Proximal Point Algorithms for a Hybrid Pair of Nonexpansive Single-Valued and Multi-Valued Mappings in Geodesic Metric Spaces

Abstract

In this paper, we propose a new proximal point algorithm for finding a common element of the set of fixed points of nonexpansive single-valued mappings, the set of fixed points of nonexpansive multi-valued mappings, and the set of minimizers of convex and lower semi-continuous functions. We obtain \(\Delta \)-convergence and strong convergence of the proposed algorithm to a common element of the three sets in CAT(0) spaces. Furthermore, we apply our convergence results to obtain in a special space of CAT(0) spaces, so-called \(\mathbb {R}\)-tree, under the gate condition. A numerical example to support our main results is also given.