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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 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
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 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
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
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 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
Condition (ii) insures that \(\{x_{n}\}\) is G-nondecreasing. 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), \(x=x_{2n-1}\) 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 ω-convergent 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 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:
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 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
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 self-mapping 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 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).
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 \(d(x,y)= \vert x-y \vert \) 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 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
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 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
Condition (ii) insures that \(\{x_{n}\}\) is G-nondecreasing. 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 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 \(x,y\in X\),
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 ω-complete;
-
(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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References
Abdou, A.A.N., Khamsi, M.A.: Fixed point results of pointwise contractions in modular metric spaces. Fixed Point Theory Appl. 2013(1), 163 (2013)
Alber, Y.A.I., Guerre-Delabriere, S.: Principle of weakly contractive maps in Hilbert spaces. Oper. Theory, Adv. Appl. 98, 7–22 (1997)
Bishta, R.K., Pant, R.P.: Contractive definitions and discontinuity at fixed point. Appl. Gen. Topol. 18(1), 173–182 (2017)
Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20(2), 458–464 (1969)
Chaira, K., Eladraoui, A., Kabil, M.: Some fixed point theorems of generalized \((\psi _{1}, \psi _{2}, \varphi ,\phi )\)-contraction in partially ordered modular metric spaces. Int. J. Pure Appl. Math. 119(4), 573–592 (2018)
Chaira, K., Eladraoui, A., Kabil, M., Lazaiz, S.: Extension of Kirk–Saliga fixed point theorem in a metric space with a reflexive digraph. Int. J. Math. Math. Sci. 2018, 1471256 (2018)
Chistyakov, V.V.: Metric Modular Spaces: Theory and Applications. Springer, Cham (2015). ISBN 978-3-319-25283-4
Khan, L.: Fixed point theorem for weakly contractive maps in metrically convex spaces under C-class function. Nonlinear Funct. Anal. Appl. 25(1), 153–160 (2020)
Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001)
Suantai, S., Chaipornjareansri, S.: Best proximity points of \(\alpha -\beta -\psi \)-proximal contractive mappings in complete metric spaces endowed with graphs. Nonlinear Funct. Anal. Appl. 24(4), 759–773 (2019)
Turkoglu, D., Manav, N.: Fixed point theorems in a new type of modular metric spaces. Fixed Point Theory Appl. 2018, 25 (2018)
Xue, Z.: The convergence theorems of fixed points for generalized φ-weakly contractive mappings. Nonlinear Funct. Anal. Appl. 22(4), 803–809 (2017)
Zhang, X.: Common fixed point theorems for some new generalized contractive type mappings. J. Math. Anal. Appl. 333, 780–786 (2007)
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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/s13663-021-00690-8
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DOI: https://doi.org/10.1186/s13663-021-00690-8