Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-02T01:41:20.526Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalizations of F. E. Browder's sharpened form of the schauder fixed point theorem

Part of: Maps and general types of spaces defined by maps Nonlinear operators and their properties

Published online by Cambridge University Press:  09 April 2009

Kok-Keong Tan
Kok-Keong Tan
Affiliation:
Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifax, Nova Scotia B3H 3J5, Canada
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let E be a Hausdorff topological vector space, let K be a nonempty compact convex subset of E and let f, g: K → 2E be upper semicontinuous such that for each x ∈ K, f(x) and g(x) are nonempty compact convex. Let Ω ⊂ 2E be convex and contain all sets of the form x − f(x), y − x + g(x) − f(x), for x, y ∈ K. Suppose p: K × Ω →, R satisfies: (i) for each (x, A) ∈ K × Ω and for ε > 0, there exist a neighborhood U of x in K and an open subset set G in E with A ⊂ G such that for all (y, B) ∈ K ×Ω with y ∈ U and B ⊂ G, | p(y, B) - p(x, A)| < ε, and (ii) for each fixed X ∈ K, p(x, ·) is a convex function on Ω. If p(x, x − f(x)) ≤ p(x, g(x) − f(x)) for all x ∈ K, and if, for each x ∈ K with f(x) ∩ g(x) = ø, there exists y ∈ K with p(x, y − x + g(x) − f(x)) < p(x, x − f(x)), then there exists an x0 ∈ K such that f(x0) ∩ g(x0) ≠ ø. Another coincidence theorem on a nonempty compact convex subset of a Hausdorff locally convex topological vector space is also given.

Keywords

fixed pointSchauder fixed point theoremset-value maplower semicontinuousupper semicontinuousultimately continuousconvex functioninward setoutward settopological vector spacelocally convex topological vector space.

MSC classification

Secondary: 47H10: Fixed-point theorems 54C60: Set-valued maps
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 42 , Issue 3 , June 1987 , pp. 390 - 398
Copyright
Copyright © Australian Mathematical Society 1987

References

[1]Berge, C., Topological Spaces (Oliver & Boyd, Edinburgh and London, 1963).Google Scholar
[2]Browder, F. E., A new generalization of the Schauder fixed point theorem, Math. Ann. 174 (1967), 285–290.CrossRefGoogle Scholar
[3]Browder, F. E., The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283–301.CrossRefGoogle Scholar
[4]Browder, F. E., On a sharpened form of the Schauder fixed-point theorem, Proc. Nat. Acad. Sci. U. S. A. 74 (1977), 4749–4751.CrossRefGoogle ScholarPubMed
[5]Caristi, J., Fixed point theorem for mapping satisfying inwardness conditions, Trans. Amer. Math. Soc. 215 (1976), 241–251.CrossRefGoogle Scholar
[6]Fan, K., A generalization of Tychonoff's fixed point theorem, Math. Ann. 142 (1961), 305–310.CrossRefGoogle Scholar
[7]Fan, K., Extensions of two fixed point theorems of F. E. Browder, Math. Z. 112 (1969), 234–240.CrossRefGoogle Scholar
[8]Halpern, B. and Bergman, G., A fixed point theorem for inward and outward maps, Trans. Amer. Math. Soc. 130 (1968), 353–358.CrossRefGoogle Scholar
[9]Jiang, J. H., Fixed point theorems for multi-valued mappings in locally convex spaces, Acta Math. Sinica 25 (1982), 365–373.Google Scholar
[10]Jiang, J. H., Generalization of two fixed point theorems of S. Reich, Acta Math. Sinica 24 (1981), 359–364.Google Scholar
[11]Reich, S., Approximate selections, best approximations, fixed points, and invariant sets, J. Math. Anal. Appl. 62 (1978), 104–113.CrossRefGoogle Scholar
[12]Reich, S., Fixed point theorems for set-valued mappings, J. Math. Anal. Appl. 69 (1979), 353–358.CrossRefGoogle Scholar
[13]Reich, S., Some problems and results in fixed point theory, Contemporary Math. 21 (1983), 179–187.CrossRefGoogle Scholar