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Extensions of Contractive Mappings and Edelstein's Iterative Test

Published online by Cambridge University Press:  20 November 2018

Jack Bryant  and
L. F. Guseman Jr.
Jack Bryant
Affiliation:
Texas A & M University, College Station, Texas
L. F. Guseman Jr.
Affiliation:
Texas A & M University, College Station, Texas
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A mapping f from a metric space (X,d) into itself is said to be contractive if x≠y implies d(f(x),f(y))<d(x,y). Theorems of Edelstein [2] state that a contractive selfmapfofa metric space X has a fixed point if, for some x0, the sequence {fn(x0)} of iterates at x0 has a convergent subsequence; moreover, the sequence {fn(x0)} converges to the unique fixed point of f. Nadler [3] observes that, from the point of view of applications, it is usually as difficult to verify the condition (for some x0 …) as it is to find the fixed point directly.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 16 , Issue 2 , June 1973 , pp. 185 - 192
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Blumenthal, L.M., Theory and application of distance geometry, Clarendon Press, Oxford, 1953.Google Scholar
2. Edelstein, M., On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79.Google Scholar
3. Nadler, Sam B. Jr., A note on an iterative test of Edelstein, Canad, Math. Bull. (3) 15 (1972), 381–386.Google Scholar