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Some Remarks Concerning Contraction Mappings

Published online by Cambridge University Press:  20 November 2018

Simeon Reich
Simeon Reich*
Affiliation:
Israel Institute of Technology, Haifa, Israel
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The following result is proved in [1, p. 6].

Theorem 1. Let X be a complete metric space, and let T and Tn(n = 1, 2,…)be contraction mappings of X into itself with the same Lipschitz constant k<1, and with fixed points u and un respectively. Suppose that limn → ∞ Tn(x) = T(x) for every x ∊ X. Then limn → ∞ un = u.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 14 , Issue 1 , January 1971 , pp. 121 - 124
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Bonsall, F. F., Lectures on some fixed point theorems of functional analysis, Tata Institute of Fundamental Research, Bombay, 1962.Google Scholar
2. Rakotch, E., A note on contractive mappings, Proc. Amer. Math. Soc. 13 (1962), 459-465.Google Scholar
3. Rakotch, E., On ε-contractive mappings, Bull. Res. Counc. of Israel, F, Math. & Phys. 10F (1962), 53-58.Google Scholar
4. Singh, S. P. and Russel, W., A note on a sequence of contraction mappings, Canad. Math. Bull. 12(1969), 513-516.Google Scholar
5. Kannan, R., Some results on fixed points?II, Amer. Math. Monthly 76 (1969), 405-408.Google Scholar