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Some Remarks on Transformations in Metric Spaces

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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Recently A. Haimovici [1] has proved a general fixed point theorem of transformations in metric spaces from which he obtained existence theorems for certain types of ordinary and partial differential equations. However, both the result and the proof are given for a rather special case. One of the purposes of this present note is to put his result on a more concrete basis and give a stronger characterization of the kind of transformations used in [l]. (Theorem 3).

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 5 , October 1965 , pp. 659 - 666
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Haimovici, A., Un théorème d' existence pour des équations fonctionelles, généralisant le théorème de Peano. Ann. St. Univ. "AI. I. Cuza" (série mona) VII (1961), pp. 65-76.Google Scholar
2. Palczewski, B., On the uniqueness of solutions and the convergence of successive approximations in the Darboux problem under the conditions of the Krasnosielski and Krein type. Ann. Polon. Math. XIV(1964), pp. 183-190.Google Scholar
3. Kolmogorov, A.N. and Fomin, S. V., Elements of the theory of functions and functional analysis. Vol. 1, Graylock Press, New York, 1957.Google Scholar
4. Wong, J. S. W., On the convergence of successive approximations in the Darboux problem. (To appear), Ann. Polon. Math., XV (1965).Google Scholar
5. Wehausen, J.V., Transformations in metric spaces and ordinary differential equations. Bull. Amer. Math. Soc. 51 (1945), pp. 113-119.Google Scholar
6. Edrei, A., On iteration of mappings of a metric space onto itself. J. London Math. Soc. 26 (1951), pp. 96-103.Google Scholar
7. Wong, J. S. W., Generalizations to the converse of contraction mapping principle, Ph. D. Thesis. California Institute of Technology. (Submitted May, 1964).Google Scholar
8. Janos, L., Converse of the theorem on contracting mapping. Amer. Math. Soc. Notices 11 (1964), p. 224.Google Scholar
9. Janos, L., Converse of the Banach theorem in the case of one-to-one contracting mapping. Amer. Math. Soc. Notices 11 (1964), p. 686.Google Scholar