KKM-type theorems for best proximity points

Let us consider two nonempty subsets A , B of a normed linear space X , and let us denote by 2 B the set of all subsets of B . We introduce a new class of multivalued mappings { T : A → 2 B } , called R-KKM mappings, which extends the notion of KKM mappings. First, we discuss some sufficient conditions for which the set ∩ { T ( x ) : x ∈ A } is nonempty. Using this nonempty intersection theorem, we attempt to prove a extended version of the Fan–Browder multivalued fixed point theorem, in a normed linear space setting, by providing an existence of a best proximity point.