Fixed Point Theory and Applications

, 2010:518243 | Cite as

Convergence of Inexact Iterative Schemes for Nonexpansive Set-Valued Mappings

Open Access
Research Article
Part of the following topical collections:
  1. Impact of Kirk's Results on the Development of Fixed Point Theory

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

Natural Number Nonexpansive Mapping Iterative Scheme Cauchy Sequence Contractive Mapping 

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, 6, 7, 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 Open image in new window we reach a point Open image in new window , then the next iterate Open image in new window is an element of Open image in new window , where Open image in new window is the given mapping, which approximates the best approximation of Open image in new window in Open image in new window . Since Open image in new window is assumed to act on a general complete metric space, we cannot, in general, choose Open image in new window to be the best approximation of Open image in new window by elements of Open image in new window . Instead, we choose Open image in new window so that it provides an approximation up to a positive number Open image in new window , such that the sequence Open image in new window 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 Open image in new window , which converges to a given invariant set Open image in new window , then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations of Open image in new window .

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, Open image in new window is a complete metric space.

For Open image in new window and a nonempty subset Open image in new window of Open image in new window , set

For each pair of nonempty Open image in new window , put

Let Open image in new window be such that Open image in new window is a closed subset of Open image in new window for each Open image in new window and

where Open image in new window is a constant.

Theorem 2.1.

Assume that Open image in new window and that for each integer Open image in new window ,

Then Open image in new window converges to a fixed point of Open image in new window .

Proof.

We first show that Open image in new window is a Cauchy sequence. To this end, let Open image in new window be an integer. Then by (2.6) and (2.5),
By (2.7),
Now we show by induction that for each integer Open image in new window ,

In view of (2.8) and (2.10), inequality (2.11) holds for Open image in new window .

Assume that Open image in new window is an integer and that (2.11) holds for Open image in new window . When combined with (2.7), this implies that

Thus (2.11) holds for Open image in new window . Therefore, we have shown by induction that (2.11) holds for all integers Open image in new window . By (2.11),
Thus Open image in new window is a Cauchy sequence and there exists

We claim that

Indeed, by (2.14), there is an integer Open image in new window such that for each integer Open image in new window ,
Let Open image in new window be an integer. By (2.3), (2.16) and (2.5),
as Open image in new window . Since Open image in new window is an arbitrary positive number, we conclude 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 Open image in new window , which converges to an invariant set Open image in new window , then a convergent trajectory also exists for a nonstationary dynamical system induced by approximations of Open image in new window .

Let Open image in new window be such that Open image in new window is a closed set for each Open image in new window and

Theorem 3.1.

Assume that for each Open image in new window , there exists a sequence Open image in new window such that
Then for each Open image in new window , there is a sequence Open image in new window such that

We begin the proof of Theorem 3.1 with two lemmata.

Lemma 3.2.

and let Open image in new window . Then there is a natural number Open image in new window and a sequence Open image in new window such that

Proof.

Choose a natural number Open image in new window such that
and a sequence Open image in new window such that
There is a sequence Open image in new window such that

We are now going to define by induction a sequence Open image in new window .

To this end, assume that Open image in new window is an integer and that we have already defined Open image in new window , Open image in new window , such that

(Clearly, this assumption holds for Open image in new window .)

By (3.11) and (3.1),

By (3.15), there is Open image in new window such that
Together with (3.3), this implies that
and there is
such that
When combined with (3.16) and (3.13), this implies that
Thus, by (3.18) and (3.20), the assumption we have made concerning Open image in new window also holds for Open image in new window . Therefore, we have indeed defined by induction a sequence Open image in new window such that
and (3.13) holds for all integers Open image in new window . By (3.11), there is an integer Open image in new window such that
Together with (3.8) and (3.13), this inequality implies that

Lemma 3.2 is proved.

Lemma 3.3.

Then Open image in new window for all integers Open image in new window .

Proof.

We intend to show by induction that for all integers Open image in new window ,
Clearly, for Open image in new window inequality (3.27) does hold. Assume now that Open image in new window is an integer and (3.27) holds. Then there is
such that
By (3.24) and (3.3), there is
such that
By (3.29) and (3.1),
and, in view of (3.30), there is
such that
By (3.33), (3.28), and (3.2),
By (3.35), (3.31), (3.34), and (3.27),
Thus, the assumption we have made concerning Open image in new window also holds for Open image in new window . Therefore, we may conclude that inequality (3.27) indeed holds for all integers Open image in new window . Together with (3.26), this implies that for all integers Open image in new window ,

Lemma 3.3 is proved.

Completion of the Proof of Theorem 3.1

Let Open image in new window . Since Open image in new window , it follows from Lemma 3.2 that there exist a sequence Open image in new window and a strictly increasing sequence of natural numbers Open image in new window , constructed by induction, such that
It now follows from (3.39) and Lemma 3.3 that

Theorem 3.1 is proved.

Notes

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.

References

  1. 1.
    Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 1922, 3: 133–181.MATHGoogle Scholar
  2. 2.
    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.CrossRefMATHGoogle Scholar
  3. 3.
    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
  4. 4.
    Kirk WA: Contraction mappings and extensions. In Handbook of Metric Fixed Point Theory. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:1–34.CrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Reich S, Zaslavski AJ: Generic existence of fixed points for set-valued mappings. Set-Valued Analysis 2002,10(4):287–296. 10.1023/A:1020602030873MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    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
  8. 8.
    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
  9. 9.
    Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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/s11228-009-0104-5MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© S. Reich and A. J. Zaslavski. 2010

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.

Authors and Affiliations

  1. 1.Department of MathematicsThe Technion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations