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.