On a Fixed Point Theorem of Ky Fan

We generalize a theorem of Ky Fan about the nearest distance between a closed convex set D in a Banach space E and its image by a function ƒ:D→E, in several directions: (1) for noncompact sets D, when ƒ(D) precompact; (2) for compact D and upper semicontinuous multifunction ƒ and more generally, (3) for noncompact D and upper semicontinuous multifunction ƒ with ƒ(D) Hausdorff precompact.In particular, we prove a version of the fixed point theorem of Kakutani-Ky Fan for multifunctions whose values are convex closed bounded, thus not necessarily compact.

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