Best Proximity Pair Theorems for Multifunctions with Open Fibres

Let A and B be non-empty subsets of a normed linear space, and f:A->B be a single valued function. A solution to the functional equation fx=x, (x@?A) will be an element x"o in A such that fx"o=x"o (i.e., such that d(fx, x)=0). In the case of non-existence of a solution to the equation fx=x, it is natural to explore the existence of an optimal approximate solution that will fulfill the requirement to some extent. In other words, an element x"o in A should be found in such a way that d(x"o, fx"o)=Min{d(x, fx):x@?A}. Thus, the crux of finding an optimal approximate solution to the aforesaid equation fx=x boils down to ascertaining a solution to the optimization problem Min{d(x, fx):x@?A}. But, d(x, fx)>=d(A, B) for all x@?A. So, in the case of seeking an optimal approximate solution to the aforesaid equation fx=x, it should be contemplated to find an element x"o in A such that d(x"o, fx"o)=d(A, B). Indeed, given a multifunction T: A->2^B with open fibres, best proximity pair theorems, furnishing the sufficient conditions for the existence of an element x"o@?A such that d(x"o, Tx"o)=d(A, B), are proved in this paper.