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Best proximity pair theorems. (English) Zbl 1021.47027

Let X and Y be any two topological spaces. A multifunction T:X2 Y is said to be

(i) upper semi-continuous if T -1 (B)={xX:(Tx)B} is closed in X whenever B is a closed subset of Y;

(ii) Kakutani multifunction if (a) T is upper semi-continuous, (b) either Tx is a singleton for each xX or Tx is a non-empty compact convex subset of Y, assuming Y to be a non-empty convex set in a Hausdorff topological vector space;

(iii) Kakutani factorizable if T can be expressed as a composition of finitely many Kakutani multifunctions.

Let E be a Hausdorff locally convex topological vector space with a continuous seminorm p. A non-empty subset A of E is said to be approximately p-compact if for each yE and each net {x α } in A satisfying d p (x α ,y)d p (y,A)inf{p(y-a):aA}, there is a subset of {x α } converging to an element of A.

In the present paper, the authors prove best proximity pair theorems which furnish sufficient conditions ensuring the existence of an element x 0 A such that

d p (gx 0 ,Tx 0 )=d p (A,B)inf{p(a-b):aA,bB},

when A is a non-empty approximately p-compact convex subset, B a non-empty closed convex subset of E, T:A2 B is a Kakutani factorizable multifunction and g:AA is a single-valued function.

47H04Set-valued operators