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A Note on an Iterative Test of Edelstein(1)

Published online by Cambridge University Press:  20 November 2018

Sam B. Nadler Jr.
Sam B. Nadler Jr.*
Affiliation:
Dalhousie University, Halifax, Nova Scotia
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Let (X1, d1) and (X2, d2) be metric spaces. A mapping/: X1→X2 is said to be a Lipschitz mapping (with respect to d1 and d2) if and only if (*)d2(f(x), f(y))≤λ⋅ d1(x, y) for all x, y∈X1, where λ is a fixed real number. The constant λ is called a Lipschitz constant for f. If (*) is satisfied for λ=l, then f is called nonexpansive (see, for example, [21]) and if (*), again with λ=1, is replaced by a strict inequality for all x≠y, then f is called contractive [1]. If x∊X1 and X1=X2, then the sequence where f1(x)=f(x) and fn(x)=f(fn-1(x)) for each n>1, is called the sequence of iterates of f at x.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 3 , September 1972 , pp. 381 - 386
Copyright
Copyright © Canadian Mathematical Society 1972

Footnotes

(1)

This work was supported by National Research Council Grant A-7346 and a Loyola Faculty Research Grant.

References

1. Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79.Google Scholar
2. Edelstein, M., On non-expansive mappings of Banach spaces, Proc. Cambridge Philos. Soc. 60 (1964), 439-447.Google Scholar