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Fixed Point Theorems for Proximately Nonexpansive Semigroups

Published online by Cambridge University Press:  20 November 2018

Mo Tak Kiang  and
Kok-Keong Tan
Mo Tak Kiang
Affiliation:
Department of Mathematics and Computing Science, Saint Mary's UniversityHalifax, N.S. Canada B3H 3C3
Kok-Keong Tan
Affiliation:
Department of Mathematics, Statistics And Computing Science, Dalhousie UniversityHalifax, N.S. Canada B3H 4H8
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Abstract

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A commutative semigroup G of continuous, selfmappings on (X, d) is called proximately nonexpansive on X if for every x in X and every (β > 0, there is a member g in G such that d(fg(x),fg(y)) ≤ (1 + β) d (x, y) for every f in G and y in X. For a uniformly convex Banach space it is shown that if G is a commutative semigroup of continuous selfmappings on X which is proximately nonexpansive, then a common fixed point exists if there is an x0 in X such that its orbit G(x0) is bounded. Furthermore, the asymptotic center of G(x0) is such a common fixed point.

Keywords

47H10Common fixed pointproximately nonexpansiveasymptotic center
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 29 , Issue 2 , 01 June 1986 , pp. 160 - 166
Copyright
Copyright © Canadian Mathematical Society 1986

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