Remarks on monotone multivalued mappings on a metric space with a graph
- Monther Rashed Alfuraidan^{1}Email author
https://doi.org/10.1186/s13660-015-0712-6
© Alfuraidan 2015
Received: 9 November 2014
Accepted: 8 May 2015
Published: 18 June 2015
Abstract
Let \((X,d)\) be a metric space and \(J: X\rightarrow2^{X}\) be a multivalued mapping. In this work, we discuss the definition of G-contraction mappings introduced by Beg et al. (Comp. Math. Appl. 60:1214-1219, 2010) and show that it is restrictive and fails to give the main result of (Beg et al. in Comp. Math. Appl. 60:1214-1219, 2010). In this work, we give a new definition of the G-contraction and obtain sufficient conditions for the existence of fixed points for such mappings.
Keywords
MSC
1 Introduction
Fixed point theorems for monotone single-valued 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: the Banach contraction principle [1], Theorem 2.1, and Tarski’s fixed point theorem [2, 3]. Generalizing the Banach contraction principle for multivalued mapping to metric spaces, Nadler [4] obtained the following result.
Theorem 1.1
([4])
A number of extensions and generalizations of Nadler’s theorem were obtained by different authors; see for instance [5, 6] and references cited therein. The Tarski theorem was extended to multivalued mappings by different authors; see [7–9]. The existence of fixed points for single-valued mappings in partially ordered metric spaces was initially considered by Ran and Reurings in [10], who proved the following result.
Theorem 1.2
([10])
- 1.There exists \(k\in[0,1)\) with$$d\bigl(f(x),f(y)\bigr)\leq k d(x,y),\quad \textit{for all }x \succeq y. $$
- 2.
There exists an \(x_{0} \in X\) with \(x_{0} \preceq f(x_{0})\) or \(x_{0} \succeq f(x_{0})\).
After this, various authors considered the problem of existence of a fixed point for contraction mappings in partially ordered metric spaces; see [11–14] and references cited therein. Nieto et al. in [14] extended the ideas of [10] to prove the existence of solutions to some differential equations. Recently, two results have appeared, giving sufficient conditions for f to be a PO, if \((X,d)\) is endowed with a graph. The first result in this direction was given by Jachymski and Lukawska [15, 16], who generalized the results of [12, 14, 17, 18] to single-valued mapping in metric spaces with a graph instead of partial ordering.
The aim of this paper is twofold: first to give a correct definition of monotone multivalued mappings, second to extend the conclusion of Theorem 1.2 to the case of monotone multivalued mappings in metric spaces endowed with a graph.
2 Preliminaries
It seems that the terminology of graph theory instead of partial ordering gives a clearer picture and yields an interesting generalization of the Banach contraction principle. Let us begin this section with such a terminology for metric spaces as will be used throughout.
Definition 2.1
([21])
Definition 2.2
([4])
Example 2.1
Example 2.2
Let \(I^{2}=\{(x,y) : 0\leq x \leq1 \mbox{ and } 0\leq y \leq 1\}\), and let \(F: I^{2} \rightarrow \mathcal {CB}(I^{2})\) be defined by \(F(x,y)\) is the line segment in \(I^{2}\) from the point \((\frac{1}{2}x,0)\) to the point \((\frac{1}{2}x,1)\) for each \((x,y)\in I^{2}\). It is easy to see that F is a multivalued contraction mapping with the set of fixed points \(\{(0,y) : 0\leq y \leq1 \}\).
Next we introduce the concept of monotone multivalued mappings. In [22], the authors offered the following definition.
Definition 2.3
([22], Def. 2.6)
Definition 2.4
Property 1
For any sequence \((x_{n})_{n\in\mathbb{N}}\) in X, if \(x_{n} \rightarrow x\) and \((x_{n}, x_{n+1})\in E(G)\) for \(n\in\mathbb{N}\), then \((x_{n}, x)\in E(G)\).
3 Main results
We begin with the following theorem, which gives the existence of a fixed point for monotone multivalued mappings in metric spaces endowed with a graph.
Theorem 3.1
- (1)
For any \(x\in X_{T}\), \(T|_{[x]_{\widetilde{G}}}\) has a fixed point.
- (2)
If G is weakly connected, then T has a fixed point in G.
- (3)
If \(X':=\bigcup\{[x]_{\widetilde{G}} : x\in X_{T}\}\), then \(T|_{X'}\) has a fixed point in X.
- (4)
If \(T(X)\subseteq E(G)\) then T has a fixed point.
- (5)
\(\operatorname {Fix}T\neq\emptyset\) if and only if \(X_{T}\neq\emptyset\).
Proof
2. Since \(X_{T}\neq\emptyset\), there exists an \(x_{0}\in X_{T}\), and since G is weakly connected, then \([x_{0}]_{\widetilde {G}}=X\) and by 1, mapping T has a fixed point.
3. It follows easily from 1 and 2.
4. \(T(X) \subseteq E(G)\) implies that all \(x\in X\) are such that there exists some \(y\in T(x)\) with \((x,y)\in E(G)\); so \(X_{T}=X\) and by 2 and 3, T has a fixed point.
5. Assume \(\operatorname {Fix}T\neq\emptyset\). This implies that there exists an \(x \in \operatorname {Fix}T\) such that \(x\in T(x)\). \(\triangle\subseteq E(G)\) therefore \((x,x)\in E(G)\), which implies that \(x\in X_{T}\). So \(X_{T}\neq \emptyset\). Conversely if \(X_{T}\neq\emptyset\), then \(\operatorname {Fix}T\neq\emptyset\), follows from 2 and 3. □
Remark 3.1
The missing information in Theorem 3.1 is the uniqueness of the fixed point. In fact, we do have a partial positive answer to this question. Indeed if \(\bar{u}\) and \(\bar{w}\) are two fixed points of T such that \((\bar {u},\bar{w})\in E(G)\), then we must have \(\bar{u} = \bar{w}\). In general T may have more than one fixed point.
Remark 3.2
If we assume G is such that \(E(G):=X\times X\) then clearly G is connected and our Theorem 3.1 gives Nadler’s theorem [4].
The following is a direct consequence of Theorem 3.1.
Corollary 3.1
Let \((X, d)\) be a complete metric space and the triple \((X,d,G)\) have the Property 1. If G is weakly connected then every G-contraction \(T: X\rightarrow \mathcal {CB}(X)\) such that \((x_{0}, x_{1})\in E(G)\), for some \(x_{1}\in T(x_{0})\), has a fixed point.
Example 3.1
Note that, for all \(x,y\in X\) with edge between x and y, there is an edge between \(T(x)\) and \(T(y)\). Also there is a path between x and y implies that there is a path between \(T(x)\) and \(T(y)\). Moreover, T is a G-contraction with all other assumptions of Theorem 3.1 satisfied and T has 0 as a fixed point.
Declarations
Acknowledgements
The author is grateful to King Fahd University of Petroleum and Minerals for supporting this research. He would also like to thank Professor M. A. Khamsi who read carefully the earlier versions of this paper and suggested some improvements.
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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