On common approximate fixed points of monotone nonexpansive semigroups in Banach spaces
- Mostafa Bachar^{1}Email author and
- Mohamed A Khamsi^{2, 3}
https://doi.org/10.1186/s13663-015-0405-3
© Bachar and Khamsi 2015
Received: 4 June 2015
Accepted: 13 August 2015
Published: 4 September 2015
Abstract
In this paper, we investigate the common approximate fixed points of monotone nonexpansive semigroups of nonlinear mappings \(\{T(t)\}_{t \geq0}\), i.e., a family such that \(T(0)x=x\), \(T(s+t)x=T(s)\circ T(t)x \), where the domain is a Banach space. In particular we prove that under suitable conditions, the common approximate fixed points are the same as the common approximate fixed points set of two mappings from the family. Then we give an algorithm of how to construct an approximate fixed point sequence of the semigroup in the case of a uniformly convex Banach space.
Keywords
MSC
1 Introduction
Nonexpansive mappings are those maps 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. The existence of fixed points for nonexpansive mappings in Banach and metric spaces has been investigated since the early 1960s; see, e.g., Belluce and Kirk [1, 2], Browder [3], Bruck [4], Lim [5].
In recent years, a new direction has been very active essentially after the publication of Ran and Reurings fixed point theorem [6] dealing with the extension of the Banach contraction principle to metric spaces endowed with a partial order. In particular, they show how this extension is useful when dealing with some special matrix equations. It is worth mentioning that similar results were discovered by Turinici [7, 8]. Another similar approach was carried out by Nieto and Rodríguez-López [9] and used such arguments in solving some differential equations. In [10] Jachymski gave a more general unified version of these extensions by considering graphs instead of a partial order.
The purpose of this paper is to prove the existence of approximate fixed points for semigroups of nonlinear monotone mappings acting in a Banach vector space endowed with a partial order. Note that from a numerical point of view, approximate fixed points are very useful since exact fixed points may be hard to find. Moreover, we will also give an algorithm of how to build such approximate fixed points. Let us recall that a family \(\{T(t)\}_{t\geq0}\) of mappings forms a semigroup if \(T(0)x = x\) and \(T(s+t)x = T(s)T(t)\). Such a situation is quite typical in mathematics and applications. For instance, in the theory of dynamical systems, the vector function space would define the state space and the mapping \((t, x)\rightarrow T(t)x\) would represent the evolution function of a dynamical system. In the setting of this paper, the state space is a Banach space. Our approach is new and different from the ideas found in [6–9, 11]. Indeed when one deals with a contraction, the focus is usually on the distance being complete. But when dealing with nonexpansive mappings, then geometric properties of the space are crucial.
For more on metric fixed point theory, the reader may consult the books [12, 13]. As for the semigroup theory, we suggest the references [14–16].
2 Preliminaries
- (i)
\([a,\rightarrow) = \{x \in X; a \preceq x\}\),
- (ii)
\((\leftarrow,a] = \{x \in X; x \preceq a\}\),
Definition 2.1
- (a)
monotone if \(T(x) \preceq T(y)\) whenever \(x \preceq y\);
- (b)monotone nonexpansive if T is monotone andfor any \(x, y \in C\) such that \(x \preceq y\).$$\bigl\| T(x)-T(y) \bigr\| \leq\|x-y\| $$
This definition is extended to the case of semigroup of mappings.
Definition 2.2
- (i)
\(T(0)x = x\) for \(x \in C\);
- (ii)
\(T(t+s) = T(t)\circ T(s)\) for \(t,s \in[0,\infty)\);
- (iii)
for each \(t \geq0\), \(T(t)\) is a monotone nonexpansive mapping.
Next we give an example of such semigroup.
Example 2.1
3 Common approximate fixed points of semigroups
Before we state our first result, we need the following definition.
Definition 3.1
- (i)continuous on C if for any \(x \in C\), the mapping \(t \rightarrow T(t)x\) is continuous, i.e., for any \(t_{0} \geq0\), we havefor any \(x \in C\);$$\lim_{t \rightarrow t_{0}} \bigl\| T(t)x- T(t_{0})x\bigr\| = 0 $$
- (ii)strongly continuous on C if for any bounded nonempty subset \(K \subset C\), we have$$\lim_{t \rightarrow t_{0}} \sup_{x \in K} \bigl(\bigl\| T(t)x- T(t_{0})x\bigr\| \bigr) = 0. $$
The following technical lemmas will be useful throughout.
Lemma 3.1
Let \((X, \|\cdot\|)\) be a Banach space and \(C \subset X\) be nonempty. Let \(J:C \rightarrow C\) be a uniformly continuous mapping. Then we have \(\operatorname{AFPS}(J) \subset \operatorname{AFPS}(J^{m})\) for any \(m \geq2\).
Proof
Lemma 3.2
Proof
The following lemma, which can be found in any introductory course on real analysis, will be crucial to proving the first result on common approximate fixed point of semigroups.
Lemma 3.3
[17]
Theorem 3.1
Proof
As a corollary to Theorem 3.1, we get the following result.
Corollary 3.1
All of the results obtained in this section may be easily stated in metric spaces. In the next section, we give an algorithm of how to construct an approximate fixed point sequence of two maps.
4 Common approximate fixed points of two monotone mappings
Before we state the main theorem of this section, let us recall the definition of a uniformly convex Banach space.
Definition 4.1
[19]
The following lemma will be needed to prove the main result of this section.
Lemma 4.1
[20]
Next we state the main result of this section.
Theorem 4.1
Proof
Remark 4.1
Declarations
Acknowledgements
The authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King Saud University for funding this Research group No. RG-1435-079.
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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