A fixed point theorem for asymptotically nonexpansive mappings

Authors:
K. Goebel and W. A. Kirk

Journal:
Proc. Amer. Math. Soc. **35** (1972), 171-174

MSC:
Primary 47H10

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298500-3

MathSciNet review:
0298500

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Abstract | References | Similar Articles | Additional Information

Abstract: Let *K* be a subset of a Banach space *X*. A mapping is said to be asymptotically nonexpansive if there exists a sequence of real numbers with as such that . It is proved that if *K* is a non-empty, closed, convex, and bounded subset of a uniformly convex Banach space, and if is asymptotically nonexpansive, then *F* has a fixed point. This result generalizes a fixed point theorem for nonexpansive mappings proved independently by F. E. Browder, D. Göhde, and W. A. Kirk.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298500-3

Keywords:
Fixed point theorem,
nonexpansive mapping,
asymptotically nonexpansive mapping,
uniformly convex Banach space,
lipschitzian mapping

Article copyright:
© Copyright 1972
American Mathematical Society