 Research
 Open Access
 Published:
The contraction principle for mappings on a modular metric space with a graph
Fixed Point Theory and Applications volume 2015, Article number: 46 (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.
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]
Let \((X,\preceq)\) be a partially ordered set such that every pair \(x,y\in X\) has an upper and lower bound. Let d be a metric on X such that \((X,d)\) is a complete metric space. Let \(f: X\rightarrow X\) be a continuous monotone (either order preserving or order reversing) mapping. Suppose that the following conditions hold:

(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 \(x_{0} \succeq f(x_{0})\).
Then f is a Picard operator (PO), that is, f has a unique fixed point \(x^{*}\in X\) and for each \(x\in X\), \(\lim_{n\rightarrow \infty} f^{n}(x)=x^{*}\).
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]
Let \((X,d)\) be a complete metric space endowed with a partial ordering ⪯. Let \(f: X \rightarrow X\) be an order preserving mapping such that there exists a \(k\in[0,1)\) with
Assume that one of the following conditions holds:

(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 \(x_{0} \succeq f(x_{0})\).
Then f has a fixed point. Moreover, if \((X,\preceq)\) is such that every pair of elements of X has an upper or a lower bound, then f is a PO.
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]
Let \((X,d)\) be a complete metric space and let the triplet \((X,d,G)\) have the following property:

(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.
Let \(f:X\rightarrow X\) be a Gcontraction. Then the following statements hold:

(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.
Preliminaries
Let X be a nonempty set. Throughout this paper for a function \(\omega: (0, \infty) \times X \times X \rightarrow(0,\infty )\) we will write
for all \(\lambda>0\) and \(x, y \in X\).
Definition 2.1
A function \(\omega :(0, \infty) \times X \times X \rightarrow[0, \infty]\) is said to be a modular on X if it satisfies the following axioms:

(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 \(x,y,z \in X\).
If instead of (i), we have only the condition (i′)
then ω is said to be a pseudomodular on X. A modular ω on X is said to be regular if the following weaker version of (i) is satisfied:
Finally, ω is said to be convex if for \(\lambda,\mu>0\) and \(x,y,z\in X\), it satisfies the inequality
Note that for a pseudomodular ω on a set X, and any \(x,y \in X\), the function \(\lambda\rightarrow\omega_{\lambda}(x,y)\) is nonincreasing on \((0, \infty)\). Indeed, if \(0 <\mu<\lambda\), then
Definition 2.2
Let ω be a pseudomodular on X. Fix \(x_{0} \in X\). The two sets
and
are said to be modular spaces (around \(x_{0}\)).
We obviously have \(X_{\omega}\subset X_{\omega}^{\ast}\). In general this inclusion may be proper. It follows from [13, 14] that if ω is a modular on X, then the modular space \(X_{\omega}\) can be equipped with a (nontrivial) distance, generated by ω and given by
for any \(x, y \in X_{\omega}\). If ω is a convex modular on X, according to [13, 14] the two modular spaces coincide, i.e. \(X_{\omega}^{*}= X_{\omega}\), and this common set can be endowed with the distance \(d^{*}_{\omega}\) given by
for any \(x, y \in X_{\omega}\). These distances will be called Luxemburg distances.
First attempts to generalize the classical function spaces of the Lebesgue type \(L^{p}\) were made in the early 1930s by Orlicz and Birnbaum in connection with orthogonal expansions. Their approach consisted in considering spaces of functions with some growth properties different from the power type growth control provided by the \(L^{p}\)norms. Namely, they considered the function spaces defined as follows:
where \(\varphi:[0,\infty] \rightarrow[0,\infty]\) was assumed to be a convex function increasing to infinity, i.e. the function which to some extent behaves similarly to power functions \(\varphi(t) = t^{p}\). A modular function spaces \(L^{\varphi}\) furnishes a wonderful example of a modular metric space. Indeed define the function ω by
for all \(\lambda>0\), and \(f,g \in L^{\varphi}\). Then ω is a modular metric on \(L^{\varphi}\). Moreover, the distance \(d_{\omega }^{*}\) is exactly the distance generated by the Luxemburg norm on \(L^{\varphi}\).
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
Let \(X_{\omega}\) be a modular metric space.

(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
$$\delta_{\omega}(M)= \sup\bigl\{ \omega_{1}(x,y);x,y\in M\bigr\} < \infty. $$
In general if \(\lim_{n \rightarrow\infty} \omega_{\lambda}(x_{n},x) = 0\), for some \(\lambda >0\), then we may not have \(\lim_{n \rightarrow\infty} \omega_{\lambda}(x_{n}, x) = 0\), for all \(\lambda >0\). Therefore, as in modular function spaces, we will say that ω satisfies the \(\Delta_{2}\)condition

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 \(\lambda>0\).
In [13] and [14], one will find a discussion of the connection between ωconvergence and metric convergence with respect to the Luxemburg distances. In particular, we have
for any \(\{x_{n}\} \in X_{\omega}\) and \(x \in X_{\omega}\). In particular we have ωconvergence and \(d_{\omega}\) convergence are equivalent if and only if the modular ω satisfies the \(\Delta_{2}\)condition. Moreover, if the modular ω is convex, then we know that \(d^{\ast }_{\omega}\) and \(d_{\omega}\) are equivalent which implies
for any \(\{x_{n}\} \in X_{\omega}\) and \(x \in X_{\omega}\) [13, 14].
Definition 2.4
Let \((X,\omega)\) be a modular metric space. We will say that ω satisfies the \(\Delta_{2}\)type condition if for any \(\alpha> 0\), there exists a \(C > 0\) such that
for any \(\lambda> 0\), \(x, y \in X_{\omega}\), with \(x \neq y\).
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
Let \((X,\omega)\) be a modular metric space. Define the growth function Ω by
for any \(\alpha> 0\).
The following properties were proved in [20].
Lemma 2.1
(Lemma 2.1, [20])
Let \((X,\omega)\) be a modular metric space. Assume that ω is a convex regular modular metric which satisfies the \(\Delta_{2}\)type condition. Then

(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]
Let \((X,\omega)\) be a modular metric space. Assume that ω is a convex regular modular metric which satisfies the \(\Delta_{2}\)type condition. Let \(\{ x_{n}\}\) be a sequence in \(X_{\omega}\) such that
where K is an arbitrary nonzero constant and \(\alpha\in(0,1)\). Then \(\{x_{n}\}\) is Cauchy for both ω and \(d_{\omega}^{*}\).
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.
By \(G^{1}\) we denote the conversion of a graph G, i.e., the graph obtained from G by reversing the direction of edges. Thus we have
A digraph G is called an oriented graph if whenever \((u,v)\in E(G)\), then \((v,u)\notin E(G)\). The letter \(\widetilde{G}\) denotes the undirected graph obtained from G by ignoring the direction of edges. Actually, it will be more convenient for us to treat \(\widetilde{G}\) as a directed graph for which the set of its edges is symmetric. Under this convention,
We call \((V',E')\) a subgraph of G if \(V'\subseteq V(G)\), \(E'\subseteq E(G)\), and for any edge \((x,y)\in E'\), \(x, y\in V'\).
If x and y are vertices in a graph G, then a (directed) path in G from x to y of length N is a sequence \((x_{i})_{i=1}^{i=N}\) of \(N + 1\) vertices such that \(x_{0} = x\), \(x_{N} = y\) and \((x_{n1},x_{n})\in E(G)\) for \(i = 1,\ldots,N\). A graph G is connected if there is a directed path between any two vertices. G is weakly connected if \(\widetilde{G}\) is connected. If G is such that \(E(G)\) is symmetric and x is a vertex in G, then the subgraph \(G_{x}\) consisting of all edges and vertices which are contained in some path beginning at x is called the component of G containing x. In this case \(V(G_{x}) =[x]_{G}\), where \([x]_{G}\) is the equivalence class of the following relation ℛ defined on \(V(G)\) by the rule:
Clearly \(G_{x}\) is connected.
Definition 2.6
Let \((X, \omega)\) be a modular metric space and M be a nonempty subset of \(X_{\omega}\). A mapping \(T: M \rightarrow M\) is called

(i)
\(G_{\omega}\)contraction if T preserves edges of \(G_{\omega}\), i.e.,
$$\forall x,y\in M \quad \bigl((x,y)\in E(G_{\omega}) \Rightarrow \bigl(T(x),T(y)\bigr)\in E(G_{\omega})\bigr), $$and if there exists a constant \(\alpha\in[0,1)\) such that
$$\omega_{1}\bigl(T(x),T(y)\bigr) \leq\alpha \omega_{1}(x,y) \quad \mbox{for any } (x,y)\in E(G_{\omega}). $$ 
(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 \(\operatorname{Fix}(T)\).
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.
We say that the triple \((M,d^{\ast}_{\omega},G_{\omega})\) has Property (P) if

(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.
Note that property (P) is precisely the Nieto et al. [8] hypothesis relaxing continuity assumption as in Theorem 1.2((2) and (3)) rephrased in terms of edges.
Theorem 3.1
Let \((X,\omega)\) be a modular metric space with a graph \(G_{\omega}\). 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 ωbounded, ωcomplete subset of \(X_{\omega}\) and the triple \((M,d^{\ast}_{\omega},G_{\omega})\) has property (P). Let \(T:M \rightarrow M\) be \(G_{\omega}\)contraction map and \(M_{T}:=\{x\in M; (x,Tx)\in E(G_{\omega}) \}\). If \((x_{0},T(x_{0}))\in E(G_{\omega})\), then the following statements hold:

(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
(1) As \((x_{0}, T(x_{0}))\in E(G_{\omega})\), then \(x_{0} \in M_{T}\). Since T is a \(G_{\omega}\)contraction, there exists a constant \(\alpha\in[0,1)\) such that \((T(x_{0}),T(T(x_{0})))\in E(G_{\omega})\) and
By induction, we construct a sequence \(\{x_{n}\}\) such that \(x_{n+1}:=T(x_{n})\), \((x_{n} ,x_{n+1})\in E(G_{\omega})\), and
for any \(n \geq1\). Since M is ωbounded, we have
for any \(n \geq1\). The technical Lemma 2.2 implies that \(\{x_{n}\}\) is ωCauchy. Since M is ωcomplete, therefore \(\{x_{n}\}\) ωconverges to some point \(x\in M\). By property (P), \((x_{n},x)\in E(G_{\omega})\) for all n, and hence
We conclude that \(\lim_{n\rightarrow\infty} \omega _{1}(x_{n+1},T(x))=0\). Using the properties of ω, we have
for all \(n\geq1\). This implies \(\omega_{2}(x,T(x))=0\). Therefore, \(x =T(x)\), i.e., x is a fixed point of T. As \((x_{0},x)\in E(G_{\omega})\), we have \(x\in[x_{0}]_{\widetilde{G}_{\omega}}\).
(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
Since M is finitely εchainable, there exists an \(N \geq 1\) such that for any \(a,b\in M\) with \((a,b)\in E(G_{\omega})\) there is a finite set of vertices \(x_{0},x_{1}, \ldots,x_{N}\in M\) 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 \(i=0,1,2,\ldots,N1\). For any \(x, y \in M\) define
where the infimum is taken over all \(N,\varepsilon\)chains \(x_{0},x_{1}, \ldots,x_{N}\) from x to y. Since M is finitely εchainable it follows that \(\omega^{*}(x,y) < \infty\), for any \(x, y \in M\). Using the basic properties of ω, we get
for any \(x,y \in M\) with \((x,y)\in E(G_{\omega})\). Moreover, if \(\omega _{1}(x,y) < \varepsilon\), then we have \(\omega^{*}(x,y) \leq\omega _{1}(x,y)\), for any \(x,y \in M\) with \((x,y)\in E(G_{\omega})\). Fix \(x \in M\). Set \(z_{0} = x\) and \(z_{1} = T(z_{0})\) with \((z_{0},z_{1})\in E(G_{\omega})\). Let \(x_{0}, \ldots, x_{N}\) be an \(N,\varepsilon\)chain from \(z_{0}\) to \(z_{1}\). Such an \(N,\varepsilon\)chain exists since M is finitely εchainable. Since T is \((\varepsilon, \alpha)\)\(G_{\omega}\)uniformly locally contraction map, there exists a constant \(\alpha\in[0,1)\) such that
for every i. Clearly this implies that \(T(x_{0}),T(x_{1}),\ldots,T(x_{N})\) is \(N,\varepsilon \)chain from \(T(z_{0})\) to \(T(z_{1})\) and
where \(z_{2}=T(z_{1})\). By induction, we construct the sequence \(\{z_{n}\} \in M\) with \((z_{n},z_{n+1})\in E(G_{\omega})\) such that
for any \(n \geq1\), where \(z_{n+1}=T(z_{n})\). Obviously we have \(\omega ^{*}(z_{n},z_{n+1}) \leq\alpha^{n} \omega^{*}(z_{0},z_{1})\), for any \(n \geq1\). Since ω satisfies the \(\Delta_{2}\)type condition, there exists \(C > 0\) such that
for any \(n \geq1\). Lemma 2.2 implies that \(\{z_{n}\}\) is ωCauchy. Since M is ωcomplete, then \(\{z_{n}\}\) ωconverges to some \(z \in M\). We claim that z is a fixed point of T. By property (P), \((z_{n},z)\in E(G_{\omega})\) for any \(n\geq 1\). Using the ideas developed above, we have
for any \(n \geq1\). Since
and
for any \(n \geq1\), we conclude that \(\omega_{N+1}(T(z),z)=0\). Indeed there exists \(n_{0}\) such that for any \(n \geq n_{0}\) we have \(\omega _{1}(z_{n},z) < \varepsilon\). By definition of \(\omega^{*}\) we have \(\omega ^{*}(z_{n},z)< \omega_{1}(z_{n},z)\), for \(n \geq n_{0}\). Hence \(\lim_{n \rightarrow\infty} \omega^{*}(z_{n},z)=0\). Therefore \(\omega _{N+1}(T(z),z)=0\) holds. Since ω is regular we get \(z = T(z)\), i.e., z is a fixed point of T as claimed. □
References
Kirk, WA, Goebel, K: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)
Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2003)
Tarski, A: A lattice theoretical fixed point and its application. Pac. J. Math. 5, 285309 (1955)
Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 14351443 (2004)
Beg, I, Butt, AR: Fixed point for set valued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal. 71, 36993704 (2009)
Drici, Z, McRae, FA, Devi, JV: Fixed point theorems in partially ordered metric space for operators with PPF dependence. Nonlinear Anal. 67, 641647 (2007)
Harjani, J, Sadarangani, K: Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations. Nonlinear Anal. 72, 11881197 (2010). doi:10.1016/j.na.2009.08.003
Nieto, JJ, Pouso, RL, RodriguezLopez, R: Fixed point theorems in ordered abstract spaces. Proc. Am. Math. Soc. 135, 25052517 (2007)
Jachymski, J: The contraction principle for mappings on a metric space with a graph. Proc. Am. Math. Soc. 1(136), 13591373 (2008)
Lukawska, GG, Jachymski, J: IFS on a metric space with a graph structure and extension of the KeliskyRivlin theorem. J. Math. Anal. Appl. 356, 453463 (2009)
O’Regan, D, Petrusel, A: Fixed point theorems for generalized contraction in ordered metric spaces. J. Math. Anal. Appl. 341, 12411252 (2008)
Petrusel, A, Rus, IA: Fixed point theorems in ordered Lspaces. Proc. Am. Math. Soc. 134, 411418 (2005)
Chistyakov, VV: Modular metric spaces, I. Basic concepts. Nonlinear Anal. 72(1), 114 (2010)
Chistyakov, VV: Modular metric spaces, II. Application to superposition operators. Nonlinear Anal. 72(1), 1530 (2010)
Nakano, H: Modulared SemiOrdered Linear Spaces, i+288 pp. Maruzen, Tokyo (1950)
Musielak, J: Orlicz Spaces and Modular Spaces. Lecture Notes in Math., vol. 1034. Springer, Berlin (1983)
Orlicz, W: Collected Papers. Part I, II. PWN, Warsaw (1988)
Abdou, AN, Khamsi, MA: Fixed points results of pointwise contractions in modular metric spaces. Fixed Point Theory Appl. 2013, 163 (2013)
Kozlowski, WM: Modular Function Spaces. Series of Monographs and Textbooks in Pure and Applied Mathematics, vol. 122. Dekker, New York (1988)
Abdou, AN, Khamsi, MA: Fixed points of multivalued contraction mappings in modular metric spaces. Fixed Point Theory Appl. 2014, 249 (2014). doi:10.1186/168718122014249
Chartrand, G, Lesniak, L, Zhang, P: Graphs & Digraphs. CRC Press, New York (2011)
Johnsonbaugh, R: Discrete Mathematics. Prentice Hall, New York (1997)
Edelstein, M: An extension of Banach’s contraction principle. Proc. Am. Math. Soc. 12, 710 (1961)
BenElMechaiekh, H: The RanReurings fixed point theorem without partial order: a simple proof. J. Fixed Point Theory Appl. (2015). doi:10.1007/s1178401502183
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Rights and permissions
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.
About this article
Cite this article
Alfuraidan, M.R. The contraction principle for mappings on a modular metric space with a graph. Fixed Point Theory Appl 2015, 46 (2015). https://doi.org/10.1186/s1366301502963
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s1366301502963
MSC
 47H09
 46B20
 47H10
 47E10
Keywords
 \(\Delta_{2}\)condition
 fixed point
 modular metric spaces
 contraction mapping
 connected graph