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Generalizations of the Converse of the Contraction Mapping Principle

Published online by Cambridge University Press:  20 November 2018

James S. W. Wong
James S. W. Wong*
Affiliation:
University of Alberta, Edmonton
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This paper is an outgrowth of studies related to the converse of the contraction mapping principle. A natural formulation of the converse statement may be stated as follows: “Let X be a complete metric space, and T be a mapping of X into itself such that for each x ∈ X, the sequence of iterates ﹛Tnx﹜ converges to a unique fixed point ω ∈ X. Then there exists a complete metric in X in which T is a contraction.” This is in fact true, even in a stronger sense, as may be seen from the following result of Bessaga (1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 18 , 1966 , pp. 1095 - 1104
Copyright
Copyright © Canadian Mathematical Society 1966

References

1. Bessaga, C., On the converse of the Banach fixed-point principle, Coll. Math., 7 (1959), 41–43.Google Scholar
2. Birkhoff, G., Lattice theory (Providence, 1948).Google Scholar
3. Wong, J. S. W., A generalization of the converse of contraction mapping principle, Amer. Math. Soc. Notices, 11 (1964), 385.Google Scholar