# Convergence of Inexact Iterative Schemes for Nonexpansive Set-Valued Mappings

- Simeon Reich
^{1}and - AlexanderJ Zaslavski
^{1}Email author

**2010**:518243

https://doi.org/10.1155/2010/518243

© S. Reich and A. J. Zaslavski. 2010

**Received: **23 October 2009

**Accepted: **10 January 2010

**Published: **19 January 2010

## Abstract

Taking into account possibly inexact data, we study iterative schemes for approximating fixed points and attractors of contractive and nonexpansive set-valued mappings, respectively. More precisely, we are concerned with the existence of convergent trajectories of nonstationary dynamical systems induced by approximations of a given set-valued mapping.

## Keywords

## 1. Introduction

The study of iterative schemes for various classes of nonexpansive mappings is a central topic in Nonlinear Functional Analysis. It began with the classical Banach theorem [1] on the existence of a unique fixed point for a strict contraction. This celebrated result also yields convergence of iterates to the unique fixed point. Since Banach's seminal result, many developments have taken place in this area. We mention, in particular, existence and approximation results regarding fixed points of those nonexpansive mappings which are not necessarily strictly contractive [2, 3]. Such results were obtained for general nonexpansive mappings in special Banach spaces, while for self-mappings of general complete metric spaces most of the results were established for several classes of contractive mappings [4]. More recently, interesting developments have occurred for nonexpansive set-valued mappings, where the situation is more difficult and less understood. See, for instance, [5–8] and the references cited therein. As we have already mentioned, one of the methods for proving the classical Banach result is to show the convergence of Picard iterations, which holds for any initial point. In the case of set-valued mappings, not all the trajectories of the dynamical system induced by the given mapping converge. Therefore, convergent trajectories have to be constructed in a special way. For example, in the setting of [9], if at the moment we reach a point , then the next iterate is an element of , where is the given mapping, which approximates the best approximation of in . Since is assumed to act on a general complete metric space, we cannot, in general, choose to be the best approximation of by elements of . Instead, we choose so that it provides an approximation up to a positive number , such that the sequence is summable. This method allowed Nadler [9] to obtain the existence of a fixed point of a strictly contractive set-valued mapping and the authors of [10] to obtain more general results.

In view of the above discussion, it is obviously important to study convergence properties of the iterates of (set-valued) nonexpansive mappings in the presence of errors and possibly inaccurate data. The present paper is a contribution in this direction. More precisely, we are concerned with the existence of convergent trajectories of nonstationary dynamical systems induced by approximations of a given set-valued mapping. In the second section of the paper, we consider an iterative scheme for approximating fixed points of closed-valued strict contractions in metric spaces and prove our first convergence theorem (see Theorem 2.1 below). Our second convergence theorem (Theorem 3.1) is established in the third section of our paper. We show there that if for any initial point, there exists a trajectory of the dynamical system induced by a nonexpansive set-valued mapping , which converges to a given invariant set , then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations of .

## 2. Convergence to a Fixed Point of a Contractive Mapping

In this section we consider iterative schemes for approximating fixed points of closed-valued strict contractions in metric spaces.

We begin with a few notations.

Throughout this paper, is a complete metric space.

For and a nonempty subset of , set

For each pair of nonempty , put

Let be such that is a closed subset of for each and

Theorem 2.1.

Then converges to a fixed point of .

Proof.

In view of (2.8) and (2.10), inequality (2.11) holds for .

Assume that is an integer and that (2.11) holds for . When combined with (2.7), this implies that

We claim that

as claimed. Theorem 2.1 is proved.

## 3. Convergence to an Attractor of a Nonexpansive Mapping

In this section we show that if for any initial point, there exists a trajectory of the dynamical system induced by a nonexpansive set-valued mapping , which converges to an invariant set , then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations of .

Let be such that is a closed set for each and

Theorem 3.1.

We begin the proof of Theorem 3.1 with two lemmata.

Lemma 3.2.

Proof.

We are now going to define by induction a sequence .

(Clearly, this assumption holds for .)

By (3.11) and (3.1),

Lemma 3.2 is proved.

Lemma 3.3.

Proof.

Lemma 3.3 is proved.

Completion of the Proof of Theorem 3.1

Theorem 3.1 is proved.

## Declarations

### Acknowledgment

This research was supported by the Israel Science Foundation (Grant no. 647/07), the Fund for the Promotion of Research at the Technion, and by the Technion President's Research Fund.

## Authors’ Affiliations

## References

- Banach S:
**Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales.***Fundamenta Mathematicae*1922,**3:**133–181.MATHGoogle Scholar - Goebel K, Kirk WA:
*Topics in Metric Fixed Point Theory, Cambridge Studies in Advanced Mathematics*.*Volume 28*. Cambridge University Press, Cambridge, UK; 1990:viii+244.View ArticleMATHGoogle Scholar - Goebel K, Reich S:
*Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 83*. Marcel Dekker, New York, NY, USA; 1984:ix+170.Google Scholar - Kirk WA:
**Contraction mappings and extensions.**In*Handbook of Metric Fixed Point Theory*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:1–34.View ArticleGoogle Scholar - Reich S, Zaslavski AJ:
**Convergence of iterates of nonexpansive set-valued mappings.**In*Set Valued Mappings with Applications in Nonlinear Analysis, Mathematical Analysis and Applications*.*Volume 4*. Taylor & Francis, London, UK; 2002:411–420.Google Scholar - Reich S, Zaslavski AJ:
**Generic existence of fixed points for set-valued mappings.***Set-Valued Analysis*2002,**10**(4):287–296. 10.1023/A:1020602030873MathSciNetView ArticleMATHGoogle Scholar - Reich S, Zaslavski AJ:
**Two results on fixed points of set-valued nonexpansive mappings.***Revue Roumaine de Mathématiques Pures et Appliqués*2006,**51**(1):89–94.MathSciNetMATHGoogle Scholar - Ricceri B:
**Une propriété topologique de l'ensemble des points fixes d'une contraction multivoque à valeurs convexes.***Atti della Accademia Nazionale dei Lincei*1987,**81**(3):283–286.MathSciNetMATHGoogle Scholar - Nadler SB Jr.:
**Multi-valued contraction mappings.***Pacific Journal of Mathematics*1969,**30:**475–488.MathSciNetView ArticleMATHGoogle Scholar - de Blasi FS, Myjak J, Reich S, Zaslavski AJ:
**Generic existence and approximation of fixed points for nonexpansive set-valued maps.***Set-Valued and Variational Analysis*2009,**17**(1):97–112. 10.1007/s11228-009-0104-5MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.