A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

In [7], Goebel, Kirk and Shimi proved the following: Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K: (1) where a i ≥0 and Then F has a fixed point in K. In this paper we shall prove that this theorem remains true in any Banach space X , provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).