- Research Article
- Open access
- Published:
Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces
Fixed Point Theory and Applications volume 2008, Article number: 528614 (2008)
Abstract
We study the influence of computational errors on the convergence to compact sets of orbits of nonexpansive mappings in Banach and metric spaces. We first establish a convergence theorem assuming that the computational errors are summable and then provide examples which show that the summability of errors is necessary for convergence.
1. Introduction
Convergence analysis of iterations of nonexpansive mappings in Banach and metric spaces 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. Banach's celebrated result also yields convergence of iterates to the unique fixed point. There are several generalizations of Banach's fixed point theorem which show that the convergence of iterates holds for larger classes of nonexpansive mappings. For instance, Rakotch [2] introduced the class of contractive mappings and showed that their iterates also converged to their unique fixed point.
In view of these results and their numerous applications, it is natural to ask if convergence of the iterates of nonexpansive mappings will be preserved in the presence of computational errors. In [3], we provide affirmative answers to this question. Related results can be found, for example, in [4, 5]. More precisely, in [3] we show that if all exact iterates of a given nonexpansive mapping converge (to fixed points), then this convergence continues to hold for inexact orbits with summable errors. In [6], we continued to study the influence of computational errors on the convergence of iterates of nonexpansive mappings in both Banach and metric spaces. We show there that if all the orbits of a nonexpansive self-mapping of a metric space converge to some closed subset
of
then all inexact orbits with summable errors also converge to this attractor set
On the other hand, we also construct examples which show that inexact orbits may fail to converge if the errors are not summable.
Our purpose in the present paper is to consider the case where different exact orbits converge to possibly different compact subsets of In Section 2, we obtain a convergence result (see Theorem 2.1 below) under the assumption that the computational errors are summable. This result is an extension of [3, Theorem 4.2]. In Sections 3 and 4, we provide examples which show that the summability of errors is necessary for convergence (see Proposition 3.1 and Theorem 4.1).
2. Convergence to Compact Sets
Let be a complete metric space. For each
and each nonempty and closed subset
put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ1_HTML.gif)
For each mapping set
for all
Theorem 2.1.
Let satisfy
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ2_HTML.gif)
Suppose that for each there exists a nonempty compact set
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ3_HTML.gif)
Assume that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ4_HTML.gif)
Then, there exists a nonempty compact subset of
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ5_HTML.gif)
Proof.
In order to prove the theorem, it is sufficient to show that any subsequence of has a convergent subsequence.
To see this, it is sufficient to show that for any the following assertion holds:
(P1) any subsequence of has a subsequence which is contained in a ball of radius
Indeed, there is an integer such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ6_HTML.gif)
Define a sequence by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ7_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ8_HTML.gif)
There exists a nonempty compact set such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ9_HTML.gif)
By (2.4), (2.7), and (2.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ10_HTML.gif)
Assume that is an integer and that for
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ11_HTML.gif)
(Note that in view of (2.10), inequality (2.11) is valid when )
By (2.2) and (2.11),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ12_HTML.gif)
When combined with (2.4), this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ13_HTML.gif)
so that (2.11) also holds for Thus, we have shown that for all integers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ14_HTML.gif)
by (2.6). In view of (2.9), we have for all large enough natural numbers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ15_HTML.gif)
By (2.15), there exist an integer and a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ16_HTML.gif)
Consider any subsequence of
Since the set
is compact, the sequence
has a convergent subsequence
We may assume without loss of generality that all elements of this convergent subsequence belong to for some
In view of (2.16),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ17_HTML.gif)
Thus, (P1) holds and this completes the proof of Theorem 2.1.
Note that Theorem 2.1 is an extension of the following result established in [3].
Theorem 2.2.
Let be a complete metric space and let
be such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ18_HTML.gif)
and for each the sequence
converges in
Assume that satisfies
and that a sequence
satisfies
Then, the sequence
converges to a fixed point of
in
3. First Example of Nonconvergence to Compact Sets
In this section, we show that both Theorems 2.1 and 2.2 cannot, in general, be improved (cf. [6, Proposition 3.1]).
Proposition 3.1.
For any normed space there exists an operator
such that
for all
the sequence
converges for each
and, for any sequence of positive numbers
there exists a sequence
with
for all nonnegative integers
which converges to a compact set if and only if the sequence
is summable, that is,
Proof.
This is a simple fact because we may take to be the identity operator:
Then, we may take
to be an arbitrary element of
with
and define by induction
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ19_HTML.gif)
Evidently, and
for all integers
so that the convergence of
to a compact set is equivalent to the summability of the sequence
Proposition 3.1 is proved.
4. Second Example of Nonconvergence to Compact Sets
In Section 3, we have shown that Theorems 2.1 and 2.2 cannot, in general, be improved. However, in Proposition 3.1 every point of the space is a fixed point of the operator and the inexact orbits tend to infinity. In this section, we construct an operator
on a certain complete metric space
(a bounded, closed, and convex subset of a Banach space) such that all of its orbits converge to its unique fixed point, and for any nonsummable sequence of errors and any initial point, there exists an inexact orbit which does not converge to any compact set (cf. [6, Theorem 4.1]).
Let be the set of all sequences
of nonnegative numbers such that
For
and
in
set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ20_HTML.gif)
Clearly, is a complete metric space.
Define a mapping as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ21_HTML.gif)
In other words, for any
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ22_HTML.gif)
Set for all
Clearly,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ23_HTML.gif)
for all
Theorem 4.1.
Let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ24_HTML.gif)
and Then, there exists a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ25_HTML.gif)
and the following property holds.
There is no nonempty compact set such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ26_HTML.gif)
In the proof of this theorem, we may assume without loss of generality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ27_HTML.gif)
We precede the proof of Theorem 4.1 with the following lemma.
Lemma 4.2.
Let let
be an integer, and let
be a natural number. Then, there exist an integer
and a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ28_HTML.gif)
with
Proof.
There is a natural number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ29_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ30_HTML.gif)
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ31_HTML.gif)
Then,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ32_HTML.gif)
By (4.5), there is a natural number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ33_HTML.gif)
By (4.14) and (4.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ34_HTML.gif)
and we may assume without loss of generality that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ35_HTML.gif)
In view of (4.14) and (4.8),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ36_HTML.gif)
For define
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ37_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ38_HTML.gif)
Clearly, for is well defined and by (4.18), (4.19), (4.10), and (4.16),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ39_HTML.gif)
Thus
Let We now estimate
If
then by (4.2), (4.3), (4.13), and (4.18),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ40_HTML.gif)
Let We first set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ41_HTML.gif)
In view of (4.14), (4.2), and (4.3), for all integers
When combined with (4.18), this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ42_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ43_HTML.gif)
By (4.18) and (4.23),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ44_HTML.gif)
for all It now follows from (4.22), (4.25), (4.18), (4.19), and (4.23) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ45_HTML.gif)
When combined with (4.12), this implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ46_HTML.gif)
By (4.17) and (4.18),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ47_HTML.gif)
This completes the proof of Lemma 4.2.
Proof.
In order to prove the theorem, we construct by induction, using Lemma 4.2, a sequence of nonnegative integers and a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ50_HTML.gif)
and for all integers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ51_HTML.gif)
In the sequel, we use the notation
Set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ52_HTML.gif)
Assume that is an integer and that we have already defined a (finite) sequence of nonnegative numbers
and a (finite) sequence of points
such that (4.33) is valid, (4.30) holds for all integers
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ53_HTML.gif)
and that (4.32) holds for all integers satisfying
(Note that for
this assumption does hold.)
Now, we show that this assumption also holds for
Indeed, applying Lemma 4.2 with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ54_HTML.gif)
we obtain that there exist an integer and a sequence
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ55_HTML.gif)
Thus, the assumption made for also holds for
Therefore, we have constructed by induction a sequence of points
and a sequence of nonnegative integers
which satisfy (4.30) and (4.31) for all integers
respectively, and (4.32) for all integers
Finally, we show that there is no nonempty compact set such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ56_HTML.gif)
Assume the contrary. Then, there does exist a nonempty compact set such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ57_HTML.gif)
This implies that any subsequence of has a convergent subsequence.
Consider such a subsequence This subsequence has a convergent subsequence
There are therefore a point
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ58_HTML.gif)
and a natural number such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ59_HTML.gif)
Hence we have, for all integers
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2008%2F528614/MediaObjects/13663_2008_Article_1090_Equ60_HTML.gif)
This, of course, contradicts the inequality The contradiction we have reached completes the proof of Theorem 4.1.
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundamenta Mathematicae 1922, 3: 133-181.
Rakotch E: A note on contractive mappings. Proceedings of the American Mathematical Society 1962, 13(3):459-465. 10.1090/S0002-9939-1962-0148046-1
Butnariu D, Reich S, Zaslavski AJ: Convergence to fixed points of inexact orbits of Bregman-monotone and of nonexpansive operators in Banach spaces. In Fixed Point Theory and Its Applications. Yokohama, Yokohama, Japan; 2006:11-32.
Butnariu D, Reich S, Zaslavski AJ: Asymptotic behavior of inexact orbits for a class of operators in complete metric spaces. Journal of Applied Analysis 2007, 13(1):1-11. 10.1515/JAA.2007.1
Ostrowski AM: The round-off stability of iterations. Zeitschrift für Angewandte Mathematik und Mechanik 1967, 47(2):77-81. 10.1002/zamm.19670470202
Pustylnik E, Reich S, Zaslavski AJ: Inexact orbits of nonexpansive mappings. Taiwanese Journal of Mathematics 2008, 12(6):1511-1523.
Acknowledgments
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Pustylnik, E., Reich, S. & Zaslavski, A.J. Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces. Fixed Point Theory Appl 2008, 528614 (2008). https://doi.org/10.1155/2008/528614
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2008/528614