will be our topic of interest, we now give a collection of notions and definitions from [12].Definition 4 (see [12]).
The given functionsF:X×X→X and g:X→X are said to be G-edge preserving if and only if the following condition is satisfied:(3)Ifgx,gu,gy,gv∈EG,thenFx,y,Fu,v,Fy,x,Fv,u∈EG.Definition 5 (see [12]).
Suppose that(X,d) is a metric space and E(G) is the set of all edges of the graph G. E(G) is said to have the transitivity property if and only if, for all x,y,  anda∈X, the following is true: (4)Ifx,a,a,y∈EG,thenx,y∈EG.Definition 6 (see [12]).
A functionF:X×X→X is said to be G-continuous if and only if the following are true. For any (x∗,y∗)∈X×X and any sequence (ni)i∈N of positive integers such that (F(xni,yni),F(xni+1,yni+1))and(F(yni,xni),F(yni+1,xni+1))∈E(G), F(xni,yni)→x∗, and F(yni,xni)→y∗ as i→∞, we have that as i→∞, (5)FFxni,yni,Fyni,xni⟶Fx∗,y∗,FFyni,xni,Fxni,yni⟶Fy∗,x∗.In 2015, Suantai et al. [12] used the above definitions to investigate some results on existence and uniqueness for coupled coincidence points and common fixed points of θ-ψ contraction mappings in complete metric spaces endowed with a directed graph. Their results also generalize others in partially ordered metric spaces.The aim of this paper is to present some existence and uniqueness results for common fixed point theorems forθ-ϕ contraction mappings with two metrics endowed with a directed graph. Furthermore, by using our main results, we are able to prove some results about coupled coincidence point endowed with a directed graph. Our results generalize the results obtained in [2, 12].


## 2. Main Results

Throughout this paper we shall say thatG will satisfy the standard conditions as mentioned in [5].In addition, we will introduce the concept ofg-Cauchy and edge preserving which are an effective tool as follows.Definition 7.
Let(X,d) and (Y,d′) be two metric spaces, and let f:X→Y and g:X→X be two mappings. The mapping f is said to be g-Cauchy on X if, for any sequence {xn} in X such that {gxn} is Cauchy sequence in (X,d), then {fxn} is Cauchy sequence in (Y,d′).It is easy to see that iff is g-uniformly continuous on X, then f is g-Cauchy on X.Definition 8.
LetG be a directed graph, and let f,g:X→X be two mappings. We say that f is g-edge preserving with respect to G if(6)gx,gy∈EG⟹fx,fy∈EG.LetΦ denote the class of all functions ϕ:[0,∞)→[0,∞) which satisfy the following conditions:(ϕ1)
ϕ is nondecreasing.(ϕ2)
ϕ is continuous.(ϕ3)
ϕ(t)=0⇔t=0.We now introduce a new class of the Geraghty type contractions in the following definition.Definition 9.
Let(X,d) be a metric space endowed with a directed graph G, and let f,g:X→X be given mappings. The pair (f,g) is called a θ-ϕ-contraction with respect to d if(1)
f is g-edge preserving with respect to G;(2)
there exist two functionsθ∈Θ and ϕ∈Φ such that, for all x,y∈X such that (gx,gy)∈E(G), (7)ϕdfx,fy≤θMgx,gyϕMgx,gy,whereM(gx,gy)  =  max{dgx,gy,dgx,fx,dgy,fy,dgx,fy+dgy,fx/2}.Let(X,d) be a metric space endowed with a directed graph G satisfying the standard conditions, and let two mappings f,g:X→X be given.We define important subsets ofX as follows:(8)Xf,g≔u∈X:gu,fu∈EG,Cf,g≔u∈X:fu=gu,that is, the set of all coincidence points of mappings f and g, and(9)Cmf,g≔u∈X:fu=gu=u,that is, the set of all common fixed points of mappings f and g.Now we are ready to present and prove the main results.Theorem 10.
Let(X,d′) be a complete metric space endowed with a directed graph G, and let d be another metric on X. Suppose that f,g:X→X and (f,g) is a θ-ϕ-contraction with respect to d. Suppose that(1)
g:(X,d′)→(X,d′) is continuous and g(X) is d′-closed;(2)
f(X)⊆g(X);(3)
E(G) satisfies the transitivity property;(4)
ifd≱d′, one assumes that f:(X,d)→(X,d′) is g-Cauchy on X;(5)
f:(X,d′)→(X,d′) is G-continuous, and f and g are d′-compatible.
Then, under these conditions,(10)Xf,g≠∅iffCf,g≠∅.Proof.
(⇐) Suppose that Cf,g≠∅. Let u∈Cf,g. We have fu=gu. Then gu,fu=gu,gu∈Δ⊂EG. Hence gu,gu=gu,fu∈EG which means that u∈Xf,g and thus Xf,g≠∅.
(⇒) Suppose nowX(f,g)≠∅. Let x0∈X such that (gx0,fx0)∈E(G). By the assumption that f(X)⊆g(X) and f(x0)∈X, it easy to construct sequences {xn} in X for which (11)gxn=fxn-1for all n∈N. If gxn0=gxn0-1 for some n0∈N, then xn0-1 is a coincidence point of the mappings g and f. Therefore, we assume that, for each n∈N,gxn≠gxn-1 holds.
Since(gx0,fx0)=(gx0,gx1)∈E(G) and f is edge preserving with respect to g, we have (fx0,fx1)=(gx1,gx2)∈E(G). Continuing inductively, we obtain that (gxn-1,gxn)∈E(G) for each n∈N. Hence it follows from the contractive condition that(12)ϕdgxn+1,gxn+2=ϕdfxn,fxn+1≤θMgxn,gxn+1ϕMgxn,gxn+1<ϕMgxn,gxn+1.On the other hand, we get (13)Mgxn,gxn+1=max⁡dgxn,gxn+1,dgxn,fxn,dgxn+1,fxn+1,dgxn,fxn+1+dgxn+1,fxn2=max⁡dgxn,gxn+1,dgxn+1,gxn+2,dgxn,gxn+22≤max⁡dgxn,gxn+1,dgxn+1,gxn+2.If M(gxn,gxn+1)=d(gxn+1,gxn+2), then, by (12), we obtain that (14)ϕdgxn+1,gxn+2<ϕdgxn+1,gxn+2which is a contradiction. So, for all n≥1, we have (15)Mgxn,gxn+1=dgxn,gxn+1.
Notice that, in view of (12), we get for all n≥1 that ϕ(d(gxn+1,gxn+2))<ϕ(d(gxn,gxn+1)). From ϕ1 of property of ϕ, we have (16)dgxn+1,gxn+2≤dgxn,gxn+1,∀n∈N.
Hence, we deduce that the sequence{d(gxn,gxn+1)} is nonnegative and nonincreasing. Consequently, there exists r≥0 such that limn→∞⁡d(gxn,gxn+1)=r. We claim that r=0. Suppose, on the contrary, that r>0. Then, due to (12), we have (17)ϕdgxn+1,gxn+2ϕdgxn,gxn+1=ϕdgxn+1,gxn+2ϕMgxn,gxn+1≤θMgxn,gxn+1<1.It follows that limn→∞⁡θ(M(gxn,gxn+1))=1. Owing to the fact that θ∈Θ, we get limn→∞⁡M(gxn,gxn+1)=0 which implies that limn→∞⁡d(gxn,gxn+1)=0, a contradiction. So, we conclude that (18)limn→∞⁡dgxn,gxn+1=0.
We assert that{gxn} is a Cauchy sequence. Suppose, on the contrary, that {gxn} is not a Cauchy sequence. Thus, there exists ϵ>0 such that, for all k∈N, there exists n(k),m(k)∈N such that n(k)>m(k)≥k with the smallest number satisfying the condition below: (19)dgxnk,gxmk≥ϵ,dgxnk-1,gxmk<ϵ.
Then, we have(20)ϵ≤dgxmk,gxnk≤dgxmk,gxnk-1+dgxnk-1,gxnk<ϵ+dgxnk-1,gxnk.
Lettingk→∞ in the above inequality, by (18), we obtain that (21)limn→∞⁡dgxmk,gxnk=ϵ>0.By the transitivity property of E(G), we get (gxm(k),gxn(k))∈E(G) for all k. Thus, we have (22)ϕdgxmk+1,gxnk+1=ϕdfxmk,fxnk≤θMgxmk,gxnkϕMgxmk,gxnk,where (23)Mgxmk,gxnk=max⁡dgxmk,gxnk,dgxmk,fxmk,dgxnk,fxnk,dgxmk,fxnk+dgxnk,fxmk2=max⁡dgxmk,gxnk,dgxmk,gxmk+1,dgxnk,gxnk+1,dgxmk,gxnk+1+dgxnk,gxmk+12.
Hence, we conclude that(24)ϕdgxmk+1,gxnk+1ϕMgxmk,gxnk≤θMgxmk,gxnk<1.
Keeping (18) and (21) in mind and letting k→∞, we derive that (25)limk→∞⁡Mgxmk,gxnk=ϵ>0.
By inequality (24), we get (26)limk→∞⁡θMgxmk,gxnk=1and hence limk→∞⁡Mgxmk,gxnk=0, a contradiction. So, we conclude that {gxn} is a Cauchy sequence in (X,d).
Next, we claim that{gxn} is a Cauchy sequence with respect to d′.
Ifd≥d′, it is trivial. Thus, suppose d≱d′. Let ε>0. Since {gxn} is a Cauchy sequence in (X,d) and f is g-Cauchy on X, we have that {fxn} is Cauchy sequence in (X,d′). Then there exists N0∈N with (27)d′gxn+1,gxm+1=d′fxn,fxm<εwhenever n,m≥N0. So {gxn} is a Cauchy sequence with respect to d′.
Sinceg(X) is a d′-closed subset of the complete metric space (X,d′), there exists u=gx∈g(X) such that (28)limn→∞⁡gxn=limn→∞⁡fxn=u.
Now, sincef:(X,d′)→(X,d′) is G-continuous, and f and g are d′-compatible, we have (29)limn→∞⁡d′gfxn,fgxn=0.Using the triangle inequality, we have (30)d′gu,fu≤d′gu,gfxn+d′gfxn,fgxn+d′fgxn,fu.Letting n→∞, from (29), the continuity of g, and the G-continuity of f, it follows that d′(gu,fu)=0 which implies that gu=fu. So u is a coincidence point of f and g.Ifd=d′, we have the following theorem.Theorem 11.
Let(X,d) be a complete metric space endowed with a directed graph G. Suppose that f,g:X→X and (f,g) is a θ-ϕ-contraction with respect to d. Moreover, suppose that(1)
g is continuous and g(X) is closed;(2)
f(X)⊆g(X);(3)
E(G) satisfies the transitivity property;(4)
(a)f is G-continuous and f and g are d-compatible or (b) (X,d,G) has property A.
Then, under these conditions,(31)Xf,g≠∅iffCf,g≠∅.Proof.
In order to avoid the repetition, following from the same proof in Theorem10, we can only consider (b) of condition (3). Since {gxn} is a Cauchy sequence in (X,d) and g(X) is closed in X, there exists u∈X such that (32)limn→∞⁡gxn=gu=limn→∞⁡fxn.Now, we show that u is a coincidence point of f and g. Suppose, on the contrary, that fu≠gu. Then d(fu,gu)>0. Since (X,d,G) has property A, we have (gxn,gu)∈E(G) for each n∈N. By using the triangle inequality, we derive (33)dgu,fu≤dgu,fxnk+dfxnk,fuwhich implies that (34)dgu,fu-dgu,fxnk≤dfxnk,fu.
By the property ofϕ, we obtain that (35)ϕdgu,fu-dgu,fxnk≤ϕdfxnk,fu≤θMgxnk,guϕMgxnk,gu,where(36)Mgxnk,gu=max⁡dgxnk,gu,dgxnk,fxnk,dgu,fu,dgxnk,fu+dgu,fxnk2.
From (32), we obtain that (37)limk→∞⁡Mgxnk,gu=dgu,fu>0.
Lettingk→∞ in inequality (35), we obtain that limk→∞⁡θ(M(gxn(k),gu))=1 so limk→∞⁡M(gxn(k),gu)=d(gu,fu)=0, a contradiction. Therefore fu=gu. Consequently, we conclude that f and g have a coincidence point.Theorem 12.
In addition to the hypotheses of Theorem10, assume that(K)
for anyx,y∈C(f,g) such that gx≠gy, we have (gx,gy)∈E(G).
Iff and g are d′-compatible and X(f,g)≠∅, then Cm(f,g)≠∅.Proof.
Theorem10 implies that there exists a coincidence point x∈X; that is, gx=fx. Suppose that there exists another coincidence point y∈X such that gy=fy. Assume that gx≠gy. By assumption (K), (gx,gy)∈E(G), and we have (38)ϕdfx,fy≤θMgx,gyϕMgx,gy<ϕMgx,gy=ϕdfx,fy,which is a contradiction. Therefore, gx=gy. Starting from x0=x, choose the sequences {xn} satisfying gxn=fxn-1 for each n∈N. Taking into account the properties of coincidence points, it is easy to see that it can be done so that xn=x; that is, (39)gxn=fx,for all n∈N. Next, set w=gx so we get gw=ggx=gfx. The definition of the sequence {xn} implies that gxn=fx=fxn-1 for all n∈N and hence (40)limn→∞⁡fxn=limn→∞⁡gxn=fxwith respect to d′. Because g and f are d′-compatible, we have (41)limn→∞⁡d′gfxn,fgxn=0;that is, gfx=fgx. Therefore, we have gw=gfx=fgx=fw which we can conclude that w is another coincidence point of the functions f and g. By the above results, we obtain fw=gw=gx=w and so w is a common fixed point of g and f. In other words, Cm(f,g)≠∅.IfM(gx,gy)=d(gx,gy), we get the following.Definition 13.
Let(X,d) be a metric space endowed with a directed graph G, and let f,g:X→X be given mappings. The pair (f,g) is called a θ-ψ-contraction with respect to d if(1)
f is g-edge preserving with respect to G;(2)
there exist two functionsθ∈Θ and ψ∈Φ such that for all x,y∈X such that (gx,gy)∈E(G), (42)ψdfx,fy≤θdgx,gyψdgx,gy.Applying the similar argument as in the proof of Theorems10 and 11, we have the following.Corollary 14.
Let(X,d′) be a complete metric space endowed with a directed graph G, and let d be another metric on X. Suppose that f,g:X→X and (f,g) is a θ-ψ-contraction with respect to d. Moreover, suppose that(1)
g:(X,d′)→(X,d′) is continuous and g(X) is d′-closed;(2)
f(X)⊆g(X);(3)
E(G) satisfies the transitivity property;(4)
ifd≱d′, one assumes that f:(X,d)→(X,d′) is g-Cauchy on X;(5)
(a)f is G-continuous, and f and g are d-compatible or (b) if d=d′, assume that (X,d,G) has property A.
Then, under these conditions,(43)Xf,g≠∅iffCf,g≠∅.Remark 15.
PutE(G)={(x,y)∈X×X:x⪯y} and ψ(t)=t in Corollary 14. In this case, we obtain the results of [2].Example 16.
LetX=[0,∞)⊆R, and let the metrics d,d′:X×X→[0,∞) be defined by (44)dx,y=x-y,d′x,y=Lx-yfor all x,y∈X, respectively, where L is a constant real number such that L∈(1,∞). It is easy to see that d<d′.
Now, we considerE(G) given by (45)EG=x,y:x=yorx,y∈0,1withx≤y.Consider the mappings f:X→X and g:X→X defined by (46)gx=x2,fx=ln⁡1+x22for all x∈X, respectively.
Next, we show that conditions(1)-(2) in Definition 13 hold as follows.
(1) Let(gx,gy)∈E(G), if x=y then (fx,fy)∈E(G), if gx,gy∈E(G)withgx≤gy, then we obtain gx=x2, gy=y2∈[0,1] and x2=gx≤gy=y2, and we have fx=ln(1+x2/2)≤fy=ln(1+y2/2) and fx,fy∈[0,1]. This implies that (fx,fy)∈E(G).
(2) Letψ(t)=t/2, and let θ∈Θ be defined by (47)θt=2ln⁡1+t/2t,ift>0,0,ift=0.Let x,y be arbitrary points in X and (gx,gy)∈E(G). If gx=gy, we have x=y and hence the contractive condition (42) holds for this case. In another case, we have gx=x2,gy=y2∈[0,1], and x2=gx≤gy=y2. Also, we have (48)ψdfx,fy=dfx,fy2=ln⁡1+x2/2-ln⁡1+y2/22=ln⁡1+y2/2-ln⁡1+x2/22=ln⁡1+y2/2/1+x2/22=ln⁡1+y2/2-x2/2/1+x2/22≤ln1+x2/2-y2/22≤2ln⁡1+1/2x2-y2x2-y2x2-y22=2ln⁡1+1/2dgx,gydgx,gyψdgx,gy=θdgx,gyψdgx,gy.Therefore, condition (3) holds for the functions θ and ψ.
Next, the proof for conditions(1)–(5) in Corollary 14 is shown as follows.
(1) The functiong:(X,d′)→(X,d′) can be easily seen to be continuous. Moreover, we immediately obtain that g(X)=[0,∞) is d′-closed.
(2) By the construction off and g, we can conclude that f(X)=g(X).
(3)E(G) is clearly satisfying the transitivity property.
(4) Sinced<d′, we need to prove that the function f:(X,d)→(X,d′) is g-Cauchy on X. Let ϵ>0, and let {xn} be a sequence in X such that {gxn} is a Cauchy sequence in (X,d). Then, there exist K∈N such that, for all n,m≥K,d(gxn,gxm)<ϵ/L. Then we have (49)d′fxn,fxm=Lfxn-fym=Lln⁡1+xn22-ln⁡1+xm22=Lln⁡1+xm2/21+xn2/2=Lln⁡1+xm2/2-xn2/21+xn2/2≤Lln⁡1+xn22-xm22≤L2ln⁡1+1/2xn2-xm2xn2-xm2xn2-xm2<Lxn2-xm2=Ldgxn,gxm<LϵL=ϵ.This implies that f:(X,d)→(X,d′) is g-Cauchy on X.
(5) It can be easily shown thatf:(X,d′)→(X,d′) is G-continuous. As a result, we will only need to prove that g and f are d′-compatible. Let {xn} be a sequence in X such that (50)limn→∞⁡gxn=limn→∞⁡fxn=a.Then we have ln1+a/2=a and so it follows that a=0. Now, we have (51)d′gfxn,fgxn=Lln⁡1+xn222-ln⁡1+xn42⟶0as n→∞.
It is easy to see that there exists a pointx0∈X such that (gx0,fx0)∈E(G), and this implies that X(f,g)≠∅.
Consequently, all the conditions of Corollary14 hold. Therefore, g and f have a coincidence point and, furthermore, a point 0 is a coincidence point of the mappings g and f.


## 3. Coupled Coincidence Point Theorems

We begin with some useful background.Definition 17 (see [4]).
Let(X,d) be a metric space, and let g:X→XandF:X×X→X. The mappings g and F are said to be d-compatible if (52)limn→∞⁡dgFxn,yn,Fgxn,gyn=0,limn→∞⁡dgFyn,xn,Fgyn,gxn=0hold whenever {xn} and {yn} are sequences in X such that limn→∞F(xn,yn)=limn→∞gxn and limn→∞F(yn,xn)=limn→∞gyn.LetX be a nonempty set, and let F:X2→Xandg:X→X be two mappings.We define two mappingsTF,Gg:X×X→X×X by (53)TFx,y=Fx,y,Fy,x,Ggx,y=gx,gyfor all x,y∈X.The following lemma guarantees that 2-dimensional notion of common fixed coincidence points can be interpreted in terms of two mappingsTF and Gg.Lemma 18 (see [2]).
Suppose thatX is a nonempty set and F:X×X→X and g:X→X are functions. We say that the point (x1,x2)∈X×X is a coupled coincidence point of F and g, that is, gx=F(x,y) and gy=F(y,x) (see [13, 14]), if and only if it is a coincidence point of the functions TF and Gg.Definition 19.
Suppose thatG is a directed graph and F:X×X→X and g:X→X are two functions. F is said to be G-edge preserving with respect to g if and only if the following holds: (54)Ifgx,gu,gy,gv∈EG,thenFx,y,Fu,v,Fy,x,Fv,u∈EG.The concept of the cross product of the graphsGi=(V(Gi),E(Gi)),i=1,2, is defined by (55)G1×G2≔VG1×G2,EG1×G2,where E(G1×G2)={((x,y),(x′,y′))∣(x,x′)∈E(G1)and(y,y′)∈E(G2)}.Now, the following lemma shows that being edge preserving and the transitivity property for 2-dimensional case can be interpreted in terms of two mappingsTF and Gg.Lemma 20.
LetF:X2→X and g:X→X be two mappings, and let (X,d) be a metric space endowed with a directed graph G. Then, F is G-edge preserving with respect to g if and only if TF is edge preserving with respect to Gg.Proof.
(
⇒
) Let F be G-edge preserving with respect to g, and let x,y,u,  andv∈X be such that (56)Ggx,y,Ggu,v∈EG×G⇓gx,gy,gu,gv∈EG×G⇓gx,gu∈EG,gy,gv∈EG⇓Fx,y,Fu,v∈EG,Fy,x,Fv,u∈EG⇓Fx,y,Fy,x,Fu,v,Fv,u∈EG×G⇓TFx,y,TFu,v∈EG×G.
(
⇐
) Let TF be edge preserving with respect to Gg, and let x,y,u,  andv∈X be such that (57)gx,gu∈EG,gy,gv∈EG⇓gx,gy,gu,gv∈EG×G⇓Ggx,y,Ggu,v∈EG×G⇓TFx,y,TFu,v∈EG×G⇓Fx,y,Fy,x,Fu,v,Fv,u∈EG×G⇓Fx,y,Fu,v∈EG,Fy,x,Fv,u∈EG.Lemma 21.
LetF:X2→Xandg:X→X be two mappings, and let (X,d) be a metric space endowed with directed graphs G1 and G2. Then, E(G1) and E(G2) satisfy the transitivity property if and only if E(G1×G2) satisfies the transitivity property.Proof.
(
⇒
) Suppose that E(G1) and E(G2) satisfy the transitivity property. Let x,y,u,v,a,  andb∈X be such that (58)x,y,u,v,u,v,a,b∈EG1×G2⇓x,u,u,a∈EG1,y,v,v,b∈EG2⇓x,a∈EG1,y,b∈EG2⇓x,y,a,b∈EG1×G2.
(
⇐
) Suppose that E(G1×G2) satisfies the transitivity property. Let x,y,u,v,a,  andb∈X be such that (59)x,u,u,a∈EG1,y,v,v,b∈EG2⇓x,y,u,v,u,v,a,b∈EG1×G2⇓x,y,a,b∈EG1×G2⇓x,a∈EG1,y,b∈EG2.Let(X,d) be a metric space endowed with a directed graph G satisfying the standard conditions, and let F:X×X→X and g:X→X be two functions. We will define the following sets: (60)X2gF=x,y∈X2:gx,Fx,y,gy,Fy,x∈EG,CcFix⁡F=x,y∈X2:Fx,y=gx,Fy,x=gy.Suantai et al. gave the notion of aθ-ψ-contraction as in [12]. We are ready to present an application of Corollary 14 in order to deduce results about coupled fixed points.Theorem 22.
Let(X,d′) be a complete metric space endowed with a directed graph G, let d:X×X→[0,+∞) be another metric on X, and let F:X×X→X and g:X→X be two functions so that the pair of F and g is a θ-ψ-contraction. Suppose that the following are true:(1)
g:(X,d′)→(X,d′) is a continuous function such that g(X) is d′-closed.(2)
F(X×X) is a subset of g(X).(3)
E(G) satisfies the transitivity property.(4)
ifd≱d′, then assume further that F:(X,d)×(X,d)→(X,d′) is g-uniformly continuous on X×X.(5)
(a)F:(X,d′)×(X,d′)→(X,d′) is G-continuous, and F and g are d′-compatible or (b) if d=d′, assume that (X,d,G) has property A.
Under these conditionsCcFix⁡F≠∅ if and only if (X2)gF≠∅.Proof.
We only need to apply Corollary14 to the functions TF and Gg on the complete metric space (X×X,D′) and the metric space (X×X,D), where the metrics D and D′ are defined as follows: (61)D′x,y,u,v=max⁡d′x,u,d′y,v,Dx,y,u,v=max⁡dx,u,dy,v.
If we putG∗=G×G, we have(62)x,y,u,v∈EG∗⟺x,u,y,v∈EGfor all x,y,u,v∈X×X.
By Lemmas20 and 21, we have that TF:X×X→X×X is Gg-edge preserving with respect to G∗ and E(G∗) satisfies the transitivity property. Since the pair of F and g is a θ-ψ-contraction, we get that (TF,Gg) is a θ-ψ-contraction with respect to d. Since F(X×X)⊆g(X), for any given x0,y0∈X, we can construct sequences {gxn},{gyn} for which(63)gxn=Fxn-1,yn-1,gyn=Fyn-1,xn-1∀n∈N.From this, it easy to see that if F is G-continuous, then TF is G-continuous. Observe that (64)Notice  thatFandgared′-compatible.⟺limn→∞⁡Fxn,yn=limn→∞⁡gxnlimn→∞⁡Fyn,xn=limn→∞⁡gyn⟹limn→∞⁡d′gFxn,yn,Fgxn,gyn=0limn→∞⁡d′gFyn,xn,Fgyn,gxn=0⟺limn→∞⁡TFxn,yn=limn→∞⁡Ggxn,yn⟹limn→∞⁡D′TFGgxn,yn,GgTFxn,yn=0⟺TFandGgareD′-compatible.
This completes the proof.Takingd=d′ in Theorem 22, we have the following Corollary which implies the result in [12].Corollary 23.
Let(X,d) be a complete metric space endowed with a directed graph G, and let F:X×X→X and g:X→X be two mappings such that the pair of F and g is a θ-ψ-contraction. Suppose that the following hold:(1)
g is continuous and g(X) is closed.(2)
F(X×X)⊆g(X).(3)
E(G) satisfies the transitivity property.(4)
(a)F:(X,d′)×(X,d′)→(X,d′) is G-continuous and F and g are compatible or (b) (X,d,G) has property A.
Under these conditionsCcFix(F)≠∅ if and only if (X2)gF≠∅.



---

*Source: 5746704-2017-11-16.xml*