Fixed points and the sequence of iterates

Abstract

In this work notions involving continuous mappings of metric and topological linear spaces are developed. It is shown that if C is a closed, bounded, convex subset o f a convex metric space and F : C - C is a commuting family of nonexpansive mappings with nonempty fixed point set such that at least one of the mapping in the family F is demicompact, then the family F has a common fixed point. If K is a compact convex subset of a strictly convex metric space and F is a left amenable semigroup of continuous generalized nonexpansive mappings having property (K) on D, then the family F has a common fixed point. The results o f R. DeMarr, A. Bahtin and W. Takahashi follow as corollaries of these results. In order to extend these results to spaces which are not necessarily metrizable, the concepts of asymptotic normal structure and limiting orbital dimaetrs are introduced. An analog of the Banach contracting mapping theorem is obtained for a continuous Banach operator. It is shown that the fixed point set of a compact, convex semigroup of nonexpansive mappings of a closed convex subset of a strictly convex locally convex Hausdorff topological linear space is a non-expansive retract. Using asymptotic normal structure we show that a generalized nonexpansive mapping of a weakly compact convex subset of a locally convex linear topological space as a fixed point. These results extend the results of Taylor, Hicks and Kubicek and Goebel, Kirk and Shimi. Finally we study the convergence of the sequence of iterates in topological linear spaces for Kirk's mappings and Ishikawa and Mann iteration process. Also we prove that under appropriate assumptions, the limit of a sequence of iterates turns out to be a fixed point.