Common fixed point for some generalized contractive mappings in a modular metric space with a graph

In this paper, we investigate the existence and the uniqueness of a common fixed point of a pair of self-mappings satisfying new contractive type conditions on a modular metric space endowed with a reflexive digraph. An application is given to show the use of our main result.


Introduction and preliminaries
More generalized contractive type conditions are considered in the study of the existence and uniqueness of the fixed point. Alber and Guerre-Delabriere in [2] introduced a class of weakly contractive maps on closed convex sets of Hilbert spaces. In [9], Rhoades extended a part of this study to an arbitrary Banach space. The notion of weak contraction has been studied by other authors in the setting of metric spaces (see [8,12] and the references therein). In [13], Zhang gave some new generalized contractive type conditions for a pair of mappings in a metric space and proved some common fixed point results for these mappings. Let F : [0, +∞[ − → R be a function satisfying the three conditions:  (iii) lim n→+∞ φ n (t) = 0 for all t > 0.
In this paper, motivated by some works as [10], we extend the following theorem to the setting of the modular metric space endowed with a reflexive digraph. Chaira  Then T and S have a unique common fixed point in X. Moreover, for each x 0 ∈ X, the iterative sequence {x n } with x 2n+1 = Tx 2n and x 2n+2 = Sx 2n+1 converges to the common fixed point of T and S.
The modular ω is said to be regular if condition (i) holds for some λ > 0. The modular ω is said to be convex if, for all λ, μ > 0 and x, y, z ∈ X, we have Let (X, ω) be a modular metric space. Fix x 0 ∈ X. Set The two linear spaces X ω and X * ω are said to be modular spaces (around x 0 ). It is clear that

Definition 1.2 ([7])
We say that ω satisfies the 2 -type condition if, for every α > 0, there exists a constant K α > 0 such that for all x, y ∈ X ω and any λ > 0. If ω satisfies the 2 -type condition, then ω is regular and X ω = X * ω = X.
It is clear that if ω satisfies the 2 -type condition, then ω satisfies the 2 -condition, and that the converse is not true. Throughout this paper, we consider the modular metrics satisfying the 2 -type condition, and we adopt the definitions of some topological notions as stated in [11].

Definition 1.5
Let ω be a modular metric on X.
1. We say that a sequence {x n } ⊂ X ω is ω-convergent to some x ∈ X ω if lim n→+∞ ω λ (x n , x) = 0 for some λ > 0. We will call x the ω-limit of {x n }.
If ω satisfies the 2 -type condition, then lim n→+∞ ω λ (x n , x) = 0 for all λ > 0. 2. We say that a sequence {x n } ⊂ X ω is ω-Cauchy if, for some λ > 0, lim n,m→+∞ If ω satisfies the 2 -type condition, then {x n } is ω-Cauchy if lim n,m→+∞ ω λ (x n , x m ) = 0 for all λ > 0. 3. We say that M ⊂ X ω is ω-closed if the ω-limit of any ω-convergent sequence of M is in M. 4. We say that M ⊂ X ω is ω-complete if any ω-Cauchy sequence in M is ω-convergent and its ω-limit belongs to M. 5. We say that ω satisfies the Fatou property if, for some λ > 0, we have for any sequence {x n } ⊂ X ω which is ω-convergent to x and for any y ∈ X ω .
Let V be an arbitrary set. A directed graph, or digraph, is a pair G = (V , E) where E is a subset of the Cartesian product V × V . The elements of V are called vertices or nodes of G, and the elements of E are the edges also called oriented edges or arcs of G. An edge of the form (v, v) is a loop on v. Another way to express that E is a subset of V × V is to say that E is a binary relation over V . Given a digraph G, the set of vertices (respectively of edges) of G is denoted by Given two vertices x, y ∈ V . A path in G, from (or joining) x to y is a sequence of vertices p = {a i } 0≤i≤n , n ∈ N * such that a 0 = x, a n = y and (a i , a i+1 ) ∈ E for all i ∈ {0, 1, . . . , n -1}. The integer n is the length of the path p. If x = y and n > 1, the path p is called a directed cycle. An acyclic digraph is a digraph which has no directed cycle.
We denote by y ∈ [x] G the fact that there is a directed path in G joining x to y. ω, G). In recent years, there has been a great interest in the study of the fixed point property in modular metric spaces endowed with a partial order, see [5] and the references therein.
In this work, we investigate the existence and uniqueness of the common fixed point of a pair of mappings satisfying a generalized contractive condition in the setting of a modular metric space with a reflexive digraph. The main result is illustrated by an example and is used to show the existence of a solution of a system of Fredholm integral equations.
As in [6], we use the property (OSC) defined as follows.
Definition 1.6 Let (X, ω, G) be a modular metric space endowed with a digraph. We say that X satisfies the property (OSC) if, for any G-nondecreasing sequence

Main result
The following technical lemmas borrowed from [5] are useful in the sequel and highlight the use of the 2 -type condition to establish the main result.
Lemma 2.2 Let s, t ∈ N * . If ω satisfies the 2 -type condition and {x n } is not ω-Cauchy, then there exist ε > 0 and two subsequences of integers {n k } and {m k } such that n k > m k ≥ k, From now on, we mean 1 instead of λ for the same reason Abdou and Khamsi used in [1]. One can see that the proof of the main result remains even if we replace 1 with any λ > 0. Theorem 2.1 Let (X, ω, G) be a modular metric space endowed with a reflexive digraph G where ω satisfies the 2 -type condition and the Fatou property. Let C be an ω-complete nonempty subset of X ω and T, S : C → C be two self-mappings. If the following conditions are satisfied:

Condition (ii) insures that {x n } is G-nondecreasing. If there exists an integer n such that
x 2n = x 2n+1 = x 2n+2 , then x 2n is a common fixed point of S and T. Otherwise, suppose that x 2n = x 2n+1 or x 2n = x 2n+2 for all n ∈ N.
The next example illustrates Theorem 2.1 and shows that the class of mappings satisfying our main result is a proper nonempty subset of the set of the mappings considered in [13]. Consider the reflexive digraph G = (X, E) represented in Fig. 1, where Consider the two self-mapping S and T defined on X by for all x ∈ X, and the two functions F and ψ defined on [0, +∞[ by We can see that 1. X is ω-complete; For this, we proceed by disjunction of the cases: • The case where x = y = 0 is avoided.
• If x = 1 3 n and y = 1 3 m for m, n ∈ N such that m > n, then F ω 1 (Sx, Ty) = 1 √ 2 • If x = 1 3 m and y = 1 3 n for m, n ∈ N such that m > n, then