Geodesic
A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi
Canadian mathematical bulletin, Tome 19 (1976) no. 1, pp. 7-12

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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). 
Bogin, Joseph. A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi. Canadian mathematical bulletin, Tome 19 (1976) no. 1, pp. 7-12. doi: 10.4153/CMB-1976-002-7
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