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Some Fixed Point Theorems in Metric and Banach Spaces

Published online by Cambridge University Press:  20 November 2018

L. P. Belluce  and
W.A. Kirk
L. P. Belluce
Affiliation:
University of British Columbia Vancouver, British Columbia
W.A. Kirk
Affiliation:
University of Iowa Iowa City, Iowa
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The purpose of this paper is two-fold. Sections 2 and 3 are motivated by an observation that certain theorems concerning "diminishing orbital diameters" (introduced in [1]) are true under weaker assumptions. Specifically, we investigate the relationship between that concept and alternate conditions such as "asymptotic regularity", and in the process we sharpen some metric space results established in [1;5]. Mention is made in these sections of examples which show that certain additional weakenings of our hypotheses cannot be made, but we include in detail only the one which seemed to us most intricate.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 4 , August 1969 , pp. 481 - 491
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Belluce, L. P. and Kirk, W.A., Fixed point theorems for certain classes of nonexpansive mappings. Proc. Amer. Math. Soc. 20 (1969) 141–146.Google Scholar
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4. Kirk, W.A., A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72 (1965) 1004–1006.Google Scholar
5. Kirk, W.A., On mappings with diminishing orbital diameters, J. London Math. Soc. 44 (1969) 107–111.Google Scholar
6. Kirk, W.A., Fixed point theorems for nonexpansive mappings. Proc. Symp. on Nonlinear Functional Analysis, April 1968. (to appear in Symp. Pure Math.)Google Scholar
7. Kirk, W. A., On a nonlinear functional equation in a Banach space (submitted).Google Scholar
8. Göhde, D., Zum Prinzip der kontraktiven Abbildung. Math. Nach. 30 (1965) 251–258.Google Scholar