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

Abstract

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.

1 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 $F:\mathopen[0,+\infty \mathclose[\longrightarrow \mathbb{R}$ 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 self-mappings satisfying

$$\begin{aligned}& F\bigl(d(T x, Sy)\bigr)\leq \phi (F\bigl(M(x, y)\bigr)\quad \textit{for each }x,y \in X, \end{aligned}$$

where

$$ M(x, y) = \max \biggl\{ d(x, y), d(T x, x), d(Sy, y), \frac{d(T x, y) + d(Sy, x)}{2}\biggr\} . $$

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

$$\begin{aligned}& \_{\lambda }(x,y)=(\lambda ,x,y)\quad \text{for all }\lambda >0 \text{ and } x,y\in X. \end{aligned}$$

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

$$ \omega _{\lambda +\mu }(x,y)\leq \frac{\lambda }{\lambda +\mu } \omega _{ \lambda }(x,z)+ \frac{\mu }{\lambda +\mu }\omega _{\mu }(z,y). $$

Let $(X,\omega )$ be a modular metric space. Fix $x_{0}\in X$. Set

$$ X_{\omega }=X_{\omega }(x_{0})=\bigl\{ x\in X: \omega _{\lambda }(x,x_{0}) \longrightarrow 0 \text{ as } \lambda \longrightarrow \infty \bigr\} $$

and

$$ X_{\omega }^{*}=X_{\omega }^{*}(x_{0})= \bigl\{ x\in X:\exists \lambda >0 , \omega _{\lambda }(x,x_{0})< \infty \bigr\} . $$

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

$$ \omega _{\frac{\lambda }{\alpha }}(x,y)\leq K_{\alpha } \omega _{\lambda }(x,y) $$

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 ω-convergent 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 ω-limit 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 ω-Cauchy 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 ω-Cauchy 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 ω-closed if the ω-limit of any ω-convergent sequence of M is in M.

• 4. We say that $M \subset X_{\omega }$ 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 $\lambda >0$, 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 ω-convergent 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 $i\in \{0,1,\ldots,n-1\}$. 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 $y\in [x]_{G}$ 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 G-nondecreasing 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 G-nondecreasing sequence $\{x_{n}\}\subseteq X$ which is ω-convergent to $x\in X$, we have $x\in [x_{n}]_{G}$ for all $n\in \mathbb{N}$.

2 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

$$\begin{aligned}& \omega _{\lambda }(x,y)< \infty \quad \textit{for all }\lambda >0\textit{ and for all } (x,y)\in X_{\omega }^{2}. \end{aligned}$$

Lemma 2.2

Let $s,t\in \mathbb{N}^{*}$. If ω satisfies the $\Delta _{2}$-type condition and $\{x_{n}\}$ is not ω-Cauchy, 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

$$ M(x,y)=\max \biggl\{ \omega _{1}(x,y),\omega _{1}(x,Sx), \omega _{1}(y,Ty), \frac{\omega _{2}(x,Ty)+\omega _{2}(y,Sx)}{2}\biggr\} $$

and

$$ \mathcal{O}_{x_{0}}(S,T)=\bigl\{ (TS)^{n}(x_{0}),S(TS)^{n}(x_{0}) : n\in \mathbb{N} \bigr\} . $$

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 ω-complete nonempty subset of $X_{\omega }$ and $T,S : C \rightarrow C$ be two self-mappings. 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

$$\begin{aligned}& x_{2n+1}=Sx_{2n}\quad \text{and} \quad x_{2n+2}=Tx_{2n+1} \quad \text{for all } n\in \mathbb{N}. \end{aligned}$$

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

$$\begin{aligned}& x_{2n}\neq x_{2n+1}\quad \text{or} \quad x_{2n}\neq x_{2n+2}\quad \text{for all } n\in \mathbb{N}. \end{aligned}$$

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

$$ F\bigl(\omega _{1}(x_{2n+1},x_{2n+2}) \bigr)\leq \psi \bigl(F\bigl(M(x_{2n},x_{2n+1})\bigr) \bigr).$$

(2)

From

$$ M(x_{2n},x_{2n+1})=\max \biggl\{ \omega _{1}(x_{2n},x_{2n+1}), \omega _{1}(x_{2n+1},x_{2n+2}), \frac{\omega _{2}(x_{2n},x_{2n+2})}{2} \biggr\} $$

and

$$ \frac{\omega _{2}(x_{2n},x_{2n+2})}{2}\leq \frac{\omega _{1}(x_{2n},x_{2n+1})+\omega _{1}(x_{2n+1},x_{2n+2})}{2}, $$

it follows that

$$ M(x_{2n},x_{2n+1})=\max \bigl\{ \omega _{1}(x_{2n},x_{2n+1}), \omega _{1}(x_{2n+1},x_{2n+2}) \bigr\} . $$

If we suppose that there exists an integer n such that

$$ \omega _{1}(x_{2n},x_{2n+1})\leq \omega _{1}(x_{2n+1},x_{2n+2}), $$

then

$$ M(x_{2n},x_{2n+1})=\omega _{1}(x_{2n+1},x_{2n+2}). $$

Thus

$$ F\bigl(\omega _{1}(x_{2n+1},x_{2n+2})\bigr)\leq \psi \bigl(F\bigl(\omega _{1}(x_{2n+1},x_{2n+2})\bigr) \bigr), $$

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

$$ \omega _{1}(x_{2n+1},x_{2n+2})\leq \omega _{1}(x_{2n},x_{2n+1}). $$

By the same argument, if we take, in inequality (1), $x=x_{2n-1}$ and $y=x_{2n}$, we obtain

$$\begin{aligned}& \omega _{1}(x_{2n},x_{2n+1})< \omega _{1}(x_{2n-1},x_{2n})\quad \text{for all } n\in \mathbb{N}^{\ast }. \end{aligned}$$

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 ω-convergent to some $r\geq 0$. Since

$$\begin{aligned} \lim_{n\rightarrow +\infty }M(x_{2n},x_{2n+1})=\lim _{n \rightarrow +\infty }\max \bigl\{ \omega _{1}(x_{2n},x_{2n+1}), \omega _{1}(x_{2n+1},x_{2n+2}) \bigr\} =r, \end{aligned}$$

by letting to limit superior in inequality (2), we obtain

$$ F(r)\leq \limsup_{n}\psi (F\bigl(M(x_{2n},x_{2n+1}) \bigr)\leq \psi \bigl(F(r)\bigr), $$

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 ω-Cauchy. For this, it is sufficient to show that the subsequence $\{x_{2n}\}$ is ω-Cauchy. 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:

$$\begin{aligned}& \omega _{8}(x_{2n_{k}},x_{2m_{k}})\geq \varepsilon \quad \text{and} \quad \omega _{\frac{1}{4}}(x_{2n_{k}-1},x_{2m_{k}})< \varepsilon . \end{aligned}$$

If we take $x=x_{2n_{k}}$ and $y=x_{2m_{k}-1}$, then $y\in [x]_{G}$ and inequality (1) becomes

$$ \psi (F\bigl(\omega _{1}(x_{2n_{k}+1},x_{2m_{k}})\bigr)\leq F\bigl(M(x_{2n_{k}},x_{2m_{k}-1})\bigr), $$

where

$$\begin{aligned} M(x_{2n_{k}},x_{2m_{k}-1}) =&\max \biggl\{ \omega _{1}(x_{2n_{k}},x_{2m_{k}-1}), \omega _{1}(x_{2n_{k}},x_{2n_{k}+1}) ,\omega _{1}(x_{2m_{k}-1},x_{2m_{k}}), \\ &\frac{\omega _{2}(x_{2n_{k}},x_{2m_{k}})+\omega _{2}(x_{2m_{k}-1},x_{2n_{k}+1})}{2} \biggr\} . \end{aligned}$$

Since

$$\begin{aligned} \varepsilon &\leq \omega _{8}(x_{2n_{k}},x_{2m_{k}})\leq \omega _{2}(x_{2n_{k}},x_{2m_{k}}) \\ &\leq \omega _{1}(x_{2n_{k}},x_{2m_{k}}) \\ &\leq \omega _{\frac{1}{2}}(x_{2n_{k}-1},x_{2m_{k}})+\omega _{ \frac{1}{2}}(x_{2n_{k}-1},x_{2n_{k}}) \\ &\leq \omega _{\frac{1}{4}}(x_{2n_{k}-1},x_{2m_{k}})+\omega _{ \frac{1}{2}}(x_{2n_{k}-1},x_{2n_{k}}) \\ &\leq \varepsilon +\omega _{\frac{1}{2}}(x_{2n_{k}-1},x_{2n_{k}}), \end{aligned}$$

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

$$ \varepsilon \leq \omega _{2}(x_{2n_{k}},x_{2m_{k}})\leq \omega _{1}(x_{2n_{k}},x_{2n_{k}+1})+ \omega _{1}(x_{2n_{k}+1},x_{2m_{k}}), $$

we get

$$\begin{aligned} \varepsilon -\omega _{1}(x_{2n_{k}},x_{2n_{k}+1})&\leq \omega _{1}(x_{2n_{k}+1},x_{2m_{k}}) \\ &\leq \omega _{\frac{1}{2}}(x_{2n_{k}-1},x_{2m_{k}})+\omega _{ \frac{1}{4}}(x_{2n_{k}-1},x_{2n_{k}}) \\ &\quad {} +\omega _{\frac{1}{4}}(x_{2n_{k}},x_{2n_{k}+1}) \\ &\leq \varepsilon +\omega _{\frac{1}{4}}(x_{2n_{k}-1},x_{2n_{k}}) + \omega _{\frac{1}{4}}(x_{2n_{k}},x_{2n_{k}+1}). \end{aligned}$$

Thus

$$ \lim_{k\rightarrow +\infty }\omega _{1}(x_{2n_{k}+1},x_{2m_{k}})= \varepsilon . $$

Similarly, using

$$ \varepsilon \leq \omega _{2}(x_{2n_{k}},x_{2m_{k}})\leq \omega _{1}(x_{2n_{k}},x_{2m_{k}-1})+ \omega _{1}(x_{2m_{k}-1},x_{2m_{k}}), $$

we get

$$\begin{aligned} \varepsilon -\omega _{1}(x_{2m_{k}-1},x_{2m_{k}})&\leq \omega _{1}(x_{2n_{k}},x_{2m_{k}-1}) \\ &\leq \omega _{\frac{1}{2}}(x_{2n_{k}},x_{2n_{k}-1})+\omega _{ \frac{1}{4}}(x_{2n_{k}-1},x_{2m_{k}}) \\ &\quad {} +\omega _{\frac{1}{4}}(x_{2m_{k}},x_{2m_{k}-1}) \\ &\leq \omega _{\frac{1}{2}}(x_{2n_{k}},x_{2n_{k}-1})+\varepsilon + \omega _{\frac{1}{4}}(x_{2m_{k}},x_{2m_{k}-1}). \end{aligned}$$

Therefore $\lim_{k\rightarrow +\infty }\omega _{1}(x_{2n_{k}},x_{2m_{k}-1})= \varepsilon$.

From

$$\begin{aligned} &\omega _{8}(x_{2n_{k}},x_{2m_{k}})-\omega _{4}(x_{2n_{k}},x_{2n_{k}+1})- \omega _{2}(x_{2m_{k}-1},x_{2m_{k}}) \\ &\quad \leq \omega _{2}(x_{2m_{k}-1},x_{2n_{k}+1}) \\ &\quad \leq \omega _{1}(x_{2m_{k}-1},x_{2n_{k}})+\omega _{1}(x_{2n_{k}},x_{2n_{k}+1}), \end{aligned}$$

we get $\lim_{k\rightarrow +\infty }\omega _{2}(x_{2m_{k}-1},x_{2n_{k}+1})= \varepsilon$. Since

$$\begin{aligned} \omega _{2}(x_{2m_{k}-1},x_{2n_{k}+1})&\leq \omega _{1}(x_{2m_{k}-1},x_{2n_{k}+1}) \\ & \leq \omega _{\frac{1}{2}}(x_{2m_{k}-1},x_{2m_{k}})+\omega _{ \frac{1}{4}}(x_{2n_{k}-1},x_{2m_{k}}) +\omega _{\frac{1}{8}}(x_{2n_{k}-1},x_{2n_{k}}) \\ &\quad{} +\omega _{\frac{1}{8}}(x_{2n_{k}},x_{2n_{k}+1}) \\ &\leq \omega _{\frac{1}{2}}(x_{2m_{k}-1},x_{2m_{k}})+ \varepsilon + \omega _{\frac{1}{8}}(x_{2n_{k}-1},x_{2n_{k}})+\omega _{\frac{1}{8}}(x_{2n_{k}},x_{2n_{k}+1}) \end{aligned}$$

and by letting $k\rightarrow +\infty$, we obtain $\lim_{k\rightarrow +\infty }\omega _{1}(x_{2m_{k}-1},x_{2n_{k}+1})= \varepsilon$. Therefore

$$ \lim_{k\rightarrow +\infty }M(x_{2n_{k}},x_{2m_{k}-1})= \varepsilon . $$

From the continuity of F and the upper semicontinuity of ψ, we have

$$ F(\varepsilon )\leq \psi \bigl(F(\varepsilon )\bigr), $$

a contradiction since $\epsilon > 0$. Therefore the sequence $\{x_{n}\}$ is ω-Cauchy. Using the ω-completeness 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

$$ F\bigl(\omega _{1}\bigl(Sx_{2n},Tx^{\ast } \bigr)\bigr)\leq \psi \bigl(F\bigl(M\bigl(x_{2n},x^{\ast }\bigr) \bigr)\bigr), $$

(3)

where

$$\begin{aligned} \begin{aligned} M\bigl(x_{2n},x^{\ast }\bigr)={}&\max \biggl\{ \omega _{1}\bigl(x_{2n},x^{\ast }\bigr),\omega _{1}(x_{2n},x_{2n+1}), \omega _{1} \bigl(x^{\ast },Tx^{\ast }\bigr), \\&{}\frac{\omega _{2}(x_{2n},Tx^{\ast })+\omega _{2}(x^{\ast },x_{2n+1})}{2} \biggr\} . \end{aligned} \end{aligned}$$

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

$$ \limsup_{n}\psi \bigl(F\bigl(M\bigl(x_{2n},x^{\ast } \bigr)\bigr)\bigr)\leq \psi (F\bigl(\omega _{1}\bigl(x^{ \ast },Tx^{\ast } \bigr)\bigr). $$

By the Fatou property, we have

$$ \omega _{1}\bigl(x^{\ast },Tx^{\ast }\bigr)\leq \liminf _{n}\omega _{1}\bigl(Sx_{2n},Tx^{ \ast } \bigr). $$

Since F is continuous and nondecreasing on $\mathopen[0,+\infty\mathclose[$, we have

$$\begin{aligned} F(\omega _{1}\bigl(x^{\ast },Tx^{\ast }\bigr)&\leq F \Bigl(\liminf_{n}\omega _{1}\bigl(Sx_{2n},Tx^{ \ast } \bigr)\Bigr) \\ &\leq F\Bigl(\liminf_{n}\omega _{1} \bigl(Sx_{2n},Tx^{\ast }\bigr)\Bigr) \\ &\leq \limsup_{n}F\bigl(\omega _{1} \bigl(Sx_{2n},Tx^{\ast }\bigr)\bigr) \\ &\leq \limsup_{n}\psi \bigl(F\bigl(M\bigl(x_{2n},x^{\ast } \bigr)\bigr)\bigr) \\ &\leq \psi (F\bigl(\omega _{1}\bigl(x^{\ast },Tx^{\ast } \bigr)\bigr), \end{aligned}$$

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

$$ M\bigl(x^{\ast },x^{\ast }\bigr)=\max \bigl\{ \omega _{1} \bigl(x^{\ast },Sx^{\ast }\bigr),\omega _{2} \bigl(x^{ \ast },Sx^{\ast }\bigr)\bigr\} =\omega _{1} \bigl(x^{\ast },Sx^{\ast }\bigr), $$

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

$$\begin{aligned}& X=[0,1]\quad \text{and}\quad \omega _{\lambda }(x,y)=\frac{ \vert x-y \vert ^{2} }{2\lambda }\quad \text{for all }\lambda \in \mathopen]0,+\infty\mathclose[\text{ and }x,y\in X. \end{aligned}$$

Consider the reflexive digraph $G=(X,E)$ represented in Fig. 1, where

$$\begin{aligned}& E=\Delta \cup \biggl\{ \biggl(\frac{1}{3^{n}},0 \biggr), \biggl( \frac{1}{3^{n}},\frac{1}{3^{n+1}} \biggr): n\in \mathbb{N} \biggr\} . \end{aligned}$$

Figure 1

The digraph G (the loops and the isolated vertices are not represented)

Consider the two self-mapping S and T defined on X by

$$\begin{aligned}& Tx=\frac{x}{3}\quad \text{and} \quad Sx=\frac{x}{9}\quad \text{for all }x\in X, \end{aligned}$$

and the two functions F and ψ defined on $\mathopen[0,+\infty\mathclose[$ by

$$\begin{aligned}& F(t)=\sqrt{t}\quad \text{and} \quad \psi (t)=\frac{t}{\sqrt{2}}\quad \text{for all } t\in \mathopen[0,+\infty\mathclose[. \end{aligned}$$

We can see that

• 1. X is ω-complete;

• 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).

  Figure 2

  

  The digraph $G[\mathcal{O}_{1}(S,T)]$ (the loops are not represented)

Let us show that, for all $x,y\in C$,

$$ \bigl( y\in [x]_{G} \text{ or } x\in [y]_{G} \bigr) \quad \Longrightarrow \quad F\bigl(\omega _{1}(Sx,Ty)\bigr)\leq \psi \bigl(F \bigl(M(x,y)\bigr)\bigr). $$

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

$$\begin{aligned}& F\bigl(d(S x,Ty)\bigr)=\frac{1}{2} > 0.8\ln \biggl(1+\frac{\sqrt{3}}{2} \biggr)=\psi (F\bigl(M'(x, y)\bigr)\quad \text{for }x=0\text{ and }y= \frac{3}{4}, \end{aligned}$$

where $d(x,y)= \vert x-y \vert$ and

$$ M'(x, y) = \max \biggl\{ d(x,y), d(T x, x), d(Sy, y), \frac{d(T x, y) + d(Sy,x)}{2} \biggr\} . $$

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 $x,y\in X$,

$$ \bigl( y\in [x]_{G} \text{ or }x\in [y]_{G} \bigr) \quad \Longrightarrow \quad F\bigl(\omega _{1}(Sx,Ty)\bigr)\leq \psi \bigl(F\bigl(M(x,y)\bigr)\bigr). $$

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 ω-complete nonempty subset of $X_{\omega }$ and $T,S : C \rightarrow C$ be two self-mappings. 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

$$\begin{aligned}& F(t)=\ln (1+t)\quad \text{and} \quad \psi (t)=kt, \end{aligned}$$

then we can verify that the second part of implication (4) is equivalent to

$$ F\bigl(\omega _{1}(Sx,Ty)\bigr)\leq \psi \bigl(F\bigl(\omega _{1}(x,y)\bigr)\bigr), $$

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 ω-Cauchy.

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 ω-complete nonempty subset of $X_{\omega }$ and $T,S : C \rightarrow C$ be two self-mappings. 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

$$\begin{aligned}& x_{2n+1}=Sx_{2n}\quad \text{and} \quad x_{2n+2}=Tx_{2n+1} \quad \text{for all } n\in \mathbb{N}. \end{aligned}$$

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

$$\begin{aligned}& x_{2n}\neq x_{2n+1}\quad \text{or} \quad x_{2n}\neq x_{2n+2}\quad \text{for all } n\in \mathbb{N}. \end{aligned}$$

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

$$ F\bigl(\omega _{1}(x_{2n+1},x_{2n+2}) \bigr)\leq \psi \bigl(F\bigl(M(x_{2n},x_{2n+1})\bigr) \bigr).$$

(6)

From

$$ M(x_{2n},x_{2n+1})=\max \bigl\{ \omega _{1}(x_{2n},x_{2n+1}), \omega _{1}(x_{2n+1},x_{2n+2}), \omega _{2}(x_{2n},x_{2n+2})\bigr\} , $$

since ω is convex,

$$ \omega _{2}(x_{2n},x_{2n+2})\leq \frac{\omega _{1}(x_{2n},x_{2n+1})+\omega _{1}(x_{2n+1},x_{2n+2})}{2}, $$

from which it follows that

$$ M(x_{2n},x_{2n+1})=\max \bigl\{ \omega _{1}(x_{2n},x_{2n+1}), \omega _{1}(x_{2n+1},x_{2n+2}) \bigr\} . $$

By the same arguments as in the proof of Theorem 2.1, we prove that

$$ \lim_{n\rightarrow +\infty }\omega _{1}(x_{n},x_{n+1})=0. $$

According to Lemma 2.5, the sequence $\{x_{n}\}$ is ω-Cauchy, and since C is ω-complete, then $\{x_{n}\}$ is ω-convergent 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. □

3 Application

Consider the space $X=\mathcal{C}^{1}([0,1],\mathbb{R})$. Let $G=(X,E)$ be the digraph such that, for all $x,y\in X$,

$$ (x,y)\in E\quad \Longleftrightarrow \quad x(t)\leq y(t) \quad \text{for each } t \in [0,1]. $$

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

$$\begin{aligned}& \omega (\lambda ,x,y)=\omega _{\lambda }(x,y)=\frac{1}{\lambda } \Vert x-y \Vert _{ \infty }^{2}=\frac{1}{\lambda } \Bigl(\sup _{t\in [0,1]} \bigl\vert x(t)-y(t) \bigr\vert \Bigr)^{2}. \end{aligned}$$

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 ω-complete;

• (iii) ω satisfies the (OSC) property.

Let us consider the following integral equations system:

$$ (\mathit{IES}): \quad \textstyle\begin{cases} x(t)=\int _{0}^{1} f(t,y(s)) \,\mathrm{d}s+a(t) \quad \forall t\in [0,1] , \\ y(t)=\int _{0}^{1} g(t,x(s)) \,\mathrm{d}s+a(t) \quad \forall t\in [0,1], \end{cases} $$

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:

$$ \textstyle\begin{cases} Tx(t)=\int _{0}^{1} f(t,x(s)) \,\mathrm{d}s+a(t), \\ Sx(t)=\int _{0}^{1} g(t,x(s)) \,\mathrm{d}s+a(t), \end{cases}\displaystyle t\in [0,1]. $$

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 $t,s\in [0,1]$,

$$ \bigl\vert f\bigl(t,x(s)\bigr)-g\bigl(t,y(s)\bigr) \bigr\vert \leq -1+ \sqrt{1+ \bigl\vert x(s)-y(s) \bigr\vert }\leq -1+ \sqrt{1+ \bigl\Vert x(s)-y(s) \bigr\Vert _{\infty }} $$

and

$$ \Vert Tx-Sy \Vert _{\infty }=\sup_{t\in [0,1]} \bigl\vert Tx(t)-Sy(t) \bigr\vert =\sup_{t\in [0,1]} \int _{0}^{1} \bigl\vert f\bigl(t,x(s)\bigr)-g \bigl(t,y(s)\bigr) \bigr\vert \,\mathrm{d}s, $$

we have

$$ \Vert Tx-Sy \Vert _{\infty }\leq -1+\sqrt{1+ \bigl\Vert x(s)-y(s) \bigr\Vert _{\infty }} . $$

Since

$$ \bigl(-1+\sqrt{1+ \bigl\Vert x(s)-y(s) \bigr\Vert _{\infty }} \bigr)^{2}\leq -1+ \sqrt{1+ \bigl\Vert x(s)-y(s) \bigr\Vert _{\infty }^{2}} , $$

we have

$$ \omega _{1}(Tx,Sy)\leq -1+ \bigl(1+\omega _{1}(x,y) \bigr)^{\frac{1}{2}}. $$

Since, for all $t\in [0,1]$,

$$ x_{0}(t)\preceq Sx_{0}(t)\preceq TSx_{0}(t) \preceq STSx_{0}(t) \preceq (TS)^{2}x_{0}(t) \preceq S(TS)^{2}x_{0}(t)\preceq \cdots, $$

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

  $$ \omega _{1}(Sx,Ty)\leq k \omega _{1}(x,y)\quad \Longrightarrow \quad \omega _{1}(Sx,Ty) \leq \bigl(1+\omega _{1}(x,y) \bigr)^{k}-1. $$