Fixed points of monotone nonexpansive mappings on a hyperbolic metric space with a graph
 Monther Rashed Alfuraidan^{1}Email author and
 Mohamed Amine Khamsi^{2}
https://doi.org/10.1186/s1366301502945
© Alfuraidan and Khamsi; licensee Springer. 2015
Received: 4 November 2014
Accepted: 27 January 2015
Published: 27 March 2015
Abstract
In this work, we define the concept of Gmonotone nonexpansive multivalued mappings defined on a metric space with a graph G. Then we obtain sufficient conditions for the existence of fixed points for such mappings in hyperbolic metric spaces. This is the first kind of such results in this direction.
Keywords
MSC
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: the Banach contraction principle ([1], Theorem 2.1) and the Tarski 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 the Nadler 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 [5, 7–9]. Investigation of the existence of fixed points for singlevalued 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$$ d\bigl(f(x),f(y)\bigr)\leq k d(x,y) \quad \textit{for all } x,y\in X \textit{ such that } x \succeq y . $$
 (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 metric spaces; see [8, 11–13] and references cited therein. Nieto et al. in [13] 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 of which was given by Jachymski [14] and the second one was given by Jachymski and Lukawska [15], generalizing the results of [11, 13, 16, 17] to a singlevalued mapping in metric spaces with a graph instead of a partial ordering.
The aim of this paper is two folds: 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 yield interesting generalization of the Banach contraction principle. Let us begin this section with terminology for metric spaces which will be used throughout.
Obviously, normed linear spaces are hyperbolic spaces. As nonlinear examples, one can consider Hadamard manifolds [23], the Hilbert open unit ball equipped with the hyperbolic metric [24], and \(\operatorname{CAT}(0)\) spaces [25–27]. We will say that a subset C of a hyperbolic metric space X is convex if \([x,y]\subset C\) whenever x, y are in C.
Definition 2.1
Next we introduce the concept of monotone multivalued mappings. In [9], the authors offered the following definition.
Definition 2.2
([9], Definition 2.6)
Definition 2.3
 (i)We say that a mapping \(T: C\rightarrow C\) is Gedge preserving if$$\forall x,y\in C,\quad (x,y)\in E(G)\quad \Rightarrow\quad \bigl(T(x),T(y)\bigr) \in E(G). $$
 (ii)We say that a mapping \(T: C\rightarrow C\) is Gcontraction if T is Gedge preserving and there exists \(k \in [0,1)\) such that$$\forall x,y\in C, \quad (x,y)\in E(G)\quad \Rightarrow\quad d\bigl(T(x),T(y) \bigr) \leq k d(x,y). $$
 (iii)We say that a mapping \(T: C\rightarrow C\) is Gnonexpansive if T is Gedge preserving and$$\forall x,y\in C,\quad (x,y)\in E(G)\quad \Rightarrow\quad d\bigl(T(x),T(y) \bigr) \leq d(x,y). $$
 (iv)A multivalued mapping \(T: C \rightarrow2^{C}\) is said to be monotone increasing (resp. decreasing) Gcontraction if there exists \(\alpha\in[0,1)\) such that for any \(x, y \in C\) with \((x,y)\in E(G)\) and any \(u \in T(x)\) there exists \(v \in T(y)\) such thatSimilarly we will say that the multivalued mapping \(T: C \rightarrow 2^{C}\) is monotone increasing (resp. decreasing) Gnonexpansive if for any \(x, y \in C\) with \((x,y)\in E(G)\) and any \(u \in T(x)\) there exists \(v \in T(y)\) such that$$(u,v) \in E(G)\qquad \bigl(\mbox{resp. }(v,u) \in E(G)\bigr) \quad \mbox{and} \quad d(u,v) \leq \alpha d(x,y). $$\(x \in C\) is called a fixed point of a singlevalued mapping T if and only if \(T(x) = x\). For a multivalued mapping T, x is a fixed point if and only if \(x \in T(x)\). The set of all fixed points of a mapping T is denoted by \(\operatorname{Fix}(T)\).$$(u,v) \in E(G)\qquad \bigl(\mbox{resp. }(v,u) \in E(G)\bigr) \quad \mbox{and} \quad d(u,v) \leq d(x,y). $$
3 Main results
We begin with the following wellknown theorem, which gives the existence of a fixed point for monotone singlevalued and multivalued contraction mappings in metric spaces endowed with a graph.
Theorem 3.1
[14]
 (∗):

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 \(n \geq1\).
 (1)
\(\operatorname{card}\operatorname{Fix} f=\operatorname{card}\{[x]_{\widetilde{G}} : x\in X_{f} \}\).
 (2)
\(\operatorname{Fix} f\neq\emptyset\) if and only if \(X_{f}\neq\emptyset\).
 (3)
f has a unique fixed point if and only if there exists an \(x_{0}\in X_{f}\) such that \(X_{f} \subseteq[x_{0}]_{\widetilde{G}}\).
 (4)
For any \(x \in X_{f}\), \(f_{[x]_{\widetilde{G}}}\) is a PO, that is, f has a unique fixed point \(x^{*}\in[x]_{\widetilde{G}}\) and for each \(x\in[x]_{\widetilde{G}}\), \(\lim_{n\rightarrow\infty} f^{n}(x)=x^{*}\).
 (5)
If \(X_{f} \neq\emptyset\) and G is weakly connected, then f is a PO, that is, f has a unique fixed point \(x^{*}\in X\) and for each \(x\in X\), \(\lim_{n\rightarrow\infty} f^{n}(x)=x^{*}\).
The multivalued version of Theorem 3.1 may be stated as follows.
Theorem 3.2
[29]
 (1)
For any \(x\in X_{T}\), \(T_{[x]_{\widetilde{G}}}\) has a fixed point.
 (2)
If \(x\in X\) with \((x,\bar{x})\in E(G)\) where \(\bar{x}\) is a fixed point of T, then \(\{T^{n}(x)\}\) converges to \(\bar{x}\).
 (3)
If G is weakly connected, then T has a fixed point in G.
 (4)
If \(X':=\bigcup\{[x]_{\widetilde{G}} : x\in X_{T}\}\), then \(T_{X'}\) has a fixed point in X.
 (5)
If \(T(X)\subseteq E(G)\) then T has a fixed point.
 (6)
\(\operatorname{Fix}T\neq\emptyset\) if and only if \(X_{T}\neq\emptyset\).
Remark 3.1
The missing information in Theorem 3.2 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 Theorem 3.2 gives the Nadler theorem [4].
The following is a direct consequence of Theorem 3.2.
Corollary 3.1
Let \((X, d)\) be a complete metric space. Let G be a graph on X such that the triple \((X,d,G)\) has the Property (∗). If G is weakly connected then every Gcontraction \(T: X\rightarrow\mathcal{CB}(X)\) such that \((x_{0}, x_{1})\in E(G)\), for some \(x_{0} \in X\) and \(x_{1}\in T(x_{0})\), has a fixed point.
Next we discuss some existence results for nonexpansive singlevalued and multivalued Gmonotone mappings. To the best of our knowledge, these results were never investigated for such mappings.
Theorem 3.3
Proof
In order to obtain a fixed point existence result for Gnonexpansive mappings, we need some extra assumptions.
Definition 3.1
We will say that G is transitive if, for any two vertices x and y that are connected by a directed finite path, we have \((x,y) \in E(G)\).
 (∗∗):

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 \(n \geq1\).
Definition 3.2
We will say that a nonempty subset C of X is Gcompact if and only if for any \((x_{n})_{n \geq1}\) in C, if \((x_{n}, x_{n+1})\in E(G)\), for \(n \geq1\), then there exists a subsequence \((x_{k_{n}})\) of \((x_{n})\) which is convergent to a point in C.
Note that Gcompactness does not necessarily imply compactness. Indeed, consider the metric set X, subset of \(\mathbb{R}^{3}\), built on a cone routed at the origin. All rays are bounded and compact. But X is unbounded. Define the graph G on X by \((x,y) \in E(G)\) if and only if x and y are on the same ray. Then any sequence \((x_{n}) \in X\) such that \((x_{n},x_{n+1}) \in E(G)\), for \(n \geq1\), will belong to a ray. Hence \((x_{n})\) has a convergent subsequence. This shows that X is Gcompact but fails to be compact.
Theorem 3.4
Let \((X,d)\) be a complete hyperbolic metric space and suppose that the triple \((X,d,G)\) has property (∗). Assume G is convex and transitive. Let C be a nonempty, Gcompact and convex subset of X. Let \(T: C\rightarrow C\) be a Gnonexpansive mapping. Assume \(C_{T}:=\{ x\in C: (x,T(x))\in E(G) \} \neq\emptyset\). Then T has a fixed point.
Proof
Next we investigate the above results for multivalued mappings. The first result for these mappings is the analog to Theorem 3.3.
Theorem 3.5
Proof
The multivalued version of Theorem 3.4 may be stated as follows.
Theorem 3.6
Let \((X,d)\) be a complete hyperbolic metric space and suppose that the triple \((X,d,G)\) has property (∗∗). Assume G is convex and transitive. Let C be a nonempty, Gcompact, and convex subset of X. Then any \(T: C\rightarrow{\mathcal{C}}(C)\) monotone increasing Gnonexpansive mapping has a fixed point provided \(C_{T}:=\{x\in C; (x,y) \in E(G)\textit{ for some }y\in T(x)\}\) is not empty.
Proof
Declarations
Acknowledgements
The first 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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