Research Article | Open Access

# Some Fixed Point Theorems in Modular Function Spaces Endowed with a Graph

**Academic Editor:**Wei-Shih Du

#### Abstract

The aim of this paper is to give fixed point theorems for -monotone -nonexpansive mappings over -compact or -a.e. compact sets in modular function spaces endowed with a reflexive digraph not necessarily transitive. Examples are given to support our work.

#### 1. Introduction

Let be a nonempty set. We denote by the set of subsets of . An element of is said to be a fixed point of a self-mapping on , if . For a set-valued mapping , we call a fixed point of the set-valued mapping every element of that verify .

The two most important results in fixed point theory, are without contest, the Banach contraction principle (BCP for short) and Tarski’s fixed point theorem. Since their appearances, they were subject of many generalizations, either by extending the contractive condition for the B.C.P., or changing the structure of the space itself. For example, in the case of B.C.P., Ran and Reurings in [1] have obtained a fixed point result for a relaxed contraction condition in metric spaces endowed with a partially order relation, i.e., a contraction only for comparable elements. Jachymski in [2] got a further generalization of Ran and Reurings result by replacing the partial order by a relaxed type of graph in metric spaces, where Nadler [3] managed to give an equivalent form of the B.C.P. for multi-valued mappings.

In the beginning of 1930’s, Orlicz and Birnbaum considered the space

where is a convex increasing function, such that . So, become a generalization of spaces, which corresponds to the particular case where and . Nevertheless, the formula that gives the norm over , does not establish a norm over . Thereby, Nakano in [4, 5], captured the essence of the good behavior of the quantity , what he called modular function, and gave some characterization of the geometry of these spaces, see [6].

Fixed point theory in modular function spaces was first studied by Khamsi et al. in [7], we find there an outline of a fixed point theory for -nonexpansive mappings defined on some subsets of modular function spaces. Recently, Alfuraidan in [8], gave some extensions of fixed point theorems in modular function spaces endowed with partial order relation, namely Ran and Reurings result in this context.

On the other hand, the combination of metric fixed point theory and graph theory allows Jachymski in [2] to give an extension of Banach Contraction Principle in a metric space endowed with a graph. Souayah et al. in [9], after introducing the notion of -contraction they were able to investigate the existence and uniqueness of the fixed point for such contractions in -metric space endowed with a graph, in particular they generalize the results of Jachymski. Recently, in [10] the authors proved the existence of a unique best proximity point for some contractive type mappings in rectangular metric spaces endowed with a graph structure.

In this paper, we generalize all the results obtained by Alfuraidan in [8]. We consider modular function spaces endowed with graph which satisfy a path connectivity instead of adjacency, and we have got new fixed point theorems for -monotone set-valued mappings. Examples are given to support our work.

#### 2. Preliminaries

Let be a modular function space where is a nonzero regular modular function see [6, 11, 12] for more details. Recall that for all , satisfies the following properties:(i);(ii) if ;(iii).

If satisfies the above conditions, we note then . Furthermore, the associated norm defined using modular function is called Luxembourg norm and we have:

The following definitions will be needed in the sequel.

*Definition 1 (see [11]). *Let .(1)We say that -converges to , and write: , if , and a sequence is called -Cauchy if .(2)A set is called -closed, if for any sequence , implies .(3)A set is called -bounded, if his diameter is finite.(4)A set is called -compact, if for any sequence there exists a subsequence and such that -converges to .

If , will denote the set of -closed subsets of , and , the set of -compact subsets of .

*Definition 2 (see [11]). *Let , We say that has the -type condition, if there exists such that: , for any .

It is known that the -convergence and -convergence are different in general, but if we assume that has the -type condition we have the equivalence. That is, the Luxembourg norm convergence and -convergence are equivalent. (see [11, Lemma 3.2]).

The following lemma will be very useful along this work.

Lemma 1 (see [13]). *Let be convex and satisfy the -type condition. Let , such that**where is arbitrary nonzero constant and , then is Cauchy for and -Cauchy.*

Theorem 1 (see [11]). *Let .*(i)* is a complete normed space, and is -complete.*(ii)* iff for every .*(iii)*If has the -property and for , then .*

We recall the definition of digraph, the interested reader can consult the book [14] for more details.

*Definition 3. *A directed graph or digraph is determined by a nonempty set of its vertices and the set of its directed edges. A digraph is reflexive if each vertex has a loop. Given a digraph .(i)If whenever , then the digraph is called an oriented graph.(ii)A digraph is transitive whenever and , for any .(iii)A dipath of is a sequence , ,…, ,…with for each .(iv)A finite dipath of length from to is a sequence of vertices with and , .(v)A closed directed path of length from to , i.e., , is called a directed cycle.(vi)A digraph is connected if there is a finite (di)path joining any two of its vertices and it is weakly connected if is connected.(vii) is the set of all vertices which are contained in some path beginning at . (i.e. ).

It seems that the graph theory when coupled with the classical metric fixed point theory leads to a new interesting theory, following the works of [2, 8] we introduce the following definitions.

*Definition 4. *Let , a nonempty subset and a set-valued mapping.(i)We say that is a -monotone -contraction, if there exists such that for any with , and any ; there exists verifying and .(ii)We say that is a -monotone -nonexpansive mapping, if for any with , and any ; there exists verifying and .

Note that if is only a single valued and the graph is transitive we get the notion of edge preserving -contraction (resp. -nonexpansive) mapping.

*Definition 5. *Let be a nonempty subset of and be a digraph such that . If for any sequence in that -converges to , we have for every , provided for every ; then we say that has the -property.

#### 3. Fixed Point Results for -Contractions Mappings

We start this section by the following result which will be useful in the sequel.

Proposition 1. *Let . Let be a nonempty -closed subset, a reflexive digraph such that . Let be a -monotone -contraction mapping and suppose that has a fixed point, then if for some where is a fixed point for such that ; there exists a sequence such that ; and -converges to .*

*Proof. *Since and where is -monotone -contraction, there exists such that and .

By induction we construct a sequence such that and for each i.e.

for all . And since , we get , -converges to .

If has the -type condition then for every . Thus, we get the following result which is a generalization of [8, Theorem 3.1].

Theorem 2. *Let be convex and satisfies the -type condition. Let be a nonempty -closed subset, a reflexive digraph such that , with the ()-property. Let be a -monotone -contraction mapping and .**If , then has a fixed point, moreover, if then has a fixed point .*

*Proof. *Let , there exists then such that and as is -monotone -contraction, there exists such that and

In the same way, there exists such that and and by induction we construct a sequence with and such that , which implies . Hence by the Lemma 1, is -Cauchy, and as is -complete then -converges to some as is -closed. And since has the ()-property then for each and as is -monotone -contraction, there exists such that for every :

And then clearly, -converges to zero in thus and both converge to zero in which mean that converges to in as is a norm on , therefore -converges to and then as and is -closed i.e., is a fixed point of .

Note that as for every , in particular thus has a fixed point.

*Example 1. *Let the set of measurable functions such that with a.e. in . It is known that is a normed linear space with the normThen, is a modular function space with the -condition. Let be the -closed ball centered at with radius , that is the setConsider the digraph defined by for every . Thus we get,It is clear that this binary relation defines a reflexive digraph without being a partially order (i.e., without antisymmetric condition).

Let and define by(i)For all , we have , which implies that . Moreover, is -closed since .(ii), if , let for any we have two cases: or . If choose , hence we have the -monotonocity:and the -contraction condition for Now, if let the -monotonocity and -contraction conditions are obvious.

For the ()-property. Let be a sequence in such that, and since we get for all ,Furthermore,implies that , thus for all , hence . That is has the ()-property. Then the set-valued mapping satisfies all the conditions of Theorem 2 hence it has a fixed point, namely .

Corollary 1. *Let be convex and satisfies the -type condition. Let be a nonempty -closed subset, a reflexive digraph such that , with the ()-property. Let be a -monotone -contraction mapping. If has a mother vertex, (a vertex such that all other vertices in can be reached by a path from ), then has a fixed point.*

*Proof. *Indeed for such , , for all and then , we then get the result using Theorem 2.

#### 4. Fixed Point Results for -Nonexpansive Mappings

Recall the notion of approximated fixed point sequence.

*Definition 6. *We say that has an approximated fixed point sequence, if there exists such that for every , there exists satisfying .

*Definition 7. *A digraph is said to be -convex if for all and we have .

If the reflexive digraph is transitive and antisymmetric, i.e., being a partially order, then the most ordered Banach spaces enjoy this property. The following result gives a sufficient condition to obtain an approximated fixed point sequence for -convex digraph.

Proposition 2. *Let be convex and satisfies the -type condition. Let be a nonempty convex, -closed, and -bounded subset, a -convex reflexive digraph such that , withe the ()-property.*

Let be a -monotone -nonexpansive mapping, if is nonempty; then has an approximated fixed point sequence.

*Proof. *Let and for each define:for any . Then is -closed since ifMoreover, is nonempty for any as .

Let such that , since is -monotone -nonexpansive, we get for all there exists such that andthus, and as is convex:i.e., is -monotone -contraction. Using Theorem 2, has a fixed point . Thus such that and then, as is convex :Choosing for gives the result.

Now with more restrictions on the graph , we obtain a stronger version of the above result (Proposition 2).

*Definition 8. *Let , a digraph is said to be compatible with the vector structure of , if for every such that then and .

Notice that if is compatible with the vector structure of then it is -convex. Indeed, for every such that and then , and thus for every .

Lemma 2. *Let be convex and satisfies the -type condition. Let be a nonempty convex, -closed, and -bounded subset, a reflexive digraph compatible with the vector structure of such that , with the ()-property. Let be a -monotone -nonexpansive mapping, if is nonempty; then there exist two sequences and such that**and .*

*Proof. *Let we define , as seen in the proof of Proposition 2 is a -monotone -contraction mapping, is -closed for every then has a fixed point such that as we have already seen in Theorem 2. Then, there exists such thatthus, it comes clearly that . In addition:as is convex.

Clearly . Let define , for all . As above admits a fixed point such that and then there exists such that , thus and .

By induction on , if we define the set-valued mapping to be

we get the existence of a fixed point of such that , thus there exists such that and then and . Since is - bounded we get the result.

We then apply Lemma 2 to get a new fixed point result for -monotone nonexpansive mapping.

Theorem 3. *Let be convex, satisfies the -type condition. Let be a nonempty convex, -compact, and -bounded subset, a reflexive digraph compatible with the vector structure of with the ()-property. Let be a -monotone -nonexpansive mapping, if is nonempty, then has a fixed point.*

*Proof. *The preceding lemma (Lemma 2) establish the existence of sequences and such that , for every , and .

Since is -compact, there exists a sub-sequence of that -converges to some , but as and the -convergence is equivalent to the convergence in (due to the -type condition, see [11, Proposition 3.13]); then -converges to . Thus one can suppose that and both -converge to , and as for every , using the ()-property we get for every .

Now, since is -monotone -nonexpansive mapping, we have for all there exists such that and and thus as . Since is -closed because it is -compact, we obtain , i.e., is a fixed point for .

*Example 2. *Consider the modular function space the space of all square-integrable functions over , and letfor every . It is clear that is convex modular function satisfying the -type condition.

Let be the set of all constant maps of such that . Thus, is -bounded and convex subset of . For the -compactness it is enough to show that satisfies the Riesz-Fréchet-Kolmogorov’s conditions (see [15, Theorem IV.25]) which is obvious. Let be the directed graph defined by:It is clear that is a reflexive digraph compatible with the vector structure of . Moreover, it has the ()-property. Indeed, let be a sequence that -converges to in with the conditionThen there exists a subsequence which is pointwise convergent almost everywhere to . Thus,Moreover, for each there exists such thatthen for all , a.e., hence has the ()-property. Note that the digraph is not even tranisitive.

Let be the mapping defined by . It is clear that is -monotone -nonexpansive mapping. Thus all the conditions of Theorem 3 hold, so has a fixed point namely .

#### 5. Fixed Point Theorem for -A.E. Compact Subsets

The -compactness assumption maybe relaxed using the weak concept of -a.e. compactness. Of course, if a subset set of is -compact then it is -a.e. compact, see [11, Proposition 3.13] for more details.

*Definition 9 (see [11]). *Let .(i)A set is called -a.e. compact, if for any sequence there exists a subsequence and such that -a.e. converges to .(ii)A set is called -a.e. closed, if for any sequence ,, -a.e. implies .

Theorem 4 (see [11]). *Let .*(i)*If there exists subsequence of such that , -a.e.*(ii)*If -a.e, then (The Fatou property).*

We need the following definition of the growth function.

*Definition 10 (see [11, 13]). *Let be a function modular, the function defined by:is called the growth of .

The growth function has the following properties.

Proposition 3 (see [11, 16]). *Let that has the -type condition, and its growth function, then:*(i)*.*(ii)* is convex, and strictly increasing, it is then also continuous.*

We then get the following lemma.

Lemma 3. *Let that has the -type condition, and and be two sequences in , such that , then:*

*Proof. *It is obvious that for and we have: . Let , then for every we have:and then,as is continuous we then get: letting .

The same arguments give: which ends the proof.

*Definition 11 (Opial property). *Let , is said satisfying the -a.e. Opial property if for every which -a.e. converges to , and such that for some , then we have:for every not equal to .

The following relaxed definition replace the ()-property.

*Definition 12. *Let be a nonempty set and a digraph such that .

If for any sequence in which -a.e. converges to , and such that for every ; we have for every ; then we say that has the -property.

*Remark 1. *If we substitute “()-Property” by “-Property” in the Theorem 2, Proposition 2, and Lemma 2; then all these results hold. This is due to the fact that for every -convergence sequence there exists a sub-sequence that -a.e. converges.

We conclude this paper by the -a.e. compact version of Theorem 3.

Theorem 5. *Let be convex, satisfies the -type condition and the -a.e. Opial property. Let be a nonempty convex, -a.e. compact, and -bounded subset, a reflexive digraph compatible with the vector structure of such that and satisfies the -property. Let be a -monotone -nonexpansive mapping, if is nonempty, then has a fixed point.*

*Proof. *By Lemma 2 and Remark 1, there exist two sequences and in such that:for every , and -a.e. converges to .

As is -a.e compact, there exists a sub-sequence that -a.e converges to some in . But then as , there exists a sub-sequence of that -a.e. converges to . Thus both and -a.e. converge to . So, without loss of generality, we may suppose that both and -a.e. converge to .

Now the -property gives . Then for every , there exists such that andMoreover, as which is -compact (since ), it admits a sub-sequence that we will keep noting that -converges to some . Applying Lemma 3 we get,So, by letting in the inequality (37), we obtainThen if the -a.e. Opial property giveswhich is a contradiction. Then necessarily and as which is -compact, thus -closed, hence , i.e., is a fixed point for .

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2020 Jaauad Jeddi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.