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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
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Abstract

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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

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