Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces
© Evgeniy Pustylnik et al. 2008
Received: 27 September 2008
Accepted: 17 November 2008
Published: 25 November 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
For each mapping set for all
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
(Note that in view of (2.10), inequality (2.11) is valid when )
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
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 .
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,
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]).
Clearly, is a complete metric space.
and the following property holds.
We precede the proof of Theorem 4.1 with the following lemma.
This completes the proof of Lemma 4.2.
In the sequel, we use the notation
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
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
This implies that any subsequence of has a convergent subsequence.
This, of course, contradicts the inequality The contradiction we have reached completes the proof of Theorem 4.1.
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.
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