# Fisher Fixed Point Results in Generalized Metric Spaces with a Graph

Authors: Karim Chaira; Abderrahim Eladraoui; Mustapha Kabil; Abdessamad Kamouss
Journal: International Journal of Mathematics and Mathematical Sciences
 (2020)

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## Abstract

We discuss Fisher’s fixed point theorem for mappings defined on a generalized metric space endowed with a graph. This work should be seen as a generalization of the classical Fisher fixed point theorem. It extends some recent works on the enlargement of Banach Contraction Principle to generalized metric spaces with graph. An example is given to illustrate our result.


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## Body

## 1. Introduction and Preliminaries

Currently, fixed point theory is a very active area of research because of its applications in multiple fields. It concerns the results which indicate that, under certain conditions, self-mapping on a set admits a fixed point. Among all the results in metric fixed point theory, the Banach Contraction Principle [1] is the most celebrated one due to its simplicity and ease of application in major areas of mathematics. Subsequently, many authors extend and generalize this principle in different directions. In 1981, Fisher [2] proved the following related fixed-point theorem involving two mappings on two complete metric spaces as follows.Theorem 1.
LetX,d and Y,δ be complete metric spaces. If T is a mapping from X into Y and S is a mapping from Y into X, satisfying the following conditions,(1)δTx,TSy≤cmaxdx,Sy,δy,Tx,δy,TSy,dSy,STx≤cmaxδy,Tx,dx,Sy,dx,STx,for all x in X and y in Y where 0≤c<1, then ST has a unique fixed point z in X and TS has a unique fixed point w in Y. Furthermore, Tz=w and Sw=z.In 2017, Chaira et al. [3] extend the Fisher theorem in the setting of partially ordered generalized metric spaces.The concept of standard metric space is a fundamental tool in topology, functional analysis, and nonlinear analysis. In recent years, several generalizations of standard metric space have appeared. In 1993, Czerwik [4] introduced the concept of a b-metric space. Since then, several works have dealt with fixed point theory in such spaces. In 2000, Hitzler and Seda [5] introduced the notion of dislocated metric spaces in which self-distance of a point need not be equal to zero. Such spaces play a very important role in topology and logical programming (see [6]). In this work, we present a new generalized metric space introduced by Jleli and Samet in [7] and that recovers a large class of topological spaces including standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces with the Fatou property. Also, several interesting results about the existence and the uniqueness of the fixed point were proved in the setting of this generalized metric space (see [3, 7–13]).An interesting approach in the theory of the fixed point in some general structures was recently given by Jachymski [14] in the setting of metric spaces endowed with a graph and Samet and Turinici [15] in the setting of metric spaces endowed with an arbitrary binary relation.In this work, inspired by the ideas given in [14–18], we investigate Fisher fixed-point theorem in generalized metric spaces with a graph. As corollary, we obtain Fisher fixed-point theorem in the setting of standard metric spaces endowed with a graph. An example is provided to illustrate our result.In the following, we describe the mathematical background materials which are necessary for establishing the results in this paper. A directed graph or digraphG is determined by a nonempty set VG of its vertices and the set EG⊂VG×VG of its arcs. Let Δ denote the diagonal of the Cartesian product VG×VG. A digraph is said to be reflexive if the set EG of its arcs contains all loops, i.e., Δ⊂EG. G is said to be transitive whenever for any x,y,z∈VG:(2)x,y∈EGandy,z∈EGimpliesx,z∈EG.We say that a vertexx in VG is isolated if for any vertex y in VG such that x≠y, and we have neither x,y∈EG nor y,x∈EG.ByG−1 we denote the converse of a digraph G, that is, the digraph obtained from G by reversing the direction of arcs. Thus, we have(3)EG−1=x,y∈VG×VG:y,x∈EG.Also,G˜ denotes the undirected graph obtained from G by ignoring direction of the edges. Thus, we have(4)EG˜=EG∪EG−1.Given a digraphG=V,E, a (di)path of G is a sequence a0,a1,…,an with ai,ai+1∈EG, for each i∈ℕ. A finite path a0,a1,…,an is said to have length n, for n∈ℕ. A closed directed path of length n>1 from x to y, i.e., x=y, is called a directed cycle. An acyclic digraph is a digraph that has no directed cycle. A digraph is connected if there is a finite (di)path joining any two of its vertices and it is weakly connected if G˜ is connected.IfG is such that EG is symmetric and x is a vertex in G, then the subgraph Gx consisting of all edges and vertices, which are contained in some path beginning at x, is called the component of G containing x. In this case, VGx=xG, where xG is the equivalence class of the following relation R defined on VG by the rule yRz if there is a path in G from y to z. Clearly, Gx is connected. We say that two vertices x and y in VG are connected if y∈xG or x∈yG.The basic concepts related to a graph may be found in any textbook on graph theory, see for example [19, 20].Definition 1.
A sequencexn in VG is said to be(i)
G-increasing, if xn+1∈xnG, for all n∈ℕ;(ii)
G-decreasing, if xn∈xn+1G, for all n∈ℕ;(iii)
G-monotone, if it is either G-increasing or G-decreasing
LetX be a nonempty set and D:X×X⟶0,+∞ be a given mapping. For every x∈X, let us define the set CD,X,x=xn⊂X:limn⟶∞Dxn,x=0. The mapping D is called a generalized metric on X if it satisfies the following conditions [7]:D1 for every x,y∈X×X, Dx,y=0⟹x=y.D2 for every x,y∈X×X, Dx,y=Dy,x.D3 there exists C>0 such that if x,y∈X×X and xn∈CD,X,x, then(5)Dx,y≤Clim  supn⟶∞Dxn,y.
In this case, we say that the pairX,D is a generalized metric space. Obviously, if the set CD,X,x is empty for every x∈X, then X,D is a generalized metric space if and only if D1 and D2are satisfied. A sequence xn in a generalized metric space X,D is said to be D-convergent to x∈X if xn∈CD,X,x. Note that if the set CD,X,x is not empty for some x∈X, then Dx,x=0. A sequence xn is said to be a D-Cauchy sequence if limm,n⟶∞Dxn,xm=0. Note that in a generalized metric space, a sequence has at most one limit and a D-convergent sequence may not be D-Cauchy sequence (see [16]). Moreover, X,D is said to be D-complete if every D-Cauchy sequence in X is D-convergent to some element in X.Definition 2.
We say thatX,D,G is G-regular if any G-increasing sequence (resp. G-decreasing sequence) xn which D-converges to some x∈VG, we have x∈xnG (resp. xn∈xG), for any n∈ℕ.


## 2. Main Results

LetX,D and Y,Δ be two generalized metric spaces such that X is endowed with a digraph G. Consider a nondecreasing function α:0,+∞⟶0,1 such that limsupt⟶0+αt<inf1,1/C with C as the positive real appeared in the condition D3 in the definition of the generalized metric D.LetT:X⟶Y and S:Y⟶X be two mappings and denote, for any x0∈X,(6)δS,T,x0,D,Δ=supDSTix0,STjx0,ΔTSTix0,TSTjx0:i,j∈ℕ.Theorem 2.
LetT:X⟶Y and S:Y⟶X be two mappings satisfying the following conditions:(1)
Let(7)DSy,STx≤αΔy,TxmaxDx,Sy,Δy,Tx,Dx,STx,ΔTx,TSy≤αDx,SymaxDx,Sy,Δy,Tx,Δy,TSy,for anyx in X and y in Y such that x and Sy are connected.(2)
X,D,G is D-complete and G-regular.(3)
There exists an elementx0∈X such that δS,T,x0,D,Δ<∞ and STi+1x0∈STix0G, for all i∈ℕ.Then, the sequenceSTnx0D-converges to some x∗ in X. If Dx∗,STx∗<∞, then there exists y∗∈Y such that y∗=Tx∗ and Sy∗=x∗; then, STx∗=x∗ and TSy∗=y∗. Moreover, Dx∗,x∗=0 and Δy∗,y∗=0.Proof.
Step 1. Suppose that there exists an elementx0∈X such that δS,T,x0,D,Δ<∞ and
(8)
S
T
i
+
1
x
0
∈
S
T
i
x
0
G
,
for all
i
∈
ℕ
.
Consider the two sequencesxn⊂X and yn⊂Y defined by
(9)
x
n
+
1
=
S
T
x
n
,
y
n
=
T
x
n
,
for  all
n
∈
ℕ
.
In (7), by taking x=xp and y=yq, where p and q in ℕ, we obtain
(10)
D
S
y
q
,
S
T
x
p
≤
α
Δ
y
q
,
T
x
p
max
D
x
p
,
S
y
q
,
Δ
y
q
,
T
x
p
,
D
x
p
,
S
T
x
p
,
Δ
T
x
p
,
T
S
y
q
≤
α
D
x
p
,
S
y
q
max
D
x
p
,
S
y
q
,
Δ
y
q
,
T
x
p
,
Δ
y
q
,
T
S
y
q
,
sincexp and Syq=xq+1 are connected.
Thus,
(11)
D
x
q
+
1
,
x
p
+
1
≤
α
Δ
y
q
,
y
p
max
D
x
p
,
x
q
+
1
,
Δ
y
q
,
y
p
,
D
x
p
,
x
p
+
1
,
Δ
y
p
,
y
q
+
1
≤
α
D
x
p
,
x
q
+
1
max
D
x
p
,
x
q
+
1
,
Δ
y
q
,
y
p
,
Δ
y
q
,
y
q
+
1
.
In (11), by taking p=q=n, we obtain
(12)
D
x
n
+
1
,
x
n
+
1
≤
α
Δ
y
n
,
y
n
max
D
x
n
,
x
n
+
1
,
Δ
y
n
,
y
n
,
Δ
y
n
,
y
n
+
1
≤
α
D
x
n
,
x
n
+
1
max
D
x
n
,
x
n
+
1
,
Δ
y
n
,
y
n
,
and by takingp−1=q=n, we obtain
(13)
D
x
n
+
1
,
x
n
+
2
≤
α
Δ
y
n
,
y
n
+
1
max
D
x
n
+
1
,
x
n
+
1
,
Δ
y
n
,
y
n
+
1
,
Δ
y
n
+
1
,
y
n
+
1
≤
α
D
x
n
+
1
,
x
n
+
1
max
D
x
n
+
1
,
x
n
+
1
,
Δ
y
n
,
y
n
+
1
.
For anyn in ℕ, let
(14)
d
n
=
D
x
n
,
x
n
,
d
n
′
=
D
x
n
,
x
n
+
1
,
s
n
=
Δ
y
n
,
y
n
,
s
n
′
=
Δ
y
n
,
y
n
+
1
.
It follows from inequalities (12) and (13) that
(15)
d
n
+
1
≤
α
s
n
max
d
n
′
,
s
n
,
s
n
′
≤
α
d
n
′
max
d
n
′
,
s
n
,
d
n
+
1
′
≤
α
s
n
′
max
d
n
+
1
,
s
n
′
,
s
n
+
1
≤
α
d
n
+
1
max
d
n
+
1
,
s
n
′
.
Now for anyn in ℕ, let kn=maxαdn′,αsn and tn=maxαdn+1,αsn′. System (15) becomes
(16)
d
n
+
1
≤
k
n
a
n
,
s
n
′
≤
k
n
a
n
,
d
n
+
1
′
≤
t
n
b
n
,
s
n
+
1
≤
t
n
b
n
,
wherean=maxdn′,sn and bn=maxdn+1,sn′. Consequently,
(17)
max
d
n
+
1
,
s
n
′
≤
k
n
a
n
,
max
d
n
+
1
′
,
s
n
+
1
≤
t
n
b
n
.
That is,bn≤knan and an+1≤tnbn, and so an+1≤knan.
Letδn be the sequence defined by δn=δS,T,STnx0,D,Δ, for any n∈ℕ. Since the function α is nondecreasing, dn′≤δ0 and sn≤δ0, for any n∈ℕ,(18)kn≤αδ0and soan+1≤αδ0an,for anyn∈ℕ,
which implies that the sequencesan and bn converge to 0. This yields(19)limn⟶∞Dxn,xn=limn⟶∞Dxn,xn+1=limn⟶∞Δyn,yn=limn⟶∞Δyn,yn+1=0.
Step 2. Let us show thatxn is a D-Cauchy sequence. Let us fix i and j in ℕ.
For anyn∈ℕ, x=xn−1+j and Sy=xn+i are connected and by using the first inequality in (7), we have
(20)
D
S
T
n
+
i
x
0
,
S
T
n
+
j
x
0
=
D
x
n
+
i
,
x
n
+
j
≤
α
Δ
y
n
+
i
−
1
,
y
n
+
j
−
1
max
D
x
n
+
j
−
1
,
x
n
+
i
,
Δ
y
n
+
i
−
1
,
y
n
+
j
−
1
D
x
n
+
j
−
1
,
x
n
+
j
,
≤
α
Δ
y
n
+
i
−
1
,
y
n
+
j
−
1
δ
n
−
1
.
For anyn in ℕ, x=xn+i and Sy=xn+j are connected and by using the second inequality in (7), we have
(21)
Δ
T
S
T
n
+
i
x
0
,
T
S
T
n
+
j
x
0
=
Δ
y
n
+
i
,
y
n
+
j
≤
α
D
x
n
+
i
,
x
n
+
j
max
D
x
n
+
i
,
x
n
+
j
,
Δ
y
n
+
i
,
y
n
−
1
+
j
Δ
y
n
−
1
+
j
,
y
n
+
j
,
≤
α
D
x
n
+
i
,
x
n
+
j
δ
n
−
1
.
Thus,δn≤αnδn−1, where αn=supαDxn+i,xn+j,αΔyn+i−1,yn+j−1:i,j∈ℕ.
Since the functionα is nondecreasing and Dxn+i,xn+j≤δ0, Δyn+i−1,yn+j−1≤δ0, for any i, j∈ℕ, we get αn≤αδ0. Thus, δn≤αδ0δn−1,and  thenδn≤αδ0nδ0,for alln∈ℕ. which implies that the sequence δn is convergent to 0.
Furthermore, for alln,m∈ℕ, we have
(22)
D
x
n
,
x
n
+
m
=
D
S
T
n
x
0
,
S
T
n
+
m
x
0
≤
δ
n
.
Thus,limn,m⟶∞Dxn,xn+m=0, and hence, xn is a D-Cauchy sequence. Since X is D-complete, there exists x∗∈X such that limn⟶∞Dxn,x∗=0.
Step 3. Lety∗=Tx∗. Since xn is a G-increasing sequence and X,D,G is G-regular, x∗∈xnG, for any n∈ℕ. Thus, x∗ and Syn−1=xn are connected.
In (7), by taking x=x∗ and y=yn−1, we obtain(23)ΔTx∗,TSyn−1≤αDx∗,Syn−1maxDx∗,Syn−1,Δyn−1,Tx∗,Δyn−1,TSyn−1.
Aslimsupt⟶0+αt<1, there exists k∈0,1 and N∈ℕ such that for all n≥N, we have αDx∗,Syn−1≤k. Thus,(24)Δy∗,yn≤kmaxDx∗,xn,Δyn−1,y∗,Δyn−1,yn.
Sincelimn⟶∞Dxn,x∗=limn⟶∞Δyn−1,yn=0, we get lim supn⟶∞Δy∗,yn≤klim supn⟶∞Δy∗,yn−1 and so lim supn⟶∞Δy∗,yn=0.
Aslim supt⟶0+αt<inf1,1/C, there exists k′∈0,inf1,1/C and N′∈ℕ such that for all n≥N′, we have αΔyn−1,y∗≤k′. Since limn⟶∞Dxn,x∗=0,(25)Dx∗,Sy∗≤Clim supn⟶∞Dxn,Sy∗=Clim supn⟶∞DSyn−1,STx∗≤Clim supn⟶∞αΔyn−1,y∗maxDxn,x∗,Δyn−1,y∗,Dx∗,Sy∗≤Ck′Dx∗,Sy∗.
We conclude thatDx∗,Sy∗=0, since Ck′<1. Then, x∗=Sy∗, and so STx∗=x∗ and TSy∗=y∗.
Sincexn∈CD,X,x∗ and yn∈CD,Y,y∗,
(26)
D
x
∗
,
x
∗
=
0
,
Δ
y
∗
,
y
∗
=
0.Remark 1.
In Theorem2, we can replace the condition STi+1x0∈STix0G by STix0∈STi+1x0G, for any i in ℕ.
For the uniqueness of the pairx∗,y∗, we establish the following.Proposition 1.
Suppose that there exists another pairx′,y′, satisfying Tx′=y′ and Sy′=x′.IfDx′,x′<∞ and Δy′,y′<∞, then Dx′,x′=0 and Δy′,y′=0. Moreover, if x∗ and x′ are connected, Dx∗,x′<∞ and Δy∗,y′<∞ then x′,y′=x∗,y∗.Proof.
By takingx=x′ and y=y′ in (7), we have(27)Dx′,x′=DSy′,STx′≤αΔy′,Tx′maxDx′,Sy′,Δy′,Tx′,Dx′,STx′,Δy′,y′=ΔTx′,TSy′≤αDx′,Sy′maxDx′,Sy′,Δy′,Tx′,Δy′,TSy′.
That is,(28)Dx′,x′≤αΔy′,y′Δy′,y′,Δy′,y′≤αDx′,x′Dx′,x′.
Therefore, we have(29)1−αΔy′,y′αDx′,x′Dx′,x′≤0,1−αΔy′,y′αDx′,x′Δy′,y′≤0.
Then,Dx′,x′=0 and Δy′,y′=0.
SinceSy∗ and x′ are connected, by taking x=x′ and y=y∗ in (7), we obtain(30)DSy∗,STx′≤αΔy∗,Tx′maxDx′,Sy∗,Δy∗,Tx′,Dx′,STx′,ΔTx′,TSy∗≤αDx′,Sy∗maxDx′,Sy∗,Δy∗,Tx′,Δy∗,TSy∗.
That is,(31)Dx∗,x′≤αΔy∗,y′maxDx′,x∗,Δy∗,y′,Δy′,y∗≤αDx′,x∗maxDx′,x∗,Δy∗,y′.
Hence,(32)Dx∗,x′≤αΔy∗,y′Δy∗,y′,Δy′,y∗≤αDx′,x∗Dx′,x∗.
Therefore, we haveDx′,x∗=0 and Δy∗,y′=0, which implies that x′,y′=x∗,y∗.Example 1.
LetX=1,3 and Y=1,2 and two mappings T:X⟶Y and S:Y⟶X be defined as follows:(33)Tx=x+12,for anyx∈X,Sy=y+1,for anyy∈Y.
Letα:0,+∞⟶0,1 such that αt=22/3. Consider the two mappings D:X×X⟶0;∞ and Δ:Y×Y⟶0;∞ defined as follows:(34)Da,b=1a−1b,Δc,d=121c−1d.
We can show thatD and Δ are generalized metrics on X and Y, respectively, with constant C≥1.
Consider the graphG on X such that VG=X and(35)EG=x,x:x∈X∪3−12n,3:n∈ℕ∪3−12n,3−12n+1:n∈ℕ.
We prove that, for anyx,y∈X×Y, such that x and Sy are connected, we have(36)DSy,STx≤αΔy,TxmaxDx,Sy,Δy,Tx,Dx,STx,ΔTx,TSy≤αDx,SymaxDx,Sy,Δy,Tx,Δy,TSy.
There are five cases.Case 1.
Fory∈1,2 and x=Sy∈2,3, we have(37)DSy,STx=1x−2x+3≤231x−1−2x+1≤223Δy,Tx,ΔTx,TSy=ΔTx,Tx=0.Case 2.
Forx=3−1/2n, for any n∈ℕ and y=2, we have (38)DSy,STx=13−2−n+1−13≤22313−2−n−13=223Dx,Sy,ΔTx,TSy=1212−2−n+1−12≤22313−2−n−13=223Dx,Sy.Case 3.
Forx=3 and y=2−1/2n, for any n∈ℕ, we have (39)DSy,STx=13−2−n−13≤2312−2−n−12=223Δy,Tx,ΔTx,TSy=1212−2−n+1−12≤2312−2−n−12=223Δy,Tx.Case 4.
Forx=3−1/2n and y=2−1/2p, for any n,p∈ℕ, such that the following can be obtained.
One can see that(40)13−α−13−β≤2312−α−12−β,for any α, β∈0,1.
By takingα=2−n+1 and β=2−p, we obtain(41)13−2−n+1−13−2−p≤2312−2−n+1−12−2−p.
Thus,(42)DSy,STx≤223Δy,Tx.
Furthermore,(43)1212−α−12−β≤22313−2α−13−2β,for any α, β∈0,1.
By takingα=2−n+1 and β=2−p+1, we obtain(44)1212−2−n+1−12−2−p+1≤22313−2−n−13−2−p.
Thus,(45)ΔTx,TSy≤223Dx,Sy.Case 5.
Forx=3−1/2n and y=2−1/2p, for any n,p∈ℕ, such that n≥p+1. In the same way as Case 4, we obtain(46)DSy,STx≤223Δy,Tx,ΔTx,TSy≤223Dx,Sy.
Letx0=2∈X, we have STnx0=3−1/2n, for any n∈ℕ. One can see that(47)δS,T,x0,D,Δ=supDSTix0,STjx0,ΔTSTix0,TSTjx0:i,j∈ℕ=supD3−12i,3−12j,Δ2−12i+1,2−12j+1:i,j∈ℕ<∞,and STi+1x0∈STix0G, for all i∈ℕ.
SinceX,D,G is D-complete and G-regular, then all assumptions of Theorem 2 are satisfied and there exists x∗,y∗=3,2 such that Tx∗=y∗ and Sy∗=x∗.Remark 2.
Theorem2 extends the main result of Chaira et al. in [3] by considering the graph G≤ on X defined by(48)x,y∈EG≤,iffx≤y.
The next corollary gives a version of Theorem2 in standard metric spaces endowed with a digraph.Corollary 1.
LetX,d and Y,δ be two metric spaces such that X is endowed with a digraph G. Consider a nondecreasing function α:0,+∞⟶0,1, such that(49)lim supt⟶0+αt<1.LetT:X⟶Y and S:Y⟶X be two mappings satisfying the following conditions:(1)
Let(50)dSy,STx≤αδy,Txmaxdx,Sy,δy,Tx,dx,STx,δTx,TSy≤αdx,Symaxdx,Sy,δy,Tx,δy,TSy,for anyx in X and y in Y, such that x and Sy are connected.(2)
X,d,G is complete and G-regular.(3)
There exists an elementx0∈X such that STi+1x0∈STix0G,for alli∈ℕ.Then, there existsx∗,y∗∈X×Y, such that Tx∗=y∗ and Sy∗=x∗; then, STx∗=x∗ and TSy∗=y∗.



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*Source: 7253759-2020-03-16.xml*