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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A generalization of Banach's contraction principle


Author: Lj. B. Ćirić
Journal: Proc. Amer. Math. Soc. 45 (1974), 267-273
MSC: Primary 54H25
MathSciNet review: 0356011
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Abstract: Let $ T:M \to M$ be a mapping of a metric space $ (M,d)$ into itself. A mapping $ T$ will be called a quasi-contraction iff $ d(Tx,Ty) \leqslant q\max \{ d(x,y);d(x,Tx);d(y,Ty);d(x,Ty);d(y,Tx)\} $ for some $ q < 1$ and all $ x,y \in M$. In the present paper the mappings of this kind are investigated. The results presented here show that the condition of quasi-contractivity implies all conclusions of Banach's contraction principle. Multi-valued quasi-contractions are also discussed.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1974-0356011-2
PII: S 0002-9939(1974)0356011-2
Keywords: Quasi-contractions, multi-valued quasi-contractions, fixed-point theorems
Article copyright: © Copyright 1974 American Mathematical Society