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 Open Access
On monotone Ćirić quasicontraction mappings with a graph
 Monther Rashed Alfuraidan^{1}Email author
https://doi.org/10.1186/s1366301503412
© Alfuraidan 2015
 Received: 13 February 2015
 Accepted: 25 May 2015
 Published: 23 June 2015
Abstract
In this paper, we obtain sufficient conditions for the existence of fixed points for monotone quasicontraction mappings in metric and modular metric spaces endowed with a graph. This is the extension of Ran and Reurings and Jachymski fixed point theorems for monotone contraction mappings in partially ordered metric spaces and in metric spaces endowed with a graph to the case of quasicontraction mappings introduced by Ćirić.
Keywords
 fixed point
 modular metric space
 monotone mappings
 quasicontraction
 directed graph
MSC
 47H09
 47H10
1 Introduction
Banach’s contraction principle [1] is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condition on the mapping is simple and easy to test, it requires only a complete metric space for its setting, and it finds almost canonical applications in the theory of differential and integral equations. Over the years, many mathematicians tried successfully to extend this fundamental theorem. Recently a version of this theorem was given in partially ordered metric spaces [2, 3] and in metric spaces with a graph [4]. In this work, we discuss the case of quasicontractive mappings defined in partially ordered metric spaces and modular metric spaces endowed with a graph.
For more on metric fixed point theory, the reader may consult the book [5].
2 Graph basic definitions
The terminology of graph theory instead of partial ordering gives a wider picture and yield interesting generalization of the Banach contraction principle. In this section, we give the basic graph theory definitions and notations which will be used throughout.
Given a digraph \(G = (V ,E)\), a (di)path of G is a sequence \(a_{0}, a_{1}, \ldots , a_{n}, \ldots\) with \((a_{i}, a_{i+1} )\in E(G)\) for each \(i = 0, 1, 2, \ldots\) . A finite path \((a_{0}, a_{1}, \ldots, a_{n})\) is said to have length \(n+1\) for \(n \in\mathbb{N}\). A closed directed path of length \(n>1\) from x to y, i.e., \(x=y\), is called a directed cycle. An acyclic digraph is a digraph that has no directed cycle. A digraph is connected if there is a finite (di)path joining any two of its vertices and it is weakly connected if \(\widetilde{G}\) is connected.
Definition 2.1
As Jachymski [4] did, we introduce the following property.
 (∗):

For any \((x_{n})_{n \geq1}\) in X, if \(x_{n} \rightarrow x\) and \((x_{n}, x_{n+1})\in E(G)\) for \(n \geq1\), then there is a subsequence \((x_{k_{n}})_{n \geq1}\) with \((x_{k_{n}}, x)\in E(G)\) for \(n \geq1\).
 (∗∗):

For any \((x_{n})_{n \geq1}\) in X, if \(x_{n} \rightarrow x\) and \((x_{n}, x_{n+1})\in E(G)\) for \(n \geq1\), then \((x_{n}, x)\in E(G)\) for every \(n \geq1\).
Let us finish this section with the following example of a transitive cyclic digraph which cannot be a partial order. Therefore our approach is different from the one used in [6] which is based on the use of a partial order in Banach and metric spaces.
Example 2.1

\((x,y)\in E(G)\) if and only if \(x_{2} \leq y_{2}\), where \(x=(x_{1}, x_{2})\) and \(y=(y_{1}, y_{2})\) are in \(\mathbb{R}^{2}\).
For more examples on the use of graph theory with fixed point theory, the reader may see [7].
3 GMonotone quasicontraction mappings in metric spaces
As a generalization to the Banach contraction principle, Ćirić [8] introduced the concept of quasicontraction mappings. In this section, we investigate monotone mappings which are quasicontraction mappings. Throughout this section we assume that \((X, d)\) is a metric space and G is a reflexive transitive digraph defined on X. Moreover, we assume that \(E(G)\) has property (∗) and Gintervals are closed. Recall that a Ginterval is any of the subsets \([a,\rightarrow) = \{x \in C; (a,x)\in E(G) \}\) and \((\leftarrow ,b] = \{x \in C; (x,b)\in E(G)\}\) for any \(a,b \in C\).
Definition 3.1
 (1)
Gmonotone if T is edge preserving, i.e., \((T(x),T(y))\in E(G)\) whenever \((x,y)\in E(G)\) for any \(x, y \in C\).
 (2)Gmonotone quasicontraction if T is Gmonotone and there exists \(k < 1\) such that for any \(x,y \in C\), \((x,y)\in E(G)\), we have$$ d\bigl(T(x),T(y) \bigr) \leq k \max \bigl(d(x,y); d\bigl(x,T(x)\bigr); d\bigl(y, T(y)\bigr); d\bigl(x,T(y)\bigr); d\bigl(y, T(x)\bigr) \bigr). $$
The point \(x \in C\) is called a fixed point of T if \(T(x) = x\). The set of fixed points of T will be denoted by \(\operatorname{Fix}(T)\).
The following technical lemma is crucial to prove the main result of this section.
Lemma 3.1
Proof
Using Lemma 3.1, we prove the main result of this section.
Theorem 3.1
 (i)\(\{T^{n}(x)\}\) converges to \(\omega\in C\) which is a fixed point of T and \((x,\omega)\in E(G)\). Moreover, we have$$d\bigl( T^{n}(x), \omega\bigr) \leq k^{n} \delta(x), \quad n \geq1. $$
 (ii)
If z is a fixed point of T such that \((x,z)\in E(G)\), then \(z = \omega\).
Proof
In the next section, we discuss the validity of Theorem 3.1 in modular metric spaces. This is a very important class of spaces since they are similar to metric spaces in their structure but without the triangle inequality and offer a wide range of applications.
4 GMonotone quasicontraction mappings in modular metric spaces
Definition 4.1
 (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 4.2
For more examples on modular function spaces, the reader my consult the book of Kozlowski [11] and for modular metric spaces [9, 10].
Definition 4.3
 (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 belongs 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$$\delta_{\omega}(M)= \sup\bigl\{ \omega_{1}(x,y);x,y\in M\bigr\} < \infty. $$
 (7)ω is said to satisfy the Fatou property if and only if for any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in \(X_{\omega}\) ωconvergent to x, we havefor any \(y \in X_{\omega}\).$$\omega_{1}(x,y) \leq\liminf_{n\rightarrow\infty} \omega_{1}(x_{n},y) $$
Let \((X,\omega)\) be a modular metric space and G be a reflexive digraph defined on X.
Definition 4.4
 (i)
Gmonotone if T is edge preserving, i.e., \((T(x),T(y))\in E(G)\) whenever \((x,y)\in E(G)\) for any \(x, y \in C\);
 (ii)Gmonotone ωquasicontraction if T is Gmonotone and there exists \(k < 1\) such that for any \(x,y \in C\), \((x,y)\in E(G)\), we have$$\begin{aligned} \omega_{1}\bigl(T(x),T(y) \bigr) \leq& k \max \bigl( \omega_{1}(x,y); \omega_{1}\bigl(x,T(x)\bigr); \omega_{1}\bigl(y, T(y)\bigr); \\ &\omega_{1}\bigl(x,T(y)\bigr); \omega_{1}\bigl(y, T(x)\bigr) \bigr). \end{aligned}$$
 (∗∗∗):

For any \((x_{n})_{n \geq1}\) in X, if \(x_{n}\) ωconverges to x and \((x_{n}, x_{n+1})\in E(G)\) for \(n \geq1\), then there is a subsequence \((x_{k_{n}})_{n \geq1}\) with \((x_{k_{n}}, x)\in E(G)\) for \(n \geq1\).

For any \((x_{n})_{n \geq1}\) in X, if \(x_{n}\) ωconverges to x and \((x_{n}, x_{n+1})\in E(G)\) for \(n \geq1\), then \((x_{n}, x)\in E(G)\) for every \(n \geq1\).
Throughout this section, we assume that \((X,\omega)\) is a modular metric space, G is a reflexive transitive digraph defined on X and \(E(G)\) has property (∗∗∗).
Lemma 4.1
Using Lemma 4.1, we prove the main result of this section.
Theorem 4.1
 (i)\(\{T^{n}(x)\}\) ωconverges to \(z \in C\) which is a fixed point of T and \((x,z)\in E(G)\), provided \(\omega_{1}(z,T(z)) < \infty\) and \(\omega_{1}(x,T(z)) < \infty\). Moreover, we have$$\omega_{1}\bigl( T^{n}(x), z\bigr) \leq k^{n} \delta_{\omega}(x), \quad n \geq1. $$
 (ii)
If w is a fixed point of T such that \((x,w)\in E(G)\) and \(\omega_{1}(T^{n}(x),w) < \infty\) for any \(n \geq1\), then \(z = w\).
Proof
Declarations
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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