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The contraction principle for mappings on a modular metric space with a graph
 Monther Rashed Alfuraidan^{1}Email author
https://doi.org/10.1186/s1366301502963
© Alfuraidan; licensee Springer. 2015
 Received: 3 December 2014
 Accepted: 6 March 2015
 Published: 1 April 2015
Abstract
We give a generalization of the Banach contraction principle on a modular metric space endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as the modular metric version of Jachymski’s fixed point result for mappings on a metric space with a graph.
Keywords
 \(\Delta_{2}\)condition
 fixed point
 modular metric spaces
 contraction mapping
 connected graph
MSC
 47H09
 46B20
 47H10
 47E10
1 Introduction
Fixed point theorems for monotone singlevalued mappings in a metric space endowed with a partial ordering have been widely investigated. These theorems are hybrids of the two most fundamental and useful theorems in fixed point theory: Banach’s contraction principle [1], Theorem 2.1, and Tarski’s fixed point theorem [2, 3]. The existence of fixed points for singlevalued mappings in partially ordered metric spaces was initially considered by Ran and Reurings in [4] who proved the following result.
Theorem 1.1
[4]
 (1)There exists a \(k\in(0,1)\) with$$d\bigl(f(x),f(y)\bigr)\preceq k d(x,y), \quad \textit{for all }x\succeq y. $$
 (2)
There exists an \(x_{0} \in X\) with \(x_{0} \preceq f(x_{0})\) or \(x_{0} \succeq f(x_{0})\).
After this, different authors considered the problem of existence of a fixed point for contraction mappings in partially ordered sets; see [5–8] and references cited therein. Nieto et al. in [8], proved the following.
Theorem 1.2
[8]
 (1)
f is continuous and there exists an \(x_{0}\in X\) with \(x_{0} \preceq f(x_{0})\) or \(x_{0} \succeq f(x_{0})\);
 (2)
\((X,d,\preceq)\) is such that for any nondecreasing \((x_{n})_{n\in N}\), if \(x_{n}\rightarrow x\), then \(x_{n} \preceq x\) for \(n \in N\), and there exists an \(x_{0} \in X\) with \(x_{0} \preceq f(x_{0})\);
 (3)
\((X,d,\preceq)\) is such that for any nonincreasing \((x_{n})_{n\in N}\), if \(x_{n}\rightarrow x\), then \(x_{n} \succeq x\) for \(n \in N\), and there exists an \(x_{0} \in X\) with \(x_{0} \succeq f(x_{0})\).
Generalizing the partial order concept of the fixed point theorems by using graphs was first established by Jachymski and Lukawska [9, 10]. Their works generalized and subsumed the works of [6, 8, 11, 12] to singlevalued mapping in metric spaces with a graph. Jachymski [9] obtained the following result.
Theorem 1.3
[9]
 (P)
for any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in X, if \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\) and \((x_{n},x_{n+1})\in E(G)\), then \((x_{n},x)\in E(G)\), for all n.
 (1)
\(F_{f} \neq\emptyset\) if and only if \(X_{f}\neq \emptyset\);
 (2)
if \(X_{f}\neq\emptyset\) and G is weakly connected, then f is a Picard operator, i.e., \(F_{f}=\{x^{\ast}\}\) and sequence \(\{f^{n}(x)\}\rightarrow x^{\ast}\) as \(n\rightarrow\infty\), for all \(x\in X \);
 (3)
for any \(x\in X_{f}\), \(f\mid_{{}[ x]_{\widetilde{G}}}\) is a Picard operator;
 (4)
if \(X_{f}\subseteq E(G)\), then f is a weakly Picard operator, i.e., \(F_{f}\neq\emptyset\) and, for each \(x\in X\), we have a sequence \(\{f^{n}(x)\}\rightarrow x^{\ast}(x)\in F_{f}\) as \(n\rightarrow\infty\).
The aim of this paper is to discuss the existence of fixed points for single Lipschitzian mappings defined on some subsets of modular metric spaces X endowed with a graph G. These modular metric spaces were introduced in [13, 14]. However, the way we approached the concept of modular metric spaces is different. Indeed we look at these spaces as the nonlinear version of the classical modular spaces as introduced by Nakano [15] on vector spaces and modular function spaces introduced by Musielack [16] and Orlicz [17]. In [18] the authors have defined and investigated the fixed point property in the framework of modular metric space and introduced the analog of the Banach contraction principle theorem in modular metric space.
2 Preliminaries
Definition 2.1
 (i)
\(x=y\) if and only if \(\omega_{ \lambda}(x,y)=0\), for all \(\lambda>0\);
 (ii)
\(\omega_{ \lambda}(x,y)= \omega_{ \lambda}(y,x) \), for all \(\lambda>0\), and \(x,y \in X\);
 (iii)
\(\omega_{ \lambda+ \mu}(x,y) \leq\omega_{ \lambda}(x,z)+ \omega_{\mu}(z,y)\), for all \(\lambda, \mu>0\) and \(x,y,z \in X\).
Definition 2.2
For more examples on modular function spaces, the reader may consult the book of Kozlowski [19], and for modular metric spaces [13, 14].
Definition 2.3
 (1)
The sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in \(X_{\omega}\) is said to be ωconvergent to \(x\in X_{\omega}\) if and only if \(\omega_{1}(x_{n},x)\rightarrow0\), as \(n\rightarrow\infty\). x will be called the ωlimit of \(\{x_{n}\}\).
 (2)
The sequence \(\{x_{n}\}_{n\in N}\) in \(X_{\omega}\) is said to be ωCauchy if \(\omega_{1}(x_{m},x_{n})\rightarrow0\), as \(m,n\rightarrow\infty\).
 (3)
A subset M of \(X_{\omega}\) is said to be ωclosed if the ωlimit of a ωconvergent sequence of M always belong to M.
 (4)
A subset M of \(X_{\omega}\) is said to be ωcomplete if any ωCauchy sequence in M is a ωconvergent sequence and its ωlimit is in M.
 (5)A subset M of \(X_{\omega}\) is said to be ωbounded if we have$$\delta_{\omega}(M)= \sup\bigl\{ \omega_{1}(x,y);x,y\in M\bigr\} < \infty. $$

If \(\lim_{n \rightarrow\infty} \omega _{\lambda}(x_{n},x) = 0\), for some \(\lambda>0\) implies \(\lim_{n \rightarrow\infty} \omega_{\lambda}(x_{n},x) = 0\), for all \(\lambda>0\).
Definition 2.4
Note that if ω satisfies the \(\Delta_{2}\)type condition, then ω satisfies the \(\Delta_{2}\)condition. The above definition will allow us to introduce the growth function in the modular metric spaces as was done in the linear case.
Definition 2.5
The following properties were proved in [20].
Lemma 2.1
(Lemma 2.1, [20])
 (1)
\(\Omega(\alpha) < \infty\), for any \(\alpha> 0\),
 (2)
Ω is a strictly increasing function, and \(\Omega(1) = 1\),
 (3)
\(\Omega(\alpha\beta) \leq\Omega(\alpha) \Omega(\beta)\), for any \(\alpha, \beta\in(0,\infty)\),
 (4)
\(\Omega^{1}(\alpha) \Omega^{1}(\beta) \leq\Omega ^{1}(\alpha\beta)\), where \(\Omega^{1}\) is the function inverse of Ω,
 (5)for any \(x, y \in X_{\omega}\), \(x \neq y\), we have$$d^{*}_{\omega}(x,y) \leq\frac{1}{\Omega^{1} (1/\omega_{1}(x,y) )}. $$
The following technical lemma will be useful later on in this work.
Lemma 2.2
[20]
Note that this lemma is crucial since the main assumption (1) on \(\{x_{n}\}\) will not be enough to imply that \(\{x_{n}\}\) is ωCauchy since ω fails the triangle inequality.
Let us finish this section with the needed graph theory terminology which will be used throughout.
Let \((X, \omega)\) be a modular metric space and M be a nonempty subset of \(X_{\omega}\). Let Δ denote the diagonal of the cartesian product \(M \times M\). Consider a directed graph \(G_{\omega}\) such that the set \(V(G_{\omega})\) of its vertices coincides with M, and the set \(E(G_{\omega})\) of its edges contains all loops, i.e., \(E(G_{\omega})\supseteq\Delta\). We assume \(G_{\omega}\) has no parallel edges (arcs), so we can identify \(G_{\omega}\) with the pair \((V(G_{\omega}),E(G_{\omega}))\). Our graph theory notations and terminology are standard and can be found in all graph theory books, like [21] and [22]. Moreover, we may treat \(G_{\omega}\) as a weighted graph (see [22], p.309) by assigning to each edge the distance between its vertices.
Definition 2.6
 (i)\(G_{\omega}\)contraction if T preserves edges of \(G_{\omega}\), i.e.,and if there exists a constant \(\alpha\in[0,1)\) such that$$\forall x,y\in M \quad \bigl((x,y)\in E(G_{\omega}) \Rightarrow \bigl(T(x),T(y)\bigr)\in E(G_{\omega})\bigr), $$$$\omega_{1}\bigl(T(x),T(y)\bigr) \leq\alpha \omega_{1}(x,y) \quad \mbox{for any } (x,y)\in E(G_{\omega}). $$
 (ii)\((\varepsilon,\alpha)\)\(G_{\omega}\)uniformly locally contraction if T preserves edges of \(G_{\omega}\) and there exists a constant \(\alpha\in[0,1)\) such that for any \((x,y)\in E(G_{\omega})\)$$\omega_{1}\bigl(T(x),T(y)\bigr) \leq\alpha \omega_{1}(x,y) \quad\mbox{whenever } \omega_{1}(x,y)< \varepsilon. $$
Definition 2.7
A point \(x \in M\) is called a fixed point of T whenever \(x= T(x) \). The set of fixed points of T will be denoted by \(\operatorname{Fix}(T)\).
3 Fixed points of \(G_{\omega}\)contractions
Throughout this section we assume that \((X, \omega)\) is a modular metric space, M be a nonempty subset of \(X_{\omega}\) and \(G_{\omega}\) is a directed graph such that \(V(G_{\omega})=M\) and \(E(G_{\omega})\supseteq \Delta\).
Our first result can be seen as an extension of Jachymski’s fixed point theorem [9] to modular metric spaces. As Jachymski [9] did, we introduce the following property.
 (P)
For any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in M, if \(x_{n}\rightarrow x\) as \(n\rightarrow\infty\) and \((x_{n},x_{n+1})\in E(G_{\omega})\), then \((x_{n},x)\in E(G_{\omega})\), for all n.
Theorem 3.1
 (1)
For any \(x\in M_{T}\), \(T_{[x]_{\widetilde{G}_{\omega}}}\) has a fixed point.
 (2)
If \(G_{\omega}\) is weakly connected, then T has a fixed point in M.
 (3)
If \(M':=\bigcup\{[x]_{\widetilde{G}_{\omega}} : x\in M_{T}\}\), then \(T_{M'}\) has a fixed point in M.
Proof
(2) Since \(M_{T}\neq\emptyset\), there exists an \(x_{0}\in M_{T}\), and since \(G_{\omega}\) is weakly connected, then \([x_{0}]_{\widetilde{G}_{\omega}}=M\) and by (1), mapping T has a fixed point.
(3) It follows easily from (1) and (2). □
Edelstein [23] has extended the classical fixed point theorem for contractions to the case when X is a complete εchainable metric space and the mapping \(T:X\rightarrow X\) is an \((\varepsilon,k)\)uniformly locally contraction. Here we investigate Edelstein’s result in modular metric spaces endowed with a graph. First let us introduce the εchainable concept in modular metric spaces with a graph. Our definition is slightly different from the one used in the classical metric spaces since the modulars fail in general the triangle inequality (see also [24]).
Definition 3.1
Let \((X, \omega)\) be a modular metric space, \(M=V(G_{\omega})\) be a nonempty subset of \(X_{\omega}\). M is said to be finitely εchainable (where \(\varepsilon>0\) is fixed) if and only if there exists an \(N \geq1\) such that for any \(a,b\in M\) with \((a,b)\in E(G_{\omega})\) there is an \(N,\varepsilon\)chain from a to b (that is, a finite set of vertices \(x_{0},x_{1}, \ldots,x_{N}\in V(G_{\omega})=M\) such that \(x_{0}=a\), \(x_{N}=b\), \((x_{i},x_{i+1})\in E(G_{\omega})\) and \(\omega_{1}(x_{i},x_{i+1})< \varepsilon\), for all \(i=0,1,2,\ldots,N1\)).
We have the following result.
Theorem 3.2
Let \((X, \omega)\) be a modular metric space. Suppose that ω is a convex regular modular metric which satisfies the \(\Delta_{2}\)type condition. Assume that \(M=V(G_{\omega})\) is a nonempty ωcomplete and ωbounded subset of \(X_{\omega }\) which is finitely εchainable, for some fixed \(\varepsilon> 0\). Suppose that the triple \((M,d^{\ast}_{\omega},G_{\omega})\) has property (P). Let \(T: M \rightarrow M\) be \((\varepsilon, \alpha)\)\(G_{\omega}\)uniformly locally contraction map. Then T has a fixed point in the vertex set of the graph M.
Proof
Declarations
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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