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A nonlinear ergodic theorem for asymptotically nonexpansive mappings

Published online by Cambridge University Press:  17 April 2009

Kok-Keong Tan  and
Hong-Kun Xu
Kok-Keong Tan
Affiliation:
Department of Mathematics, Statistics and Computing Science, Dalhousie University, Halifax, Nova Scotia, CanadaB3H 3J5
Hong-Kun Xu
Affiliation:
Institute of Applied Mathematics, East China University of Chemical Technology Shanghai 200237 People's, Republic of China
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Abstract

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Let X be a real uniformly convex Banach space satisfying the Opial's condition, C a bounded closed convex subset of X, and T: C → C an asymptotically non-expansive mapping. Then we show that for each x in C, the sequence {Tnx} almost converges weakly to a fixed point y of T, that is,

This implies that {Tnx} converges weakly to y if and only if T is weakly asymptotically regular at x, that is, weak- . We also present a weak convergence theorem for asymptotically nonexpansive semigroups.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 45 , Issue 1 , February 1992 , pp. 25 - 36
Copyright
Copyright © Australian Mathematical Society 1992

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