KKM-type theorems for best proximity points

Let us consider two nonempty subsets A,BA,B of a normed linear space XX, and let us denote by 2B2B the set of all subsets of BB. We introduce a new class of multivalued mappings {T:A→2B}{T:A→2B}, 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}∩{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.