Convergence of Ishikawa iterations on noncompact sets

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.