## Fixed Point Theorem for Non-Expansive Mappings on Banach Spaces with Unifformly Normal Structure

##### Abstract

In [1] Kirk proved that if D is a bounded, closed, and convex subset of a reflexive Banach space that has normal structure, then every non-expansive mapping of D into D has a fixed point. This was also proved for a uniformly convex space by Browder [2]. In this paper we replace uniformly convex, or reflexive and normal structure, by uniformly normal structure to obtain this result.
Let S be a bounded subset of the Banach space X and [see pdf for notation]
(1) [see pdf for notation]
(2) [see pdf for notation]
(3) [see pdf for notation]
A space X is said to have uniformly normal structure if for some [see pdf for notation], every bounded closed and convex subset S of X, [see pdf for notation].
In [3] Edelstein shows that a space that is uniformly convex will have uniformly normal structure. An example of a Banach space that has uniformly normal structure that is not uniformly convex is the space [see pdf for notation].
Later in this paper we give an example of a Banach space that is uniformly convex in all directions but does not have uniformly normal structure.
Hence all uniformly convex spaces will have uniformly normal structure but spaces that have uniformly normal structure will not necessarily be uniformly convex. All spaces that are uniformly convex in all directions will have normal structure but not all these spaces will have uniformly normal structure. All spaces with uniformly normal structure will have normal structure.