Best Proximity Pair Theorems for Multifunctions with Open Fibres

Abstract

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 xo in A such that fxo=xo (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 xo in A should be found in such a way that d(xo, fxo)=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 xo in A such that d(xo, fxo)=d(A, B). Indeed, given a multifunction T: A→2B with open fibres, best proximity pair theorems, furnishing the sufficient conditions for the existence of an element xo∈A such that d(xo, Txo)=d(A, B), are proved in this paper. © 2000 Academic Press.