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Uniformly Lipschitzian Families of Transformations in Banach Spaces

Published online by Cambridge University Press:  20 November 2018

K. Goebel ,
W. A. Kirk  and
R. L. Thele
K. Goebel
Affiliation:
Maria Curie Sklodowska University, Lublin, Poland
W. A. Kirk
Affiliation:
The University of Iowa, Iowa City, Iowa
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The observations of this paper evolved from the concept of 'asymptotic nonexpansiveness' introduced by two of the writers in a previous paper [10]. Let X be a Banach space and K ⊆ X. A mapping T : K → K is called asymptotically nonexpansive if for each x, y ∊ K

where {ki} is a fixed sequence of real numbers such that ki→1 as i → ∞ . It is proved in [10] that if K is a bounded closed and convex subset of a uniformly convex space X then every asymptotically nonexpansive mapping T : K → K has a fixed point. This theorem generalizes the fixed point theorem of Browder-Göhde-Kirk [2 ; 12 ; 16] for nonexpansive mappings (mappings T for which ||T(x) — T(y)|| ≦ ||x — y||, x, y ∊ K) in a uniformly convex space. (A generalization along similar lines also has been obtained by Edelstein [4].)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 26 , Issue 5 , October 1974 , pp. 1245 - 1256
Copyright
Copyright © Canadian Mathematical Society 1974

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