Vol. 66, No. 1, 1976

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Existence of fixed points of nonexpansive mappings in a space without normal structure

Les Andrew Karlovitz

Vol. 66 (1976), No. 1, 153–159

A mapping T : C X defined on a subset C of a Banach space X, with norm ∥⋅∥, is said to be nonexpansive if Tx Tyx yfor all x,y C. If C is assumed to be convex and weakly compact and if T : C C then one of the main open questions is whether T has a fixed point in C, i.e., whether there exists x C so that Tx = x. If X is reflexive and uniformly convex or, more generally, if X is reflexive and has normal structure then the answer is affirmative. Our purpose is to give an example of a classical reflexive space which does not have normal structure and for which the answer is nevertheless affirmative.

Mathematical Subject Classification 2000
Primary: 47H10
Received: 4 June 1976
Published: 1 September 1976
Les Andrew Karlovitz