Generalizations of the Converse of the Contraction Mapping Principle
Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 1095-1104
Voir la notice de l'article provenant de la source Cambridge University Press
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).
Wong, James S. W. Generalizations of the Converse of the Contraction Mapping Principle. Canadian journal of mathematics, Tome 18 (1966) no. 1, pp. 1095-1104. doi: 10.4153/CJM-1966-110-1
@article{10_4153_CJM_1966_110_1,
author = {Wong, James S. W.},
title = {Generalizations of the {Converse} of the {Contraction} {Mapping} {Principle}},
journal = {Canadian journal of mathematics},
pages = {1095--1104},
year = {1966},
volume = {18},
number = {1},
doi = {10.4153/CJM-1966-110-1},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-110-1/}
}
TY - JOUR AU - Wong, James S. W. TI - Generalizations of the Converse of the Contraction Mapping Principle JO - Canadian journal of mathematics PY - 1966 SP - 1095 EP - 1104 VL - 18 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1966-110-1/ DO - 10.4153/CJM-1966-110-1 ID - 10_4153_CJM_1966_110_1 ER -
[1] 1. Bessaga, C., On the converse of the Banach fixed-point principle, Coll. Math., 7 (1959), 41–43. Google Scholar
[2] 2. Birkhoff, G., Lattice theory (Providence, 1948). Google Scholar
[3] 3. Wong, J. S. W., A generalization of the converse of contraction mapping principle, Amer. Math. Soc. Notices, 11 (1964), 385. Google Scholar
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