Proximinal Retracts and Best Proximity Pair Theorems

Abstract

This note is concerned with proximinality and best proximity pair theorems in hyperconvex metric spaces and in Hilbert spaces. Given two subsets A and B of a metric space and a mapping best proximity pair theorems provide sufficient conditions that ensure the existence of an such that

Thus such theorems provide optimal approximate solutions in the case that the mapping T does not have fixed points.

Keywords:

• Best proximity pairs
• Hyperconvex spaces
• Nonexpansive mappings
• Proximinal sets
• AMS Subject Classification: 47H09, 41A65, 46B20

Acknowledgment

The work of the second author was partially supported by the Fund for the Promotion of Research at the Technion and by the Technion VPR Fund.

Notes

^aA Banach space is hyperconvex if and only if it is isometrically isomorphic to a space of continuous functions on a stonian space K (see, e.g., Schaefer (1974))