On monotone nonexpansive mappings in \(L_{1}([0,1])\)
- Mohamed Amine Khamsi^{1} and
- Abdul Rahim Khan^{2}Email author
https://doi.org/10.1186/s13663-015-0346-x
© Khamsi and Khan 2015
Received: 3 March 2015
Accepted: 17 May 2015
Published: 25 June 2015
Abstract
Let the set \(C \subset L_{1}([0,1])\) be nonempty, convex and compact for the convergence almost everywhere and \(T: C \rightarrow C\) be a monotone nonexpansive mapping. In this paper, we study the behavior of the Krasnoselskii-Ishikawa iteration sequence \(\{f_{n}\}\) defined by \(f_{n+1} = \lambda f_{n} + (1-\lambda)T(f_{n})\), \(n = 1, 2, \ldots\) , \(\lambda\in(0,1)\). Then we prove a fixed point theorem for these mappings. Our result is new and was never investigated.
Keywords
convergence almost everywhere fixed point Ishikawa iteration Krasnoselskii iteration Lebesgue measure monotone mapping nonexpansive mappingMSC
46B20 47E101 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 in a complete metric space. The principle itself finds almost canonical applications in the theory of differential and integral equations. Over the years, many mathematicians successfully extended 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 monotone nonexpansive mappings defined in \(L_{1}([0,1])\). Nonexpansive mappings are those which have Lipschitz constant equal to 1. The fixed point theory for such mappings is very rich [5–9] and has many applications in nonlinear functional analysis [10]. It is worth mentioning that the work presented here is new and has never been carried out.
For more on metric fixed point theory, the reader may consult the book of Khamsi and Kirk [11].
2 Preliminaries
Next we give the definition of monotone mappings.
Definition 2.1
- (a)
monotone if \(T(f) \leq T(g)\) whenever \(f \leq g\);
- (b)monotone K-Lipschitzian, \(K \in\mathbb{R}^{+}\), if T is monotone andwhenever \(f \leq g\). If \(K = 1\), then T is said to be a monotone nonexpansive mapping.$$\bigl\Vert T(g) - T(f)\bigr\Vert \leq K \|g - f\|, $$
Remark 2.1
It is not difficult to show that a monotone nonexpansive mapping may not be continuous. Therefore it is quite difficult to expect any nice behavior that will imply the existence of a fixed point for this class of mappings.
The following lemma will be crucial to prove the main result of this paper.
Lemma 2.1
[13]
In particular, this result holds when \(p =1\).
3 Iteration process for monotone nonexpansive mappings
Let us first recall the definition of the Krasnoselskii iteration [14] (see also [6, 15, 16]).
Definition 3.1
We have the following technical lemma.
Lemma 3.1
Proof
First note that if \(f \leq g\) holds, then we have \(f \leq \lambda f + (1-\lambda)g \leq g\) since order intervals are convex. Therefore it is just enough to prove \(f_{n} \leq T(f_{n})\), for any \(n \geq 1\). By assumption, we have \(f_{1} \leq T(f_{1})\). Assume that \(f_{n} \leq T(f_{n})\), for \(n \geq1\). Then we have \(f_{n} \leq\lambda f_{n} + (1-\lambda ) T(f_{n})\), i.e., \(f_{n} \leq f_{n+1} \leq T(f_{n})\). Since T is monotone, we get \(T(f_{n}) \leq T(f_{n+1})\), which implies \(f_{n+1} \leq T(f_{n+1})\). By induction, we conclude that the inequalities (KI) hold for any \(n \geq1\). Next let \(\{f_{\phi(n)}\}\) be a subsequence of \(\{ f_{n}\}\) which converges almost everywhere to g. Fix \(k \geq1\). Then the order interval \([f_{k}, \rightarrow)\) contains the sequence \(\{f_{\phi (n)}\}\) except maybe for finitely many elements. Since order intervals are almost everywhere closed and convex, we conclude that \(g \in[f_{k}, \rightarrow)\), i.e., \(f_{k} \leq g\) for any \(k \geq1\). Consequently, if \(\{f_{n}\}\) has another subsequence which converges almost everywhere to h, then we must have \(h = g\). Indeed, since \(f_{n} \leq g\), for any \(n \geq1\), we get \(h \leq g\). Similarly, we have \(g \leq h\), which implies \(h = g\). □
Remark 3.1
In the general theory of nonexpansive mappings, the iteration sequence defined by (KIS) provides an approximate fixed point sequence of T, i.e., \(\lim_{n \rightarrow+\infty} \|f_{n} - T(f_{n})\| = 0\) (see e.g. [6, 17]). It is amazing that this result holds for monotone nonexpansive mappings as well.
Theorem 3.1
Let \(C \subset L_{1}\) be nonempty, convex and compact for the convergence almost everywhere. Let \(T: C \rightarrow C\) be a monotone nonexpansive mapping. Assume there exists \(f_{1} \in C\) such that \(f_{1}\) and \(T(f_{1})\) are comparable. Then \(\{ f_{n}\}\), defined by (KIS), converges almost everywhere to some \(f \in C\) and \(\lim_{n \rightarrow\infty} \|f_{n} - T(f_{n})\|=0\). Moreover, f and \(f_{n}\) are comparable for any \(n \geq1\).
Proof
Next we state the main result of this paper.
Theorem 3.2
Let \(C \subset L_{1}\) be nonempty, convex and compact for the convergence almost everywhere. Let \(T: C \rightarrow C\) be a monotone nonexpansive mapping. Assume there exists \(f_{1} \in C\) such that \(f_{1}\) and \(T(f_{1})\) are comparable. Then \(\{f_{n}\} \), defined by (KIS), converges almost everywhere to some \(f \in C\) which is a fixed point of T, i.e., \(T(f) = f\). Moreover, f and \(f_{1}\) are comparable.
Proof
This result is a generalization of the original existence theorem in [19, 20] for nonexpansive mappings which are not monotone. As we said earlier, monotone nonexpansive mappings are not necessarily continuous. Therefore this class of mappings is bigger and is used to answer questions about the existence of positive or negative solutions in some applications considered in [2, 3].
Declarations
Acknowledgements
The authors acknowledge gratefully the support of KACST, Riyad, Saudi Arabia, for supporting Research Project ARP-32-34.
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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