Best proximity points in partially ordered metric spaces

Abstract

The existence of best proximity point is an important aspect of optimization theory. We define the concept of proximally monotone Lipschitzian mappings on a partially ordered metric space. Then we obtain sufficient conditions for the existence and uniqueness of best proximity points for these mappings in partially ordered CAT(0) spaces. This work is a continuation of the work of Ran and Reurings [Proc. Amer. Math. Soc. 132 (2004), 1435–1443] and Nieto and Rodr´ ıguez-López [Order, 22 (2005), 223–239] for the new class of mappings introduced herein.