ORIGINAL ARTICLES

Acta Mathematica Sinica

, Volume 18, Issue 2, pp 363-374

On a Fixed Point Theorem of Ky Fan

  • Francesco S. De BlasiAffiliated withDepartment of Mathematics, University of Roma II 'Tor Vergata' Email author 
  • , Pando Gr. GeorgievAffiliated withDepartment of Mathematics and Informatics, Sofia University "St. Kl. Ohridski"Laboratory for Advanced Brain Signal Processing,, Brain Science Institute, The Institute of Physical and Chemical Resarch (RIKEN) 2-1, Hirosawa, Wako-shi

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Abstract

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:DE, 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.

Keywords

Ky Fan, Compact hcompact Upper semicontinuous multifunction

MR (2000) Subject Classification

47H10 54H25