- Research Article
- Open Access
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)
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.
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  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  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 , we provide affirmative answers to this question. Related results can be found, for example, in [4, 5]. More precisely, in  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 , 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
For each mapping set for all
Suppose that for each there exists a nonempty compact set such that
Then, there exists a nonempty compact subset of such that
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
Define a sequence by
There exists a nonempty compact set such that
By (2.4), (2.7), and (2.8),
Assume that is an integer and that for
(Note that in view of (2.10), inequality (2.11) is valid when )
By (2.2) and (2.11),
When combined with (2.4), this implies that
so that (2.11) also holds for Thus, we have shown that for all integers
by (2.6). In view of (2.9), we have for all large enough natural numbers
By (2.15), there exist an integer and a sequence such that
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),
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 .
Let be a complete metric space and let be such that
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]).
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,
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
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
Clearly, is a complete metric space.
Define a mapping as follows:
In other words, for any
Set for all Clearly,
and Then, there exists a sequence such that
and the following property holds.
There is no nonempty compact set such that
In the proof of this theorem, we may assume without loss of generality that
We precede the proof of Theorem 4.1 with the following lemma.
Let let be an integer, and let be a natural number. Then, there exist an integer and a sequence such that
There is a natural number such that
By (4.5), there is a natural number such that
By (4.14) and (4.8),
and we may assume without loss of generality that
In view of (4.14) and (4.8),
For define as follows:
Clearly, for is well defined and by (4.18), (4.19), (4.10), and (4.16),
Let We now estimate If then by (4.2), (4.3), (4.13), and (4.18),
Let We first set
In view of (4.14), (4.2), and (4.3), for all integers When combined with (4.18), this implies that
By (4.18) and (4.23),
for all It now follows from (4.22), (4.25), (4.18), (4.19), and (4.23) that
When combined with (4.12), this implies that
By (4.17) and (4.18),
This completes the proof of Lemma 4.2.
In order to prove the theorem, we construct by induction, using Lemma 4.2, a sequence of nonnegative integers and a sequence such that
and for all integers
In the sequel, we use the notation
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
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
we obtain that there exist an integer and a sequence such that
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
Assume the contrary. Then, there does exist a nonempty compact set such that
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
and a natural number such that
Hence we have, for all integers
This, of course, contradicts the inequality The contradiction we have reached completes the proof of Theorem 4.1.
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.
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.
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
- Natural Number
- Compact Subset
- Point Theorem
- Convex Subset
- Fixed Point Theorem