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Sequences of Contractions in a Generalized Metric Space

Published online by Cambridge University Press:  20 November 2018

C. W. Norris
C. W. Norris*
Affiliation:
Memorial University, St. John's, Newfoundland
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The main aim of this paper is to study the convergence of a sequence of contractions in a generalized metric space. More specifically, we investigate the following question:

"If a sequence of contractions {fr} with fixed points ur (r = 1,2,...) converges to a mapping f with a fixed point u, under what conditions will the sequence ur converge to u?"

A partial answer to the above question has been given in metric spaces by Bonsall [1]. This result has since been improved by Russell and Singh [6]. Further results will now be given in a generalized metric space.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 1 , March 1970 , pp. 55 - 58
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Bonsall, F., Lecture notes on some fixed point theorems in analysis, Tata Institute of Fundamental Research, Bombay, India, 1962.Google Scholar
2. Diaz, and Margolis, , A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74, No. 2 (1968), 305-309.Google Scholar
3. Luxemburg, W. A. J., On the convergence of successive approximations in the theory of ordinary differential equations, II, Nederl. Akad. Wetensch. Proc. (ser. A61) Indag. Mathematicae 20 (1958), 540-546.Google Scholar
4. Luxemburg, W. A. J., On the convergence of successive approximations in the theory of ordinary differential equations III, Nieuw Archief Voor Wiskunde 6, No. 3 (1958), 93-98.Google Scholar
5. Russel, and Singh, , A remark on a fixed point theorem for contractions, Canad. Math. Bull, (to appear).Google Scholar