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Common fixed point for some generalized contractive mappings in a modular metric space with a graph
Fixed Point Theory and Algorithms for Sciences and Engineering volume 2021, Article number: 4 (2021)
Abstract
In this paper, we investigate the existence and the uniqueness of a common fixed point of a pair of selfmappings 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 GuerreDelabriere 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 \(F:\mathopen[0,+\infty \mathclose[\longrightarrow \mathbb{R}\) be a function satisfying the three conditions:

(i)
\(F(0)=0\) and \(F(t)> 0\) for all \(t> 0\);

(ii)
F is nondecreasing on \(\mathopen[0,+\infty\mathclose[\);

(iii)
F is continuous on \(\mathopen[0,+\infty\mathclose[\).
Consider the function \(\phi : \mathopen[0,+\infty\mathclose[\longrightarrow \mathopen[0,+\infty\mathclose[ \) such that

(i)
\(\phi (t)< t\) for all \(t> 0\);

(ii)
ϕ is nondecreasing and right upper semicontinuous on \(\mathopen[0,+\infty\mathclose[\);

(iii)
\(\lim_{n\rightarrow +\infty }\phi ^{n}(t)=0\) 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.
Theorem
([13])
Let X be a complete metric space, and let \(T,S: X \longrightarrow X\) be two selfmappings satisfying
where
Then T and S have a unique common fixed point in X. Moreover, for each \(x_{0} \in X\), 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.
In the sequel, we recall some basic notions: Let X be a nonempty set. For a function \(\mathopen]0,+\infty\mathclose[\times X\times X\rightarrow [0,+\infty ]\), we will use the notation
Definition 1.1
([7])
A function \(\omega :\mathopen]0,+\infty\mathclose[\times X\times X\rightarrow [0,+\infty ]\) is said to be modular metric on X if it satisfies the following conditions:

(i)
Given \(x,y\in X\), \(x=y\) if and only if \(\omega _{\lambda }(x,y)=0\) for all \(\lambda >0\);

(ii)
For all \(x,y\in X\), for all \(\lambda >0\), \(\omega _{ \lambda }(x,y)=\omega _{\lambda }(y,x)\);

(iii)
For all \(x,y,z\in X\) and for all \(\lambda ,\mu >0\), \(\omega _{\lambda +\mu }(x,y)\leq \omega _{\lambda }(x,z)+\omega _{\mu }(z,y)\).
In this case, \((X,\omega )\) is called modular metric space.
The modular ω is said to be regular if condition (i) holds for some \(\lambda >0\).
The modular ω is said to be convex if, for all \(\lambda ,\mu >0\) and \(x,y,z\in X\), we have
Let \((X,\omega )\) be a modular metric space. Fix \(x_{0}\in X\). Set
and
The two linear spaces \(X_{\omega }\) and \(X_{\omega }^{*}\) are said to be modular spaces (around \(x_{0}\)). It is clear that \(X_{\omega }\subseteq X_{\omega }^{*}\).
Definition 1.2
([7])
We say that ω satisfies the \(\Delta _{2}\)type condition if, for every \(\alpha >0\), there exists a constant \(K_{\alpha }>0\) such that
for all \(x,y\in X_{\omega }\) and any \(\lambda >0\).
Remark 1.3
If ω satisfies the \(\Delta _{2}\)type condition, then ω is regular and \(X_{\omega }=X_{\omega }^{*}=X\).
A condition weaker than the \(\Delta _{2}\)type condition is often used in the literature:
Definition 1.4
We say that ω satisfies the \(\Delta _{2}\)condition if \(\lim_{n\rightarrow +\infty }\omega _{\lambda }(x_{n},x)=0\) for some \(\lambda >0\) implies that \(\lim_{n\rightarrow +\infty }\omega _{\lambda }(x_{n},x)=0\) for all \(\lambda >0\).
It is clear that if ω satisfies the \(\Delta _{2}\)type condition, then ω satisfies the \(\Delta _{2}\)condition, and that the converse is not true. Throughout this paper, we consider the modular metrics satisfying the \(\Delta _{2}\)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 \(\{x_{n}\}\subset X_{\omega }\) is ωconvergent to some \(x\in X_{\omega }\) if \(\lim_{n\rightarrow +\infty }\omega _{\lambda }(x_{n},x)=0\) for some \(\lambda >0\). We will call x the ωlimit of \(\{x_{n}\}\).
If ω satisfies the \(\Delta _{2}\)type condition, then \(\lim_{n\rightarrow +\infty }\omega _{\lambda }(x_{n},x)=0\) for all \(\lambda >0\).

2.
We say that a sequence \(\{x_{n}\}\subset X_{\omega }\) is ωCauchy if, for some \(\lambda >0\),
$$ \lim_{n,m\rightarrow +\infty }\omega _{\lambda }(x_{n},x_{m})=0. $$If ω satisfies the \(\Delta _{2}\)type condition, then \(\{x_{n}\}\) is ωCauchy if \(\lim_{n,m\rightarrow + \infty }\omega _{\lambda }(x_{n},x_{m})=0\) for all \(\lambda >0\).

3.
We say that \(M \subset X_{\omega }\) is ωclosed if the ωlimit of any ωconvergent sequence of M is in M.

4.
We say that \(M \subset X_{\omega }\) is ωcomplete if any ωCauchy sequence in M is ωconvergent and its ωlimit belongs to M.

5.
We say that ω satisfies the Fatou property if, for some \(\lambda >0\), we have
$$ \omega _{\lambda }(x,y)\leq \liminf_{n\rightarrow +\infty } \omega _{\lambda }(x_{n},y) $$for any sequence \(\{x_{n}\}\subset X_{\omega }\) which is ωconvergent to x and for any \(y\in X_{\omega }\).
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 \(V \times V\). 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 \(V \times V\) 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 \(V(G)\) (respectively \(E(G)\)). A digraph \(G'=(V',E')\) is said to be an induced subgraph of a digraph \(G=(V,E)\) on \(V'\) if \(V'\subseteq V\) and \(E'=E\cap (V'\times V')\). We denote \(G'\) by \(G[V']\).
The digraph \(G=(V,E)\) is said to be

(i)
transitive if whenever \((x,y)\in E\) and \((y,z)\in E\), then \((x,z)\in E\).

(ii)
reflexive if \(\Delta := \{ (v, v) : v \in V \}\) is a subset of E.
A vertex x is said to be

(i)
a start point of G if there exists no vertex y such that \((y,x)\in E\).

(ii)
isolated if, for each vertex \(y\neq x\), we have neither \((x,y)\in E\) nor \((y,x)\in E\).
Given two vertices \(x,y\in V\). A path in G, from (or joining) x to y is a sequence of vertices \(p=\{a_{i}\}_{0\leq i\leq n}\), \(n\in \mathbb{N}^{\ast }\) such that \(a_{0}=x\), \(a_{n}=y\) and \((a_{i},a_{i+1})\in E\) for all \(i\in \{0,1,\ldots,n1\}\). 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 \(y\in [x]_{G}\) the fact that there is a directed path in G joining x to y.
A sequence \(\{ x_{n} \}_{n\in \mathbb{N}}\) is said to be Gnondecreasing if \(x_{n+1}\in [x_{n}]_{G}\) for all \(n\in \mathbb{N}\).
A modular metric space \((X,\omega )\) endowed with a digraph G such that \(V(G)=X\) is denoted by \((X,\omega ,G)\). 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 \((X,\omega ,G)\) be a modular metric space endowed with a digraph. We say that X satisfies the property (OSC) if, for any Gnondecreasing sequence \(\{x_{n}\}\subseteq X\) which is ωconvergent to \(x\in X\), we have \(x\in [x_{n}]_{G}\) for all \(n\in \mathbb{N}\).
Main result
The following technical lemmas borrowed from [5] are useful in the sequel and highlight the use of the \(\Delta _{2}\)type condition to establish the main result.
Lemma 2.1
If ω satisfies the \(\Delta _{2}\)type condition, then
Lemma 2.2
Let \(s,t\in \mathbb{N}^{*}\). If ω satisfies the \(\Delta _{2}\)type condition and \(\{x_{n}\}\) is not ωCauchy, then there exist \(\varepsilon >0\) and two subsequences of integers \(\{n_{k}\}\) and \(\{m_{k}\}\) such that \(n_{k}> m_{k}\geq k\), \(\omega _{2^{s}}(x_{n_{k}},x_{m_{k}})\geq \varepsilon \), and \(\omega _{\frac{1}{2^{t}}}(x_{n_{k}1},x_{m_{k}})< \varepsilon \).
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 \(\lambda >0\).
Let \(\psi : \mathopen[0,+\infty\mathclose[\longrightarrow \mathopen[0,+\infty\mathclose[\) be a function satisfying the two conditions:

(i)
\(\psi (t)< t\) for all \(t> 0\);

(ii)
ψ is right upper semicontinuous on \(\mathopen[0,+\infty\mathclose[\).
Let
and
Theorem 2.1
Let \((X,\omega , G)\) be a modular metric space endowed with a reflexive digraph G where ω satisfies the \(\Delta _{2}\)type condition and the Fatou property. Let C be an ωcomplete nonempty subset of \(X_{\omega }\) and \(T,S : C \rightarrow C\) be two selfmappings. If the following conditions are satisfied:

(i)
for all \(x,y\in C\),
$$ \bigl( y\in [x]_{G} \textit{ or }x\in [y]_{G} \bigr) \quad \Longrightarrow \quad F\bigl(\omega _{1}(Sx,Ty)\bigr)\leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr); $$(1) 
(ii)
there exists an element \(x_{0}\in C\) such that the induced subgraph \(G[\mathcal{O}_{x_{0}}(S,T)]\) is a directed path with a unique starting point \(x_{0}\);

(iii)
ω satisfies the property (OSC),
then S and T have a common fixed point in C.
Proof
Let \(x_{0}\) be an element of C such that \(G[\mathcal{O}_{x_{0}}(S,T)]\) is a directed path. Consider the sequence \(\{x_{n}\}\) defined by
Condition (ii) insures that \(\{x_{n}\}\) is Gnondecreasing. If there exists an integer n such that
then \(x_{2n}\) is a common fixed point of S and T. Otherwise, suppose that
Let \(n\in \mathbb{N}\). From \(x_{2n+1}\in [x_{2n}]_{G}\) and applying (1) for \(x=x_{2n}\) and \(y=x_{2n+1}\), we obtain
From
and
it follows that
If we suppose that there exists an integer n such that
then
Thus
which implies that \(F(\omega _{1}(x_{2n+1},x_{2n+2}))=0\). Hence, \(x_{2n+1}=x_{2n+2}\) and, from (2), \(x_{2n}=x_{2n+1}\), a contradiction. Hence, for each integer n, we have
By the same argument, if we take, in inequality (1), \(x=x_{2n1}\) and \(y=x_{2n}\), we obtain
Then \(\omega _{1}(x_{n+1},x_{n+2})<\omega _{1}(x_{n},x_{n+1})\) for all \(n\in \mathbb{N}\). Thus, the sequence \(\{\omega _{1}(x_{n},x_{n+1})\}\) is decreasing and bounded below. Therefore it is ωconvergent to some \(r\geq 0\). Since
by letting to limit superior in inequality (2), we obtain
which implies that \(r=0\). Thus, \(\lim_{n\rightarrow +\infty }\omega _{1}(x_{n},x_{n+1})=0\).
Let us prove that the sequence \(\{x_{n}\}\) is ωCauchy. For this, it is sufficient to show that the subsequence \(\{x_{2n}\}\) is ωCauchy. Assume the contrary. Then, according to Lemma 2.2, there exists \(\varepsilon >0\) such that we can find two subsequences \(\{m_{k}\}\) and \(\{n_{k}\}\) of positive integers satisfying \(n_{k}>m_{k}\geq k\) such that the following inequalities hold:
If we take \(x=x_{2n_{k}}\) and \(y=x_{2m_{k}1}\), then \(y\in [x]_{G}\) and inequality (1) becomes
where
Since
it follows that \(\lim_{k\rightarrow +\infty }\omega _{2}(x_{2n_{k}},x_{2m_{k}})= \lim_{k\rightarrow +\infty }\omega _{1}(x_{2n_{k}},x_{2m_{k}})= \varepsilon \).
From
we get
Thus
Similarly, using
we get
Therefore \(\lim_{k\rightarrow +\infty }\omega _{1}(x_{2n_{k}},x_{2m_{k}1})= \varepsilon \).
From
we get \(\lim_{k\rightarrow +\infty }\omega _{2}(x_{2m_{k}1},x_{2n_{k}+1})= \varepsilon \). Since
and by letting \(k\rightarrow +\infty \), we obtain \(\lim_{k\rightarrow +\infty }\omega _{1}(x_{2m_{k}1},x_{2n_{k}+1})= \varepsilon \). Therefore
From the continuity of F and the upper semicontinuity of ψ, we have
a contradiction since \(\epsilon > 0\). Therefore the sequence \(\{x_{n}\}\) is ωCauchy. Using the ωcompleteness of C, there exists \(x^{\ast }\in C\) such that \(\lim_{n\rightarrow +\infty }\omega _{1}(x_{n},x^{\ast })=0\). The property (OSC) insures that \(x^{\ast }\in [x_{n}]\) for all \(n\in \mathbb{N}\). Then
where
Since \(\omega _{2}(x_{2n},Tx^{\ast })\leq \omega _{1}(x_{2n},x^{\ast })+ \omega _{1}(x^{\ast },Tx^{\ast })\), \(\lim_{n}M(x_{2n},x^{\ast })=\omega _{1}(x^{\ast },Tx^{\ast })\).
Using the continuity of F and the upper continuity of ψ, we obtain
By the Fatou property, we have
Since F is continuous and nondecreasing on \(\mathopen[0,+\infty\mathclose[\), we have
which implies that \(\omega _{1}(x^{\ast },Tx^{\ast })=0\), and according to the regularity of ω, we have \(Tx^{\ast }=x^{\ast }\). Since \(x^{\ast }\in [x^{\ast }]_{G}\), \(F(\omega _{1}(Sx^{\ast },Tx^{\ast })) \leq \psi (F(M(x^{\ast },x^{\ast })))\) where
which implies that \(F(\omega _{1}(Sx^{\ast },x^{\ast }))\leq \psi (F(\omega _{1}(Sx^{\ast },x^{ \ast })))\). Hence \(\omega _{1}(Sx^{\ast },x^{\ast })=0\) and the regularity of ω insures that \(Sx^{\ast }=x^{\ast }\). □
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].
Example 2.3
Consider the modular metric space \((X,\omega )\) where
Consider the reflexive digraph \(G=(X,E)\) represented in Fig. 1, where
Consider the two selfmapping S and T defined on X by
and the two functions F and ψ defined on \(\mathopen[0,+\infty\mathclose[\) by
We can see that

1.
X is ωcomplete;

2.
ω satisfies the \(\Delta _{2}\)type condition and the Fatou property;

3.
\(G[\mathcal{O}_{1}(S,T)]\) is a directed path with a unique starting point \(x_{0}\) (see Figure 2).
Let us show that, for all \(x,y\in C\),
For this, we proceed by disjunction of the cases:

The case where \(x=y=0\) is avoided.

If \(x=\frac{1}{3^{n}}\) for \(n\in \mathbb{N}\) and \(y=0\), then
$$ F\bigl(\omega _{1}(Sx,Ty)\bigr)=\frac{1}{\sqrt{2}.3^{n+2}}\leq \frac{1}{2.3^{n}}= \psi \bigl(F\bigl(M(x,y)\bigr)\bigr). $$ 
If \(x=0\) and \(y=\frac{1}{3^{n}}\) for \(n\in \mathbb{N}\), then
$$ F\bigl(\omega _{1}(Sx,Ty)\bigr)=\frac{1}{\sqrt{2}.3^{n+1}}\leq \frac{1}{2.3^{n}} \leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr). $$ 
If \(x=y=\frac{1}{3^{n}}\) for \(n\in \mathbb{N}\), then
$$ F\bigl(\omega _{1}(Sx,Ty)\bigr)=\frac{\sqrt{2}}{3^{n+2}}\leq \frac{4}{3^{n+2}} \leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr). $$ 
If \(x=\frac{1}{3^{n}}\) and \(y=\frac{1}{3^{m}}\) for \(m,n\in \mathbb{N}\) such that \(m> n\), then
$$\begin{aligned} F\bigl(\omega _{1}(Sx,Ty)\bigr)=\frac{1}{\sqrt{2}} \biggl( \frac{1}{3^{n+2}} \frac{1}{3^{m+1}} \biggr)&\leq \frac{4}{3^{n+2}} \leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr). \end{aligned}$$ 
If \(x=\frac{1}{3^{m}}\) and \(y=\frac{1}{3^{n}}\) for \(m,n\in \mathbb{N}\) such that \(m> n\), then
$$\begin{aligned} F\bigl(\omega _{1}(Sx,Ty)\bigr)=\frac{1}{\sqrt{2}} \biggl( \frac{1}{3^{m+2}} \frac{1}{3^{n+1}} \biggr)\leq \frac{\sqrt{2}}{3^{n+1}} \leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr). \end{aligned}$$
All assumptions of Theorem 2.1 are satisfied and S and T have a fixed point \(x^{\ast }=0\).
Remark 2.4
In Example 2.3, if we consider the function \(\psi (t)=0.8\times \ln (1+t)\) for all \(t\in \mathopen[0,+\infty\mathclose[\), we get
where \(d(x,y)= \vert xy \vert \) and
Theorem on page 2 is not applicable, but by Theorem 2.1, we obtain the existence of a common fixed point of S and T. Indeed, we have, for all \(x,y\in X\),
Corollary 2.2
Let \((X,\omega , G)\) be a modular metric space endowed with a reflexive digraph G where ω satisfies the \(\Delta _{2}\)type condition and the Fatou property. Let C be an ωcomplete nonempty subset of \(X_{\omega }\) and \(T,S : C \rightarrow C\) be two selfmappings. If the following conditions are satisfied:

(i)
there exists \(k\in \mathopen[0,1\mathclose[\) such that, for all \(x,y\in C\),
$$ \bigl( y\in [x]_{G} \textit{ or }x\in [y]_{G} \bigr) \quad \Longrightarrow \quad \omega _{1}(Sx,Ty)\leq \bigl(1+ \omega _{1}(x,y) \bigr)^{k}1; $$(4) 
(ii)
there exists an element \(x_{0}\in C\) such that \(G[\mathcal{O}_{x_{0}}(S,T)]\) is a directed path with a unique starting point \(x_{0}\);

(iii)
ω satisfies the property (OSC),
then S and T have a common fixed point in C.
Proof
If we consider the two functions F and ψ defined on \(\mathopen[0,+\infty\mathclose[\) by
then we can verify that the second part of implication (4) is equivalent to
which implies that \(F(\omega _{1}(Sx,Ty))\leq \psi (F(M(x,y)))\), since F and ψ are nondecreasing on \(\mathopen[0,+\infty\mathclose[\). By applying Theorem 2.1, we terminate the demonstration. □
In the sequel, we use the following lemma.
Lemma 2.5
([5])
Let \((X,\omega )\) be a modular space such that ω is convex and satisfies the \(\Delta _{2}\)condition. If \(\{x_{n}\}\) is a sequence in \(X_{\omega }\) such that \(\lim_{n \rightarrow +\infty }\omega _{1}(x_{n},x_{n+1})=0\), then \(\{x_{n}\}\) is ωCauchy.
Theorem 2.3
Let \((X,\omega , G)\) be a modular metric space endowed with a reflexive digraph G where ω is convex and satisfies the \(\Delta _{2}\)type condition and the Fatou property. Let C be an ωcomplete nonempty subset of \(X_{\omega }\) and \(T,S : C \rightarrow C\) be two selfmappings. If the following conditions are satisfied:

(i)
for all \(x,y\in C\),
$$ \bigl( y\in [x]_{G} \textit{ or }x\in [y]_{G} \bigr)\quad \Longrightarrow \quad F\bigl( \omega _{1}(Sx,Ty)\bigr)\leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr), $$(5)where
$$ M(x,y)=\max \bigl\{ \omega _{1}(x,y),\omega _{1}(x,Sx), \omega _{1}(y,Ty), \omega _{2}(x,Ty)+\omega _{2}(y,Sx)\bigr\} ; $$ 
(ii)
there exists an element \(x_{0}\in C\) such that \(G[\mathcal{O}_{x_{0}}(S,T)]\) is a directed path with a unique starting point \(x_{0}\);

(iii)
ω satisfies the property (OSC),
then S and T have a common fixed point in C and \(\mathfrak{F}(S,T)=\mathfrak{F}(S)=\mathfrak{F}(T)\), where \(\mathfrak{F}(T)\) is the set of fixed points of T.
Proof
Let \(x_{0}\) an element of C such that \(G[\mathcal{O}_{x_{0}}(S,T)]\) is a directed path. Consider the sequence \(\{x_{n}\}\) defined by
Condition (ii) insures that \(\{x_{n}\}\) is Gnondecreasing. If there exists an integer n such that
then \(x_{2n}\) is a common fixed point of S and T. Otherwise, suppose that
Let \(n\in \mathbb{N}\). From \(x_{2n+1}\in [x_{2n}]_{G}\) and applying (5) for \(x=x_{2n}\) and \(y=x_{2n+1}\), we obtain
From
since ω is convex,
from which it follows that
By the same arguments as in the proof of Theorem 2.1, we prove that
According to Lemma 2.5, the sequence \(\{x_{n}\}\) is ωCauchy, and since C is ωcomplete, then \(\{x_{n}\}\) is ωconvergent to an element \(x^{\ast }\in C\). Again similar to the proof of Theorem 2.1, we prove that \(x^{\ast }\) is a common fixed point of S and T. □
Application
Consider the space \(X=\mathcal{C}^{1}([0,1],\mathbb{R})\). Let \(G=(X,E)\) be the digraph such that, for all \(x,y\in X\),
Consider the function \(\omega :\mathopen]0,+\infty\mathclose[\times X\times X\longrightarrow [0,+\infty ]\) defined, for each \(\lambda \in \mathopen]0,+\infty\mathclose[\) and \(x,y\in X\), by
It is easy to check the following result.
Lemma 3.1
The function ω is a modular metric satisfying the following:

(i)
ω satisfies the \(\Delta _{2}\)type condition and the Fatou property;

(ii)
\(X_{\omega }=X\) is ωcomplete;

(iii)
ω satisfies the (OSC) property.
Let us consider the following integral equations system:
where \(a\in X\) and \(f,g:[0,1]\times \mathbb{R}\rightarrow \mathbb{R}\) are two mappings such that f and g are of the class \(C^{1}\) on \([0,1]\times \mathbb{R}\).
Let us consider the two mappings T and S defined in X as follows:
One can see that Tx and Sx are in X for all \(x\in X\).
Theorem 3.2
If the following two conditions are satisfied:

(i)
for every \(s,t\in [0,1]\) and for all comparable elements \(x,y\in X\),
$$ \bigl\vert f\bigl(t,x(s)\bigr)g\bigl(t,y(s)\bigr) \bigr\vert \leq 1+ \sqrt{1+ \bigl\vert x(s)y(s) \bigr\vert }, $$ 
(ii)
there exists \(x_{0}\in X\) such that, for all \(t\in [0,1]\), we have
$$ x_{0}(t)\preceq Sx_{0}(t)\preceq TSx_{0}(t) \preceq STSx_{0}(t) \preceq (TS)^{2}x_{0}(t) \preceq S(TS)^{2}x_{0}(t)\preceq \cdots, $$
then the system (IES) admits at least a solution which belongs to the diagonal of \(X^{2}\).
Proof
Let x and y be two comparable elements in X, that is, \(x\in [y]_{G}\) or \(y\in [x]_{G}\). Since, for each \(t,s\in [0,1]\),
and
we have
Since
we have
Since, for all \(t\in [0,1]\),
the induced subgraph \(G[\mathcal{O}_{x_{0}}(S,T)]\) is a directed path with the unique starting point \(x_{0}\).
According to Corollary 2.2, T and S have a common fixed point in X, i.e., there exists an element \(x^{\ast }\in X\) such that \((x^{\ast },x^{\ast })\) verifies the system (IES). Then the system (IES) admits at least a solution in \(X^{2}\) which belongs to \(\Delta (X\times X)= \{(u,u)/u\in X \}\) the diagonal of \(X^{2}\). □
Conclusion
Our results improve, extend, and generalize some classical results:

(i)
In Theorem 2.3, if we take \(\omega _{\lambda }(x,y)=\frac{d(x,y)}{\lambda }\) for all \(\lambda \in \mathopen]0,+\infty\mathclose[\), we get an improved version of the main result of Zhang [13, Theorem 1] by removing condition (iii) verified by the function ϕ and the monotony of ϕ.

(ii)
In Theorem 2.1, if the function F is the identity and the function ψ is nondecreasing, we obtain an analogue of [4, Theorem 2] but for a common fixed point in the setting of modular metric spaces with graph.

(iii)
Theorem 2.3generalizes and extends [3, Theorem 2.1] in the setting of a modular metric space with graph.

(iv)
Corollary 2.2generalizes and extends [1, Theorem 3.1] in the setting of modular metric spaces with graph, since
$$ \omega _{1}(Sx,Ty)\leq k \omega _{1}(x,y)\quad \Longrightarrow \quad \omega _{1}(Sx,Ty) \leq \bigl(1+\omega _{1}(x,y) \bigr)^{k}1. $$
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Chaira, K., Eladraoui, A., Kabil, M. et al. Common fixed point for some generalized contractive mappings in a modular metric space with a graph. Fixed Point Theory Algorithms Sci Eng 2021, 4 (2021). https://doi.org/10.1186/s13663021006908
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MSC
 46A80
 47H10
Keywords
 Common fixed point
 Weak contraction
 Digraph
 Modular metric space