# Caristi-Type Fixed Point Theorem over Száz Principle in Quasi-Metric Space with a Graph

Authors: Karim Chaira; Abderrahim Eladraoui; Mustapha Kabil; Samih Lazaiz
Journal: Journal of Mathematics
 (2019)

---

## Abstract

The aim of this paper is to generalize Caristi’s fixed point theorem in a K-complete quasi-metric space endowed with a reflexive digraph by using Száz maximum principle. An example is given to support our main result.


---

## Body

## 1. Introduction

LetX be a nonempty set. A binary relation “≼” on X is said to be a preorder on X if it is reflexive and transitive. In this case X,≼ is called a preordered set. An element x∈X is said to be maximal in X if for all y∈X, (1)x≼y⇒x=y.The set S≼+x=z∈X;x≼z is called the final segment generated by x.In 2007, Á. Száz (see [1]) generalized the Brézis-Browder principle in the setting of the preorder sets and gave a generalized version of Caristi’s theorem.Theorem 1 (Száz [1]).
LetX,≼ be a preordered set and let Φ:X×X→R¯ be a function satisfying: (S1)
x↦supy∈S≼+x⁡Φx,y is decreasing;(S2)
-∞<supy∈S≼+x⁡Φx,y for all x∈X;(S3)
supy∈S≼+a⁡Φa,y<∞ for some a∈X;(S4)
For every nondecreasing sequencexnn∈N⊂X with x0=a, there exists some x∈X such that xn≼x for all n∈N and liminfn→∞⁡Φxn,xn+1=0;(S5)
0<Φx,y for all x,y∈X with x≺y.
Then, there exists a maximal elementx^∈X.In 1976, Caristi (see [2]) gave a generalization of Banach contraction principle where the assumption “T:X→X is continuous” is dropped and replaced by a weak assumption. Since then, various proofs, extensions, and generalizations are given by many authors (see [3–7]). It is worth mentioning that Caristi’s fixed point theorem is equivalent to the Ekeland’s variational principle [8].In this work, we use the Száz principle to give a more generalized version of Caristi’s fixed point theorem in the setting of the quasimetric space with a graph. For that, we introduce a new class of functions calledK-functions and M-functions which generalize the notion of dominated function in Caristi’s theorem. Also, we give an improved result in the framework of set-valued mappings and we derive some known results as corollaries.


## 2. Preliminaries

Definition 2.
LetX be a nonempty set; a function δ:X×X→0,∞ is quasidistance if we have (1)
δx,y=0 if and only if x=y(2)
δx,z≤δx,y+δy,z for each x,y,z∈X.
The pairX,δ is called a quasimetric space.Sinceδx,y=δy,x is not necessarily satisfied in such spaces, there are many characterizations of completeness in this setting (e.g., [9]). Following the framework of [8], we have the following.Definition 3.
A sequencexnn in X,δ is (1)
left K-Cauchy if for everyε>0 there exists n0∈N such that ∀n,m∈N, with n0≤n≤m, δxn,xm<ε;(2)
left K-converges tox, if limn→∞⁡δx,xn=0;(3)
X,δ will be called left K-complete quasimetric if any left K-Cauchy sequence is left K-convergent.Definition 4.
Let(X,δ) be a quasimetric space. A mapping φ:X→R+ is said to be (1)
lower semicontinuous if given any sequencexnn in X, whenever limnxn=x¯ and limnφ(xn)=r, then φ(x¯)≤r;(2)
upper semicontinuous if-φ is lower semicontinuous.In the sequel, we recall some basic notions on graphs borrowed from [10].LetV be an arbitrary set. A directed graph, or digraph, is a pair G=(V,E) where E is a subset of the Cartesian product V×V. The elements of V are called vertices or nodes of G and the elements of E are the edges, also called oriented edges or arcs of G. An edge of the form (v,v) is a loop on v. Another way to express that E is a subset of V×V is to say that E is a binary relation over V. Given a digraph G, the set of vertices (respectively, of edges) of G is denoted by V(G) (respectively, E(G)). The digraph G=(V,E) is said to be(i)
transitive if whenever(x,y)∈E and (y,z)∈E, then (x,z)∈E;(ii)
reflexive ifΔ≔{(v,v)∣v∈V} is a subset of E.A vertexx is said to be isolated if for all vertex y≠x, we have neither (x,y)∈E nor (y,x)∈E. Given two vertices x,y∈V. A path in G, from (or joining) x to y is a sequence of vertices p={ai}0≤i≤n, n∈N∗ such that a0=x, an=y and (ai,ai+1)∈E, for all i∈{0,1,…,n-1}. The integer n is the length of the path p. If x=y and n>1, the path p is called a directed cycle. An acyclic digraph is a digraph which has no directed cycle.We denote byy∈[x]G the fact that there is a directed path in G joining x to y.A quasimetric space(X,δ) endowed with a digraph G such that V(G)=X is denoted by (X,δ,G).As in [4], we use the following.Definition 5.
Let(X,δ,G) be a quasimetric space endowed with a digraph. We say that X satisfies the property (OSC) if for any sequence {xn}⊆X that is convergent to x∈X and for all n∈N, xn+1∈[xn]G, then x∈[xn]G for all n∈N.LetX,G be a reflexive digraph, and a function Φ:X×X→R¯. Having in mind Φx,y=φx-φy for each x,y∈X, most dominated Caristi’s functions satisfied the following conditions:(C1)
N-superadditivity:Φx,x=0 and Φx,y+Φy,z≤Φx,z for each x,y,z∈X with y∈[x]G and z∈[y]G.(C2)
y↦Φx,y is upper semicontinuous for each x∈X.(C3)
There existsa∈X such that supy∈[a]G⁡Φa,y<∞.(C4)
There exists a functionψ:X→R such that, for all x,y∈X, (2)y∈xG⇒Φx,y≤ψx.Next, we introduce a new class of functions.Definition 6.
LetX,G be a reflexive digraph; a real function Φ:X×X→R¯ is said to be (i)
K-function if (C1), (C2), and (C3) hold.(ii)
M-function if (C1), (C2), and (C4) hold.Remark 7.
Obviously, eachM-function is a K-function. Indeed, let a∈X. Then for each y∈[a]G, we have Φa,y≤ψa which implies (3)supy∈aG⁡Φa,y≤ψa<∞.


## 3. Main Results

Let(X,δ,G) be a quasimetric space endowed with a digraph. Define a binary relation on X by (4)x≼Φy⟺δx,y≤Φx,yy∈xG.We will use particularly the following fact.Lemma 8.
Let(X,δ,G) be a quasimetric space with a digraph and Φ:X×X→R¯ a function satisfying condition (C1); then (X,δ,≼Φ) is a preordered quasimetric space.The following result is a generalization and an extension of Caristi’s theorem in the setting of the quasimetric spaces with a graph.Theorem 9.
LetX,δ,G be a left K-complete quasimetric space endowed with a reflexive digraph satisfying the (OSC) property and T:X→2X a set-valued map. If there exists a K-function Φ:X×X→R¯ such that for each x∈X, there exists y∈Tx∩[x]G with δ(x,y)≤Φ(x,y), then T has a fixed point in X.Proof.
Firstly, we show that for each increasing sequencexnn with respect to ≼Φ where x0=a, there exists x¯ such that for each n∈N we get xn≼Φx¯.
According to the definition of≼Φ, we have xn≼Φxn+1⇒δxn,xn+1≤Φxn,xn+1; hence, (5)∑i=0nδxi,xi+1≤∑i=0nΦxi,xi+1≤Φa,xn+1≤supy∈S≼Φ+aΦa,y
Then∑i=0n‍Φxi,xi+1 is real convergent sequence, so limn→∞⁡Φxn,xn+1=0. And xnn is a left K-Cauchy sequence in X since ∑i=0∞‍δxi,xi+1 is convergent. Then, xnn left K-converges to some x¯. The property (OSC) guarantees that x¯∈[xn]G for all n∈N.
Note that for eachn,m∈N with n≤m, we get (6)xn≼Φxm⇒δxn,xm≤Φxn,xmwhich by left K-convergence of xnn and upper semicontinuity of y↦Φx,y, we have limsupm→∞⁡Φxn,xm≤Φxn,x¯. Then, δxn,x¯≤Φxn,x¯ and x¯∈[xn]G for all n∈N, which leads to (7)xn≼Φx¯,for  everyn∈N.Thus, (S4) holds.
Define a functionφ:X→0,∞ for each x∈X by(8)φx=supy∈xG⁡Φx,yφ is nonnegative function, since for each x∈X, there is y∈Tx such that δx,y≤Φx,y; then 0≤φx that is (S2) holds.
Ifx≺Φy; that is, (x≼Φyandx≠y), then 0<δx,y≤Φx,y, which implies (S5).
Letx≼Φy; then for each z∈[y]G, we get (9)Φx,y+Φy,z≤Φx,zand since Φx,y≥0, we have (10)Φy,z≤Φx,zThen(11)supz∈yG⁡Φy,z≤supz∈xG⁡Φx,zi.e., φy≤φx, which implies that φ is nonincreasing function; that is, (S1) holds.
All assumptions of Száz principle hold; thenX has a maximal element x^. By hypothesis, there exists y^∈Tx^ such that x^≼Φy^; then we get x^=y^, which implies that x^∈Tx^.We support our result by the following example.Example 10.
Consider the digraphG=(X,E) represented in Figure 1, where (12)X=0,1∪1-12n:n∈N,E=Δ∪0,1∪1-12n,1,1-12n,1-12n+1:n∈N∗
Define onX the quasimetric δ as follows: (13)δx,y=y-xify∈xG;1ifnot.
Consider theK-function Φ:X×X→R¯ defined by (14)Φx,y=ln2y+1x+1+y-xand the set-valued mapping T:X→2X defined by (15)T0=1;T1-12n=1-12nk:k∈N;T1=X.One can see that(i)
x∈[1]G for all x∈X.(ii)
For alln,m∈N∗ with n>m, we have 1-1/2n∈1-1/2mG.(iii)
(X,d) is a complete left K-quasimetric space.(iv)
Φ is obviously K-function.(v)
For allx∈X, there exists y∈Tx∩[x]G such that(16)dx,y≤ln2y+1x+1+y-x
All assumptions of Theorem9 are satisfied and 1/2∈T1/2.Figure 1
Graph G (the loops and the isolated vertices are not represented).Corollary 11.
Under assumptions of Theorem9, with T:X→X is only a single-valued map and for all x∈X, (17)Tx∈xG⇒δx,Tx≤Φx,Tx,then T has a fixed point in X.Using Remark7, we have immediately the following.Corollary 12.
Under assumptions of Theorem9, if Φ:X×X→R¯ is an M-function, then T has a fixed point in X.The following theorem improves the results of [2, 11, 12] and generalizes the main theorem of Chaira et al. [4]Theorem 13.
LetX,δ,G be a left K-complete quasimetric space with a reflexive digraph satisfying the (OSC) property and φ:X→0,∞ a lower semicontinuous function. If the mapping T:X→X satisfies, for all x∈X,(18)Tx∈xG⇒δx,Tx≤φx-φTx,then T has a fixed point in X.Proof.
We consider the functionϕ:X×X→R¯ defined by (19)ϕx,y=φx-φy,∀x,y∈X
It is clear thatϕ is a K-function. Applying Theorem 9, the proof is complete.



---

*Source: 8956083-2019-07-01.xml*