Best proximity pair and coincidence point theorems for nonexpansive set-valued maps in Hilbert spaces

Amini-Harandi, A.

Best proximity pair and coincidence point theorems for nonexpansive set-valued maps in Hilbert spaces

Document Type : Research Paper

Author

A. Amini-Harandi

Abstract

This paper is concerned with the best proximity pair problem in
Hilbert spaces. Given two subsets $A$ and $B$ of a Hilbert space
$H$ and the set-valued maps $F:A o 2^ B$ and $G:A_0 o 2^{A_0}$,
where $A_0={xin A: |x-y|=d(A,B)~~~mbox{for some}~~~ yin
B}$, best proximity pair theorems provide sufficient conditions
that ensure the existence of an $x_0in A$ such that
$$d(G(x_0),F(x_0))=d(A,B).$$

Keywords

Best proximity pair

coincidence point

nonexpansive map

Hilbert space