#### 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.

#### 1. Introduction

Let be a nonempty set. A binary relation “” on is said to be a preorder on if it is reflexive and transitive. In this case is called a preordered set. An element is said to be maximal in if for all , The set is called the final segment generated by .

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]). Let be a preordered set and let be a function satisfying: (S1) is decreasing;(S2) for all ;(S3) for some ;(S4)For every nondecreasing sequence with , there exists some such that for all and ;(S5) for all with . Then, there exists a maximal element .

In 1976, Caristi (see [2]) gave a generalization of Banach contraction principle where the assumption “ is continuous” is dropped and replaced by a weak assumption. Since then, various proofs, extensions, and generalizations are given by many authors (see [37]). 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 called -functions and -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. Let be a nonempty set; a function is quasidistance if we have (1) if and only if (2) for each . The pair is called a quasimetric space.

Since 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 sequence in is (1)left K-Cauchy if for every there exists such that , with , ;(2)left K-converges to , if ;(3) will be called left K-complete quasimetric if any left K-Cauchy sequence is left K-convergent.

Definition 4. Let be a quasimetric space. A mapping is said to be (1)lower semicontinuous if given any sequence in , whenever and , then ;(2)upper semicontinuous if is lower semicontinuous.

In the sequel, we recall some basic notions on graphs borrowed from [10].

Let be an arbitrary set. A directed graph, or digraph, is a pair where is a subset of the Cartesian product . The elements of are called vertices or nodes of and the elements of are the edges, also called oriented edges or arcs of . An edge of the form is a loop on . Another way to express that is a subset of is to say that is a binary relation over . Given a digraph , the set of vertices (respectively, of edges) of is denoted by (respectively, ). The digraph is said to be(i)transitive if whenever and , then ;(ii)reflexive if is a subset of .

A vertex is said to be isolated if for all vertex , we have neither nor . Given two vertices . A path in , from (or joining) to is a sequence of vertices , such that , and , for all . The integer is the length of the path . If and , the path is called a directed cycle. An acyclic digraph is a digraph which has no directed cycle.

We denote by the fact that there is a directed path in joining to .

A quasimetric space endowed with a digraph such that is denoted by .

As in [4], we use the following.

Definition 5. Let be a quasimetric space endowed with a digraph. We say that satisfies the property (OSC) if for any sequence that is convergent to and for all , , then for all .

Let be a reflexive digraph, and a function . Having in mind for each , most dominated Caristi’s functions satisfied the following conditions:(C1)N-superadditivity: and for each with and .(C2) is upper semicontinuous for each .(C3) There exists such that .(C4) There exists a function such that, for all ,

Next, we introduce a new class of functions.

Definition 6. Let be a reflexive digraph; a real function is said to be (i)-function if (C1), (C2), and (C3) hold.(ii)-function if (C1), (C2), and (C4) hold.

Remark 7. Obviously, each -function is a -function. Indeed, let . Then for each , we have which implies

#### 3. Main Results

Let be a quasimetric space endowed with a digraph. Define a binary relation on by We will use particularly the following fact.

Lemma 8. Let be a quasimetric space with a digraph and a function satisfying condition (C1); then 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. Let be a left K-complete quasimetric space endowed with a reflexive digraph satisfying the (OSC) property and a set-valued map. If there exists a -function such that for each , there exists with , then has a fixed point in .

Proof. Firstly, we show that for each increasing sequence with respect to where , there exists such that for each we get
According to the definition of , we have ; hence, Then is real convergent sequence, so . And is a left K-Cauchy sequence in since is convergent. Then, left K-converges to some The property (OSC) guarantees that for all .
Note that for each with , we get which by left K-convergence of and upper semicontinuity of , we have . Then, and for all , which leads to Thus, (S4) holds.
Define a function for each by is nonnegative function, since for each , there is such that ; then that is (S2) holds.
If ; that is, (), then , which implies (S5).
Let ; then for each , we get and since , we have Theni.e., , which implies that is nonincreasing function; that is, (S1) holds.
All assumptions of Száz principle hold; then has a maximal element . By hypothesis, there exists such that ; then we get , which implies that

We support our result by the following example.

Example 10. Consider the digraph represented in Figure 1, where Define on the quasimetric as follows: Consider the -function defined by and the set-valued mapping defined by One can see that(i) for all .(ii)For all with , we have .(iii) is a complete left -quasimetric space.(iv) is obviously -function.(v)For all , there exists such thatAll assumptions of Theorem 9 are satisfied and .

Corollary 11. Under assumptions of Theorem 9, with is only a single-valued map and for all , then has a fixed point in .

Using Remark 7, we have immediately the following.

Corollary 12. Under assumptions of Theorem 9, if is an -function, then has a fixed point in .

The following theorem improves the results of [2, 11, 12] and generalizes the main theorem of Chaira et al. [4]

Theorem 13. Let be a left K-complete quasimetric space with a reflexive digraph satisfying the (OSC) property and a lower semicontinuous function. If the mapping satisfies, for all ,then has a fixed point in .

Proof. We consider the function defined by It is clear that is a K-function. Applying Theorem 9, the proof is complete.

#### Data Availability

There were no data used to support this study.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.