Abstract
AbstractIn 1965, Kirk proved that if "Equation missing" is a nonempty weakly compact convex subset of a Banach space with normal structure, then every nonexpansive mapping "Equation missing" has a fixed point. The purpose of this paper is to outline various generalizations of Kirk's fixed point theorem to semigroup of nonexpansive mappings and for Banach spaces associated to a locally compact group.
Highlights
A closed convex subset C of a Banach space E has normal structure if for each bounded closed convex subset D of C which contains more than one point, there is a point x ∈ D which is not a diametral point of D, that is, sup { x − y : y ∈ D} < δ D, where δ D the diameter of D
Fixed Point Theory and Applications. It is the purpose of this paper to outline the relation of normal structure and fixed point property for semigroup of nonexpansive mappings
Belluce and Kirk 14 improved DeMarr’s result in 10 and proved that if C is a nonempty weakly compact convex subset of a Banach space and if C has complete normal structure, every family of commuting nonexpansive self-maps on C has a common fixed point. This result was extended to the class of left reversible semitopological semigroup by Holmes and Lau in 15
Summary
A closed convex subset C of a Banach space E has normal structure if for each bounded closed convex subset D of C which contains more than one point, there is a point x ∈ D which is not a diametral point of D, that is, sup { x − y : y ∈ D} < δ D , where δ D the diameter of D. Compact convex subset of a Banach space E always has normal structure see 2 It was an open problem for over 15 years whether every weakly compact convex subset of E has normal structure. This problem was answered negatively by Alspach 3 when he showed that there is a weakly compact convex subset C of L1 0, 1 which does not have the fixed point property. It is the purpose of this paper to outline the relation of normal structure and fixed point property for semigroup of nonexpansive mappings.
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