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Best proximity pair theorems for multifunctions with open fibres. (English) Zbl 0965.41020
Let A and B be non-empty subsets of a normed linear space E, and let T:A2 B be a convex multi-valued function with open fibres T -1 (y) (i.e.) {xX:yTx}. For an element x 0 A sufficient conditions are found so that dist(x 0 ,Tx 0 )=dist(A,B). This is the case if, say, A is a non-empty approximately compact, and convex proximinal subset of E, and B is a non-empty, closed and convex subset of E, and A 0 is compact, while T(A 0 )B 0 . Here A 0 ={aA:dist(a,b)=dist(A,B) for some bB}. Consequences include special cases of the Brouwer’s fixed point theorem.
41A65Abstract approximation theory