The existence of best proximity points for multivalued non-self-mappings

Let A, B be nonempty subsets of a metric space (X, d) and T : A → 2B be a multivalued non-self-mapping. The purpose of this paper is to establish some theorems on the existence of a point $${x^*\in A}$$ , called best proximity point, which satisfies $${{\rm inf}\{d(x^*,y):y\in Tx^*\}=dist(A,B).}$$ This will be done for contraction multivalued non-self-mappings in metric spaces, as well as for nonexpansive multivalued non-self-mappings in Banach spaces having appropriate geometric property.