Spherical nonspreadingness of resolvents of convex functions in geodesic spaces

Abstract

In this paper, we propose the concepts of a spherically nonspreading mapping and a firmly spherically nonspreading mapping in a complete geodesic space with curvature bounded above by one and we prove that the resolvent of a proper lower semicontinuous convex function in that space is both well defined and firmly spherically nonspreading. We further discuss the existence and approximation of fixed points of such mappings and apply our results to convex optimization in geodesic spaces.