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A generalized contraction principle

Published online by Cambridge University Press:  17 April 2009

Ivan L. Reilly
Ivan L. Reilly
Affiliation:
Department of Mathematics, University of Auckland, Auckland, New Zealand.
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This paper presents an extension of Banach's contraction mapping principle to Hausdorff spaces, in fact to the larger class of topological spaces in which convergent sequences have unique limits. This is achieved by considering topologies on X generated by families of quasi-pseudo-metrics on X. An extension of the concept of Cauchy sequence to this non-metric setting is given.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 3 , June 1974 , pp. 359 - 363
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Banach, Stefan, “Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales”, Fund. Math. 3 (1922), 133–181.CrossRefGoogle Scholar
[2]Brown, Thomas A. and Comfort, W.W., “New method for expansion and contraction maps in uniform spaces”, Proc. Amer. Math. Soc. 11 (1960), 483–486.Google Scholar
[3]Dugundji, James, Topology (Allyn and Bacon, Boston, 1966).Google Scholar
[4]Edelstein, Michael, “On nonexpansive mappings of uniform spaces”, Proc. K. Nederl. Akad. Wetensah. Ser. A 68 (1965), 47–51.CrossRefGoogle Scholar
[5]Kammerer, W.J. and Kasriel, R.H., “On contractive mappings in uniform spaces”, Proc. Amer. Math. Soc. 15 (1964), 288–290.Google Scholar
[6]Kelly, J.C., “Bitopological spaces”, Proc. London Math. Soc. (3) 13 (1963), 71–89.Google Scholar
[7]Knill, Ronald J., “Fixed points of uniform contractions”, J. Math. Anal. Appl. 12 (1965), 449–455.CrossRefGoogle Scholar
[8]Naimpally, Somashekhar Amrith, “Contractive mappings in uniform spaces”, Proc. K. Nederl. Akad. Wetensoh. Ser. A 68 (1965), 477–481.Google Scholar
[9]Reilly, Ivan L., “Quasi-gauge spaces”, J. London Math. Soc. (2) 6 (1973), 481–487.Google Scholar
[10]Wilansky, Albert, “Between T 1 and T 2”, Amer. Math. Monthly 74 (1967), 261–266.Google Scholar