Common Fixed-Point Theorems in Modular Function Spaces Endowed with Reflexive Digraph

&e 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 chaincomplete 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. &ey 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. &eir starting point was to approach the fixed point by iterative techniques and successive approximations. Recently, Espı́nola and Wiśnicki [8] generalized the above results in Hausdorff topological spaces endowed with partial order. &e 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 G-regular monotone mapping and we give some applications in modular function spaces of the obtained results.


Introduction
e 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 chaincomplete 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. ey 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. eir 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.
e 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 G-regular monotone mapping and we give some applications in modular function spaces of the obtained results.
and [(y, z) ∈ E(G)]⇒(x, z) ∈ E(G), for any x, y, z ∈ V(G). (iii) A dipath of G is a sequence a 0 , a 1 ,. . .a n ,. . .with (a i , a i+1 ) ∈ E(G) for each i ∈ N. (iv) A finite dipath of length n from x to y is a sequence of n + 1 vertices (a 0 , a 1 , . . . , a n ) with (a i , a i+1 ) ∈ E(G) and x � a 0 and y � a n . (v) A closed directed path of length n > 1 from x to y, i.e., x � y, is called a directed cycle.
joining any two of its vertices, and it is weakly connected if G is connected. (vi) [x] G is the set of all vertices which are contained in some path beginning at x, i.e., y ∈ [x] G ⇔ there exist (a 0 , a 1 , . . . , a n ) with (a i , a i+1 ) ∈ E(G) and x � a 0 .
We extend the notions of upper and lower bound and supremum and infimum known in the case of ordered sets to graph structures.
(i) We define the G-intervals as follows: and thus, 0 is the only G-lower bound of A 1 . Moreover, since 0 ∈ A 1 it is the only G-minimal element and it is the G-infimum.
Since A 1 has no G-upper bound the sets of G-maximal and G-supremum are empty.
(2) Unlike the case of partially order, the G-supremum of set may be not unique. Indeed, we consider Recall that a collection of sets E has the finite intersection property (f.i.p.), if, for every family F of members of E, the intersection of F is nonempty provided that the intersection of all finite subfamilies of F are nonempty.
We then get the following generalization of the result obtained in [8] for graphs.
. . x n and as G has the finite intersection property, and M is nonempty.
Let us consider now the set . . x n ∈ L and y 1 , y 2 , . . . y n ∈ M is nonempty (also it contains any G-upper bound that is in L of the finite subset x 1 , x 2 , . . . x n ) and G has the finite intersection property M ′ which is nonempty. And, it is clear that every element of M ′ is a G-supremum of L.
Recall that a map T: Next, we introduce the notion of G-regular monotone. □ Definition 4. Let X be a set endowed with a graph G � (V(G), E(G)) a map T: X ⟶ X is said to be G-regular monotone; if T is G-monotone and for every e following theorem is the cornerstone of what follows.
then the set of fixed points of T is not empty and has a G-maximal element.
is a G-directed subset of X that contains strictly L 0 (and thus J as well) and such that, for all x ∈ L, T(x) ∈ L and T(x) ∈ [x] G , which is in contradiction with maximality of L 0 . As s ∈ L 0 and T(s) ∈ L 0 too, we have s ∈ [T(s)] G and as T(s) ∈ [s] G ; we get then T(T(s)) � T(s), that is, T(s) is a fixed point of T.
Notice that if z is a fixed point of T, z ∈ L 0 , and thus s ∈ [z] G and T(s) ∈ [T(z)] as T is G-monotone.
Consider T: X ⟶ X defined T(x) � |x| for all x ∈ X with the usual topology.
It is easy then to check that G-intervals are compact, and that T is G-regular monotone mapping and that 1 is a fixed point for T (every real positive number in X is indeed a fixed point for T).
In the same way, we obtain a common fixed point for commuting family of G-monotone mappings.
Theorem 2. Let X be a topological space endowed with a reflexive digraph G � (V(G), E(G)) such that G-intervals are compact, and let T λ : X ⟶ X, λ ∈ Λ be a family of commuting G-regular monotone mappings; if there exists for all λ ∈ Λ then the set of common fixed points of the family (T λ ) λ∈Λ is nonempty and has a G-maximal element.
Proof. Let L � T λ 1 · T λ 2 · . . . · T λ n (x 0 ): n ∈ N and λ 1 . . . λ n ∈ Λ ; then, L is G-directed and we have T λ (x) ∈ L and T λ (x) ∈ [x] G for all x ∈ L and λ ∈ Λ. Let en, F is a nonempty inductive set with respect to the inclusion order. Indeed, if (J i ) i∈I is a chain in F, then ∪ i∈I J i is an upper bound of (J i ) i∈I in F. By Zorn's lemma, there exists a maximal G-directed set L 0 such that L ⊂ L 0 , and for all x ∈ L 0 and λ ∈ Λ, we get T λ (x) ∈ L 0 and T λ (x) ∈ [x] G . As G has the finite intersection property for G-intervals, L 0 has a G-supremum s.
For all x ∈ L 0 , we have s ∈ [x] G and T λ (s) ∈ [T λ (x)] G and T λ (x) ∈ [x] G for all x ∈ L 0 and for all λ ∈ Λ. Hence, T λ (s) is a G-upper bound of L 0 , for any λ ∈ Λ.
Finally, if z is a common fixed point of the family of mappings (T λ ) λ∈Λ then z ∈ L thus s ∈ [z] G and T μ (s) ∈ [T μ (z)] G , for all μ ∈ Λ. en, T μ (s) is G-maximal element of the set of common fixed points of (T λ ) λ∈Λ .

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 Ω be a nonempty set and P a nontrivial δ-ring of subsets of Ω, and let Σ be the smallest σ-algebra of subsets of Ω such that Σ contains P such that E∩A ∈ P for every E ∈ P and A ∈ Σ; K n ↑Ω, where K n ∈ P, for all n.
E is the linear space of P-simple functions; M ∞ is the set of measurable functions. We denote by 1 A the characteristic function of A, where A ⊂ Ω.
Let ρ be a regular convex function pseudomodular; we then introduce these notions: (ii) A property (P) is said to hold ρ almost everywhere if the exceptional set is ρ-null. (iii) We will identify pair of measurable sets whose symmetric difference is ρ-null, as well as pair of measurable function differing only on a ρ-null set.
briefly noted M. (i) We say that (f n ) n ∈ L ρ ρ-converges to f, and write Definition 7. Let ρ ∈ R. e modular function space is the vector space L ρ (Ω, Σ) or briefly L ρ defined by e map ‖.‖ ρ : L ρ ⟶ [0, +∞) is defined by which is called norm of Luxembourg on L ρ . e following properties play a prominent role in the study of modular function spaces.

Definition 8. Let ρ ∈ R:
(i) We say that ρ has the Δ 2 -property, if ρ(2f n ) ⟶ 0 whenever We need the following definition of the growth function.
A modular ρ is said σ-finite if there exists an increasing sequence of sets K n ∈ P such that, for every n ∈ N, 0 < ρ(K n ) < ∞ and Ω � ∪ n∈N K n .
Let d: Theorem 4. Let ρ ∈ R if ρ is σ-finite and has the Δ 2 -type condition; then, for any f, g ∈ L ρ , and if (f n ) n is a sequence in L ρ that is ρ-a.e. convergent to f, then lim n⟶+∞ d(f n , f) � 0. Moreover, if lim n⟶+∞ d(f n , f) � 0, then there exists a subsequence (f n k ) k that converges ρ-a.e. to f.
Here is the first application of our main result to modular function spaces.
Theorem 5. Let ρ ∈ R, that has the Δ 2 -property and G a digraph on L ρ such that G-intervals are ρ-compact. Let T λ : L ρ ⟶ L ρ , λ ∈ Λ be a family of commuting G-regular monotone mappings; if there exists f 0 ∈ L ρ such that T λ (f 0 ) ∈ [f 0 ] G , ∀λ ∈ Λ, then the set of common fixed points of the family (T λ ) λ∈Λ is not empty and has a G-maximal element.
Proof. As ρ has the Δ 2 -property, then the ρ-convergence is equivalent to convergence in the Banach space (L ρ , ‖.‖ ρ ), which means that every ρ-compact subset of L ρ is a compact in (L ρ , ‖.‖ ρ ); we can then apply eorem 2.
Requiring more conditions on the function modular ρ one can suppose that G-intervals are only ρ-a.e. compact. □ Theorem 6. Let ρ ∈ R, a σ-finite function modular, that has the Δ 2 -type condition and G a digraph on L ρ such that G-intervals are ρ-a.e.-compact. Let T λ : L ρ ⟶ L ρ , λ ∈ Λ be a family of commuting G-regular monotone mappings; if there exists f 0 ∈ L ρ such that T λ (f 0 ) ∈ [f 0 ] G , ∀λ ∈ Λ, then the set of common fixed points of the family (T λ ) λ∈Λ is not empty and has a G-maximal element.
Proof. Indeed, eorem 4 states that (L ρ , d) 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 K of L ρ is ρ.a.e.-compact, then from every sequence of elements of K one can extract a subsequence that converges ρ-a.e.; then, by eorem 4, one can extract a subsequence that converges in the b-metric space (L ρ , d). at is, K is sequentially compact in (L ρ , d) and thus compact, which implies that G-intervals are compact for the topology of the b-metric space (L ρ , d), then using eorem 2 we get the result.

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