On Dugundji’s notion of positive definiteness
HTML articles powered by AMS MathViewer
- by Chi Song Wong PDF
- Proc. Amer. Math. Soc. 46 (1974), 443-450 Request permission
Abstract:
Dugundji’s notion of positive definiteness is generalized to nonnegative real-valued functions on a uniform space. Its relations with completeness and various notions of compactness are investigated. For an arbitrary uniform space $X$, there may be lack of the right kind of lower semicontinuous real-valued functions on $X$ and so a further generalization of Dugundji’s notion of positive definiteness is needed for the development of the fixed point (or coincidence) theory. With such an extension, a very general fixed point theorem is obtained to include a recent result of the author, which contains, as special cases, some results of S. Banach, F.E. Browder, D. W. Boyd and J. S. W. Wong, M. Edelstein and R. Kannan.References
- D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458–464. MR 239559, DOI 10.1090/S0002-9939-1969-0239559-9
- Felix E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71=Indag. Math. 30 (1968), 27–35. MR 0230180, DOI 10.1016/S1385-7258(68)50004-0
- James Dugundji, Topology, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0193606 —, Positive definite functions and coincidences, Séminaire de Mathématiques Supérieures (fixed point theory and its applications), Université de Montréal (June, 1973).
- M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74–79. MR 133102, DOI 10.1112/jlms/s1-37.1.74
- R. Kannan, Fixed point theorems in reflexive Banach spaces, Proc. Amer. Math. Soc. 38 (1973), 111–118. MR 313896, DOI 10.1090/S0002-9939-1973-0313896-2
- John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144 C. S. Wong, Fixed point theorems for nonexpansive mappings, Ph. D. Thesis, University of Illinois, Urbana, Ill., 1969. —, A fixed point theorem for certain functions on a complete Hausdorff uniform space, Notices Amer. Math. Soc. 18 (1971), 191. Abstract #628-47-1.
- Chi Song Wong, Fixed point theorems for nonexpansive mappings, J. Math. Anal. Appl. 37 (1972), 142–150. MR 293614, DOI 10.1016/0022-247X(72)90263-6
- Chi Song Wong, A fixed point theorem for a class of mappings, Math. Ann. 204 (1973), 97–103. MR 367964, DOI 10.1007/BF01433408 —, Semicontinuous functions.
Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 46 (1974), 443-450
- MSC: Primary 54H25
- DOI: https://doi.org/10.1090/S0002-9939-1974-0418076-9
- MathSciNet review: 0418076