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A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

Published online by Cambridge University Press:  20 November 2018

Joseph Bogin
Joseph Bogin*
Affiliation:
Haifa, Israel
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In [7], Goebel, Kirk and Shimi proved the following:

Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:

(1)

where ai≥0 and Then F has a fixed point in K.

In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 19 , Issue 1 , 01 March 1976 , pp. 7 - 12
Copyright
Copyright © Canadian Mathematical Society 1976

Footnotes

Technion Preprint series NO. MT-205.

(1)

This paper is a part of the author’s M.Sc. thesis which was prepared under the guidance of Professor M. Marcus, Dept. of Mathematics, Technion, Israel Institute of Technology, Haifa.

References

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