Convergence of Ishikawa iterations on noncompact sets
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Date
Authors
Mutangadura, Simba A.
Journal Title
Journal ISSN
Volume Title
Publisher
Taylor & Francis
Abstract
Recall that Ishikawa’s theorem [4] provides an iterative procedure that yields a sequence
which converges to a fixed point of a Lipschitz pseudocontrative map T : C ! C, where
C is a compact convex subset of a Hilbert space X. The conditions on T and C, as well
as the fact that X has to be a Hilbert space, are clearly very restrictive. Modifications
of the Ishikawa’s iterative scheme have been suggested to take care of, for example, the
case where C is no longer compact or where T is only continuous. The purpose of this
paper is to explore those cases where the unmodified Ishikawa iterative procedure still
yields a sequence that converges to a fixed point of T, with C no longer compact. We
show that, if T has a fixed point, then every Ishikawa iteration sequence converges in
norm to a fixed point of T if C is boundedly compact or if the set of fixed points of T
is “suitably large”. In the process, we also prove a convexity result for the fixed points
of continuous pseudocontractions.
Description
Keywords
Convergence, Ishikawa iterative, Fixed point, Noncompact sets
Sustainable Development Goals
Citation
Mutangadura, SA 2014, 'Convergence of Ishikawa iterations on noncompact sets', Quaestiones Mathematicae, vol. 37, no. 2, pp. 191-198..