On Dugundji's notion of positive definiteness

Author:
Chi Song Wong

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

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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 , there may be lack of the right kind of lower semicontinuous real-valued functions on 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.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1974-0418076-9

Keywords:
Bolzano-Weierstrass property,
compactness,
completeness,
countable compactness,
fixed point,
lower semicontinuity,
pseudo compactness,
positive definiteness,
sequential compactness,
uniformity,
uniform continuity

Article copyright:
© Copyright 1974
American Mathematical Society