Existence of fixed points of nonexpansive mappings in a space without normal structure
Les Karlovitz · Pacific Journal of Mathematics · 1976
A mapping T: C->X defined on a subset C of a Banach space X, with norm || ||, is said to be nonexpansive if || Tx -Ty || ^ || x -y || for all x, y E 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 E C so that Tx = x.It 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.