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Abstract and Applied Analysis
Volume 2003, Issue 2, Pages 67-74
http://dx.doi.org/10.1155/S1085337503206060

A weak ergodic theorem for infinite products of Lipschitzian mappings

Simeon Reich and Alexander J. Zaslavski

Department of Mathematics, The Technion-Israel Institute of Technology, Haifa 32000, Israel

Received 16 May 2002

Copyright © 2003 Hindawi Publishing Corporation. 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.

Abstract

Let K be a bounded, closed, and convex subset of a Banach space. For a Lipschitzian self-mapping A of K, we denote by Lip(A) its Lipschitz constant. In this paper, we establish a convergence property of infinite products of Lipschitzian self-mappings of K. We consider the set of all sequences {At }t=1∞ of such self-mappings with the property limsupt→∞Lip(At )≤1. Endowing it with an appropriate topology, we establish a weak ergodic theorem for the infinite products corresponding to generic sequences in this space.