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An ergodic theorem for asymptotically nonexpansive mappings

Published online by Cambridge University Press:  14 November 2011

Manfred Krüppel  and
Jaroslaw Górnicki
Manfred Krüppel
Affiliation:
Universität Rostock, Auβenstelle Güstrow, Fachbereich Mathematik, Goldberger Str. 12, O-2600 Güstrow, Germany
Jaroslaw Górnicki
Affiliation:
Department of Mathematics, Rzeszów Technical University, P.O. Box 85, 35-959 Rzeszów, Poland
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Abstract

The purpose of this paper is to prove the following (nonlinear) mean ergodic theorem: Let E be a uniformly convex Banach space, let C be a nonempty bounded closed convex subset of E and let T: C → C be an asymptotically nonexpansive mapping. If

exists uniformly in r = 0, 1, 2,…, then the sequence {Tnx} is strongly almost-convergent to a fixed point y of T, that is,

uniformly in i = 0, 1, 2, ….

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 1 , 1994 , pp. 23 - 31
Copyright
Copyright © Royal Society of Edinburgh 1994

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