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Fixed points of monotone nonexpansive mappings with a graph
- Monther Rashed Alfuraidan^{1}Email author
https://doi.org/10.1186/s13663-015-0299-0
© Alfuraidan; licensee Springer. 2015
- Received: 5 January 2015
- Accepted: 19 March 2015
- Published: 9 April 2015
Abstract
In this paper, we study the existence of fixed points of monotone nonexpansive mappings defined in Banach spaces endowed with a graph. This work is a continuity of the previous results of Ran and Reurings, Nieto et al., and Jachimsky done for contraction mappings defined in metric spaces endowed with a graph.
Keywords
- Banach space
- directed graph
- fixed point
- monotone nonexpansive mappings
MSC
- 06F30
- 46B20
- 47E10
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, because it requires only a complete metric space for its setting, and because it is a powerful result with a wide range of applications, including iterative methods for solving linear, nonlinear, differential, integral, and difference equations. Due to its applications in mathematics and other related disciplines, the Banach contraction principle has been generalized in many directions. Recently a version of this theorem was given in partially ordered metric spaces [2, 3] and in metric spaces endowed with a graph [4–6].
In this paper, we study the case of nonexpansive mappings defined in Banach spaces endowed with a graph. Nonexpansive mappings are those which have Lipschitz constant equal to 1. The fixed point theory for such mappings is rich and varied. It finds many applications in nonlinear functional analysis [7]. It is worth mentioning that such investigation is, to the best of our knowledge, new and was never carried out. This work was inspired by [8].
2 Graph basic definitions
The terminology of graph theory instead of partial ordering gives a wider picture and yields interesting generalization of the Banach contraction principle. In this section, we give the basic graph theory definitions and notations which will be used throughout.
3 Monotone nonexpansive mappings
Throughout we assume that \((X, \|\cdot\|)\) is a Banach space and τ is a Hausdorff topological vector space topology on X which is weaker than the norm topology. Let C be a nonempty, convex and bounded subset of X not reduced to one point. Let G be a directed graph such that \(V(G)=C\) and \(E(G)\supseteq\Delta\). Assume that G-intervals are convex. Recall that a G-interval 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)
G-monotone if \((T(x),T(y))\in E(G)\) whenever \((x,y)\in E(G)\) for any \(x, y \in C\);
- (2)G-monotone nonexpansive if T is G-monotone andfor any \(x, y \in C\).$$\bigl\Vert T(x)-T(y)\bigr\Vert \leq\|x-y\|,\quad \mbox{whenever }(x,y)\in E(G) $$
Remark 3.1
For examples of metric spaces endowed with a graph and G-monotone mappings which are Lipschitzian with respect to the graph, we refer the reader to the examples found in [5].
- (i)
\(x_{n+1} = \lambda x_{n} + (1-\lambda) T(x_{n})\),
- (ii)
\((x_{n}, x_{n+1})\), \((x_{n}, T(x_{n}))\) and \((T(x_{n}),T(x_{n+1}))\) are in \(E(G)\),
- (iii)
\(\|T(x_{n+1}) - T(x_{n})\| \leq\|x_{n+1} - x_{n}\|\).
Proposition 3.1
Remark 3.2
Before we state the main result of this paper, let us recall the definition of τ-Opial condition.
Definition 3.2
Definition 3.3
The triple \((C, \|\cdot\|,G)\) has property (P) if and only if for any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in C such that \((x_{n},x_{n+1})\in E(G)\) for any \(n\geq0\), and if a subsequence \(\{ x_{k_{n}}\}\) τ-converges to x, then \((x_{k_{n}},x)\in E(G)\) for all n.
Theorem 3.1
Let X be a Banach space which satisfies the τ-Opial condition. Let C be a bounded convex τ-compact nonempty subset of X not reduced to one point. Assume that \((C, \|\cdot\|,G)\) has property (P) and the G-intervals are convex. Let \(T: C \rightarrow C\) be a G-monotone nonexpansive mapping. Assume that there exists \(x_{0} \in C\) such that \((x_{0},T(x_{0}))\in E(G)\). Then T has a fixed point.
Proof
Remark 3.3
The triple \((C, \|\cdot\|,G)\) has property (P^{∗}) if and only if for any sequence \(\{x_{n}\}_{n\in\mathbb{N}}\) in C such that \((x_{n+1},x_{n})\in E(G)\) for any \(n\geq0\), and if a subsequence \(\{ x_{k_{n}}\}\) τ-converges to x, then \((x,x_{k_{n}})\in E(G)\) for all n.
The following results are direct consequences of Theorem 3.1.
Corollary 3.1
Let C be a bounded closed convex nonempty subset of \(l_{p}\), \(1 < p < +\infty\). Let τ be the weak topology. Let G be the digraph defined on \(l_{p}\) by \((\{\alpha_{n}\} ,\{\beta_{n}\})\in E(G)\) iff \(\alpha_{n} \leq\beta_{n}\) for any \(n \geq1\). Then any G-monotone nonexpansive mapping \(T: C \rightarrow C\) has a fixed point provided there exists a point \(x_{0} \in C\) such that \((x_{0},T(x_{0}))\in E(G)\) or \((T(x_{0}),x_{0})\in E(G)\).
Remark 3.4
The case of \(p=1\) is not interesting for the weak-topology since \(l_{1}\) is a Schur Banach space. But if we consider the weak^{∗}-topology \(\sigma(l_{1}, c_{0})\) on \(l_{1}\) or the pointwise convergence topology, then \(l_{1}\) satisfies the weak^{∗}-Opial condition. In this case we have a similar conclusion of Corollary 3.1 for \(l_{1}\).
Corollary 3.2
Let C be a bounded closed convex nonempty subset of \(L_{p}\), \(1 \leq p < +\infty\). Let τ be the almost everywhere convergence topology. Let G be the digraph defined on \(L_{p}\) by \((f , g)\in E(G)\) if and only if \(f(t) \leq g(t)\) almost everywhere. Assume that C is almost everywhere compact. Then any G-monotone nonexpansive mapping \(T: C \rightarrow C\) has a fixed point provided there exists a point \(f_{0} \in C\) such that \((f_{0},T(f_{0}))\in E(G)\) or \((T(f_{0}),f_{0})\in E(G)\).
Declarations
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research. The author thanks the referees for their valuable suggestions including the reference [5].
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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