tailieunhanh - Báo cáo hóa học: " Research Article Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008 Article ID 528614 10 pages doi 2008 528614 Research Article Convergence to Compact Sets of Inexact Orbits of Nonexpansive Mappings in Banach and Metric Spaces Evgeniy Pustylnik Simeon Reich and Alexander J. Zaslavski Department of Mathematics The Technion-Israel Institute of Technology 32000 Haifa Israel Correspondence should be addressed to Simeon Reich sreich@ Received 27 September 2008 Accepted 17 November 2008 Recommended by Brailey Sims 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. Copyright 2008 Evgeniy Pustylnik et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction 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 1 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 2 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.