#### Abstract

The purpose of this work is to extend the Knaster–Tarski fixed-point theorem to the wider field of reflexive digraph. We give also a DeMarr-type common fixed-point theorem in this context. We then explore some interesting applications of the obtained results in modular function spaces.

#### 1. Introduction

The study of fixed point results in partially ordered sets finds its root in the work of Knaster [1] and Tarski [2]. In 1955, Tarski published his work in the context of complete lattices; the result states that each monotone mapping from a complete lattice to itself has a fixed point. In [3], Abian and Brown extended the result of Knaster–Tarski to chain-complete poset with a least or largest element and showed that every order-preserving map has a fixed point.

On the contrary, several fixed-point theorems in metric spaces endowed with a partial order have been stated and studied. In 2004, Ran and Reuring (see [4]) combined successfully the Banach Contraction Principle and Knaster–Tarski fixed point. They managed to prove that every monotone mapping in a complete metric space has a fixed point provided that it satisfies contraction condition only for comparable elements. Jachymski [5] managed to prove an equivalent result of Ran and Reuring’s in a metric space endowed with a graph.

In the same vein, the authors in [6, 7] extended the result of Ran and Reuring to the case of monotone nonexpansive mappings. Their starting point was to approach the fixed point by iterative techniques and successive approximations. Recently, Espínola and Wiśnicki [8] generalized the above results in Hausdorff topological spaces endowed with partial order. The key ingredient in such a generalization is the compactness of the order intervals mixed with Knaster–Tarski fixed point. For more details, see [9, 10].

In this work, we generalize several known results in the context of topological spaces endowed with a digraph instead of partial order. For this purpose, we introduce the concept of -regular monotone mapping and we give some applications in modular function spaces of the obtained results.

#### 2. Main Results

Since the main result of this work relates topological properties to graphs, the following definition is needed. The interested reader can consult [11], for more details.

*Definition 1. *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 G is transitive whenever and , for any .(iii)A dipath of is a sequence , ,…,…with for each .(iv)A finite dipath of length from *x* to *y* is a sequence of vertices with and and .(v)A closed directed path of length from to , i.e., , is called a directed cycle.(v)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.(vi) is the set of all vertices which are contained in some path beginning at , i.e., there exist with and .We extend the notions of upper and lower bound and supremum and infimum known in the case of ordered sets to graph structures.

*Definition 2. *Let be a reflexive digraph and .(i)We define the -intervals as follows:and(ii)For a subset of , we say that is a -upper bound of if *-*lower bound of if (iii)A -upper bound of that belongs to is called -maximal element of , and a -lower bound of that belongs to is called -minimal element of (iv)We say that is a -supremum of if is a -upper bound For every -upper bound of , (v)We say that is a -infimum of if *is a**-lower bound and* *For every**-lower bound**of**,*The following example illustrates this last definition.

*Example 1. *(1)Let and . is a reflexive digraph. Set . is a -upper bound if for each thus, has no -*upper* bound. is a -lower bound if for each thus, 0 is the only -lower bound of . Moreover, since it is the only -minimal element and it is the -infimum. Since has no -upper bound the sets of -maximal and -supremum are empty.(2)Unlike the case of partially order, the -supremum of set may be not unique. Indeed, we consider , thus we have The set of -upper bound of is 1 is the *only*-maximal element of is -supremum of if and only if Recall that a collection of sets has the finite intersection property (f.i.p.), if, for every family of members of , the intersection of is nonempty provided that the intersection of all finite subfamilies of are nonempty.

*Definition 3. *Let be a reflexive digraph; a subset of is said -directed, if every finite subset of has a -upper bound in .

We then get the following generalization of the result obtained in [8] for graphs.

Lemma 1. *Let be a reflexive digraph; if has the finite intersection property for -intervals, then every -directed subset of has a -supremum.*

*Proof 1. *We consider the set , as is -directed; every finite intersection , where , is nonempty since it contains every -upper bound that is in *L* of the finite subset and as has the finite intersection property, and is nonempty.

Let us consider now the set , then, again, every finite intersection , where and is nonempty (also it contains any -upper bound that is in *L* of the finite subset ) and has the finite intersection property which is nonempty. And, it is clear that every element of is a -supremum of .

Recall that a map is said to be -monotone if for all , whenever , then . Next, we introduce the notion of -regular monotone.

*Definition 4. *Let be a set endowed with a graph a map is said to be -regular monotone; if is -monotone and for every , if and , then .

The following theorem is the cornerstone of what follows.

Theorem 1. *Let be a topological space endowed with a reflexive digraph such that -intervals are compact, and let be -regular monotone map; if there exists such that , then the set of fixed points of is not empty and has a -maximal element.*

*Proof. *Let . Then, is -directed and for all , and . Set nowthen is nonempty inductive set with respect to the order (if is a chain in then is an upper bound of in ).

By Zorn’s lemma, there exists a maximal -directed set such that and for all , , and . Since the -intervals are compacts, has the finite intersection property for -intervals; thus, has a -supremum .

Now, for all , we have and and for all . Hence, is a -upper bound of , that is, . We then claim that . Indeed if , thenis a -directed subset of that contains strictly (and thus as well) and such that, for all , and , which is in contradiction with maximality of .

As and too, we have and as ; we get then , that is, is a fixed point of .

Notice that if is a fixed point of , , and thus and as is -monotone.

*Example 2. *Let the unit disc in , i.e., , endowed with the graph , where and *iff**.*

Consider defined for all with the usual topology.

It is easy then to check that -intervals are compact, and that is -regular monotone mapping and that 1 is a fixed point for (every real positive number in is indeed a fixed point for ).

In the same way, we obtain a common fixed point for commuting family of -monotone mappings.

Theorem 2. *Let be a topological space endowed with a reflexive digraph such that -intervals are compact, and let be a family of commuting -regular monotone mappings; if there exists such that for all then the set of common fixed points of the family is nonempty and has a -maximal element.*

*Proof. *Let ; then, is -directed and we have for all and . LetThen, is a nonempty inductive set with respect to the inclusion order. Indeed, if is a chain in , then is an upper bound of in . By Zorn’s lemma, there exists a maximal -directed set such that , and for all and , we get and . As has the finite intersection property for -intervals, has a -supremum .

For all , we have and and for all and for all . Hence, is a -upper bound of , for any . That is, . We then claim that , indeed if , thenis a -directed subset of that contains strictly (and thus as well) and such that , , which is in contradiction with maximality of .

Now again, as , then, for each , and ; hence, . By commutativity, is a common fixed point for the family of mappings .

Finally, if is a common fixed point of the family of mappings then thus and , for all . Then, is -maximal element of the set of common fixed points of .

#### 3. Application to Modular Function Spaces

For the sake of completeness, we begin by recalling some definitions and properties of modular function spaces that we used later. For more details, see [12].

Let be a nonempty set and a nontrivial -ring of subsets of , and let be the smallest -algebra of subsets of such that contains such that for every and ; , where , for all .

is the linear space of -simple functions; is the set of measurable functions. We denote by the characteristic function of , where .

*Definition 5. *An even convex function is called regular convex function pseudomodular if(i)*.*(ii) is monotone, i.e., if for , for all , then *.*(iii) is orthogonally subadditive, i.e., , whenever and and *.*(iv) has the Fatou property, i.e., if , for all , then .(v) is order continuous in , i.e., and implies .Let be a regular convex function pseudomodular; we then introduce these notions:(i)A set is said -null, if .(ii)A _{property} is said to hold almost everywhere if the exceptional set is -null.(iii)We will identify pair of measurable sets whose symmetric difference is -null, as well as pair of measurable function differing only on a -null set.(iv) briefly noted .(v) is said a regular convex function modular if implies -a.e.(vi)We denote by the set of all nonzero regular convex function modulars on .

*Definition 6. **Let*:(i)We say that -converges to , and write , if , and a sequence is called -Cauchy if .(ii)A set is called -closed, if for any sequence implies .(iii)A set is called -bounded, if his diameter is finite.(iv)A set is called -compact, if for any sequence there exists a subsequence and such that -converges to .(v)A set is called -a.e.-compact, if for any sequence there exists a subsequence and such that -a.e.-converges to .(vi)A set is called -a.e.-closed, if for any sequence , -a.e. implies .

*Definition 7. *Let .

The modular function space is the vector space or briefly defined byThe *map* is defined bywhich is called norm of Luxembourg on .

The following properties play a prominent role in the study of modular function spaces.

*Definition 8. *Let :(i)We say that has the -property, if whenever , .(ii)We say that has the -type condition, if there exists such that , for any .We need the following definition of the growth function.

*Definition 9. *(see [12]). Let be a function modular; the function is defined bywhich is called the growth of .

The growth function has the following properties.

Proposition 1. *(see [12]). Let that has the -type condition, and its growth function; then,*(i)* .*(ii)

*is convex, and strictly increasing, it is then also continuous.*(iii)

*, for all .*(iv)

*, for all , where is the inverse function of .*(v)

*, for any .*

*Proofs of following theorems could be found in [12].*

Theorem 3. *Let :*(i)* is a complete normed space, and is -complete.*(ii)* iff for every .*(iii)*If there exists subsequence of such that , -a.e.*(iv)*If -a.e, then (the Fatou property).*(v)*If has the -property and for , then .**A modular is said -finite if there exists an increasing sequence of sets such that, for every and .**Let defined by*

Theorem 4. *Let if is -finite and has the - type condition; then, for any ,*(1)* if f -a.e.*(2)*(3)**and if is a sequence in that is -a.e. convergent to , then .**Moreover, if , then there exists a subsequence that converges -a.e. to .**Here is the first application of our main result to modular function spaces.*

Theorem 5. *Let , that has the -property and a digraph on such that -intervals are -compact. Let be a family of commuting -regular monotone mappings; if there exists such that , then the set of common fixed points of the family is not empty and has a -maximal element.*

*Proof. *As has the -property, then the -convergence is equivalent to convergence in the Banach space , which means that every -compact subset of is a compact in ; we can then apply Theorem 2.

Requiring more conditions on the function modular one can suppose that -intervals are only -a.e. compact.

Theorem 6. *Let , a -finite function modular, that has the -type condition and a digraph on such that -intervals are -a.e.-compact. Let be a family of commuting -regular monotone mappings; if there exists such that , then the set of common fixed points of the family is not empty and has a -maximal element.*

*Proof. *Indeed, Theorem 4 states that is a b-metric space, and then sequential compactness is equivalent to compactness (the usual argument, which proves that fact for metric spaces, still holds in b-metric spaces). Now, if a subset of is .a.e.-compact, then from every sequence of elements of one can extract a subsequence that converges -a.e.; then, by Theorem 4, one can extract a subsequence that converges in the b-metric space . That is, is sequentially compact in and thus compact, which implies that -intervals are compact for the topology of the b-metric space , then using Theorem 2 we get the result.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

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