Mann iteration process for monotone nonexpansive mappings
- Buthinah Abdullatif Bin Dehaish^{1}Email author and
- Mohamed Amine Khamsi^{2, 3}
https://doi.org/10.1186/s13663-015-0416-0
© Bin Dehaish and Khamsi 2015
Received: 9 April 2015
Accepted: 1 September 2015
Published: 29 September 2015
Abstract
Keywords
MSC
1 Introduction
Nonexpansive mappings are those mappings which have Lipschitz constant equal to one. Their investigation remains a popular area of research in various fields. In 1965, Browder [1] and Göhde [2] independently proved that every nonexpansive self-mapping of a closed convex and bounded subset of a uniformly convex Banach space has a fixed point. This result was also obtained by Kirk [3] under slightly weaker assumptions. Since then several fixed point theorems for nonexpansive mappings in Banach spaces have been derived [4–6].
Recently a new direction has been developed when the Lipschitz condition is satisfied only for comparable elements in a partially ordered metric space. This direction was initiated by Ran and Reurings [7] (see also [8]) who proved an analogue of the classical Banach contraction principle [9] in partially ordered metric spaces. In both papers [7, 8], the motivation for this new direction is the problem of the existence of a solution which is positive. In other words, the classical approaches only deal with the existence of solutions, while here we ask whether a positive or negative solution exists. It is a natural question to ask since most of the classical metric spaces are endowed with a natural partial order.
When we relax the contraction condition to the case of the Lipschitz constant equal to 1, i.e., nonexpansive mapping, the completeness of the distance will not be enough as it was in the original case. We need some geometric assumptions to be added. But in general the Lipschitz condition on comparable elements is a weak assumption. In particular, we do not have the continuity property. Therefore, one has to be very careful when dealing with such mappings. In this work we use the iterative methods [10] to prove the existence of fixed points of such mappings.
For more on metric fixed point theory, the reader may consult the books [4, 6].
2 Basic definitions
Next we give the definition of monotone mappings.
Definition 2.1
- (a)
monotone if \(T(x) \leq T(y)\) whenever \(x \leq y\);
- (b)monotone K-Lipschitzian, \(K \in\mathbb{R}^{+}\), if T is monotone andwhenever \(x \leq y\). If \(K = 1\), then T is said to be a monotone nonexpansive mapping.$$\bigl\| T(x) - T(y)\bigr\| \leq K \|x - y\|, $$
Remark 2.1
It is not difficult to see 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.
Definition 2.2
Uniformly convex Banach spaces enjoy many nice geometric properties. The interested reader may consult the book [11].
3 Iteration process for monotone nonexpansive mappings
Let \((X, \|\cdot\|)\) be a Banach space.
Definition 3.1
The following technical lemma will be useful to prove the main result of this work.
Lemma 3.1
Proof
Remark 3.1
Lemma 3.2
Let C be a nonempty, closed, and convex subset of a Banach space \((X,\|\cdot\|)\). Let \(T:C\rightarrow C\) be a monotone nonexpansive mapping. Let \(\{t_{n}\} \subset[0,1]\) be the sequence associated to the Mann iteration process defined by (3.1). Then, for any \(\omega\in \operatorname{Fix}(T)\), \(\lim_{n \rightarrow\infty} \|x_{n}-\omega\|\) exists provided \(x_{1}\) and ω are comparable.
Proof
In the general theory of nonexpansive mappings, the main property of the Mann iterative sequence is an approximate fixed point property. Recall that \(\{x_{n}\}\) is called an approximate fixed point sequence of the mapping T if \(\lim_{n \rightarrow +\infty} \|x_{n} - T(x_{n})\| = 0\). We have a similar conclusion for monotone nonexpansive mappings if we assume that X is uniformly convex.
Theorem 3.1
Proof
The conclusion of Theorem 3.1 is strongly dependent on the assumption that a fixed point of T which is comparable to \(x_{1}\) exists. In fact, we may relax such assumption and obtain a similar conclusion. First we will need the following technical lemma.
Lemma 3.3
Proof
Without loss of any generality, we may assume \(x_{1} \leq T(x_{1})\). In this case, we have \(x_{n} \leq x_{n+1} \leq T(x_{n}) \leq T(x_{n+1})\) for any \(n \geq1\). In particular, we have \(\|T(x_{n+1}) - T(x_{n})\| \leq\|x_{n+1} - x_{n}\|\) for any \(n \geq1\). Moreover, from the definition of \(\{x_{n}\}\) we have \(\|x_{n+1} - x_{n} \| = t_{n} \|x_{n} - T(x_{n})\|\) for any \(n \geq1\). Therefore all the assumptions of Proposition 1 of [10] are satisfied, which implies the conclusion of Lemma 3.3. □
Using this lemma, we have a similar conclusion to Theorem 3.1 with less stringent assumptions. This result is similar to the one found in [13].
Theorem 3.2
Let C be a nonempty, closed, convex and bounded subset of a Banach space \((X,\|\cdot\|)\). Let \(T:C\rightarrow C\) be a monotone nonexpansive mapping. Let \(x_{1} \in C\) be such that \(x_{1}\) and \(T(x_{1})\) are comparable. Let \(\{x_{n}\}\) be the Mann iterative sequence defined by (3.1) such that \(\{ t_{n}\} \subset[a,b]\), with \(a > 0\) and \(b < 1\). Then we have \(\lim_{n \rightarrow+\infty} \|x_{n} - T(x_{n})\| = 0\).
Proof
Before we state the main fixed point result of this work, let us recall the definition of the weak-Opial condition.
Definition 3.2
[17]
Theorem 3.3
Let \((X, \|\cdot\|)\) be a Banach space which satisfies the weak-Opial condition. Let C be a nonempty weakly compact convex subset of X. Let \(T:C\rightarrow C\) be a monotone nonexpansive mapping. Assume that there exists \(x_{1} \in C\) such that \(x_{1}\) and \(T(x_{1})\) are comparable. Let \(\{x_{n}\}\) be the Mann iterative sequence defined by (3.1) such that \(\{t_{n}\} \subset [a,b]\), with \(a > 0\) and \(b < 1\). Then \(\{x_{n}\}\) is weakly convergent to x which is a fixed point of T, i.e., \(T(x) = x\). Moreover, x and \(x_{1}\) are comparable.
Proof
Declarations
Acknowledgements
This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks technical and financial support of DSR.
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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