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    STUDIA MATHEMATICA - Issue no. 1 / 2010  
         
  Article:   SET-VALUED APPROXIMATION OF MULTIFUNCTIONS.

Authors:  MARIAN MUREŞAN. 		 
       
         
  Abstract:  This survey paper introduces several results on approximation of multifunctions with convex and non-convex values. We consider multifunctions having at least nonempty and compact values in Rn. The convex case (when the multifunctions have convex values) is closer to the point-topoint case. The non-convex case (the values of the multifunctions are not longer assumed to be convex) is more challenging. In the convex case we present results on the Bernstein approximation, the Stone-Weierstrass approximation theorem, and the Korovkin-type approximation. In the nonconvex case we present results on linear operators on multifunctions based on a metric linear combination of ordered sets, metric piecewise linear approximations of multifunctions, and approximation by metric Bernstein, Schoenberg, and interpolation operators. The present survey paper was introduced at University of Duisburg–Essen located in Duisburg while the author was a visiting scientist under a grant of “Center of Excellence for Applications of Mathematics” supported by DAAD. The author expresses his gratitude to Prof. H. Gonska for his invitation and warm hospitality in Duisburg. The author also appreciates the valuable comments and remarks of Mr. Michael Wozniczka from the same University.

Key words and phrases. compact sets, Minkowski linear combination, metric average, set-valued functions, piecewise linear set-valued functions, linear approximation operators, metric Bernstein approximation, metric Schoenberg approximation, metric polynomial interpolation.
  
         
     
         
         
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