Resolvents of convex functions in complete geodesic metric spaces with negative curvature

Abstract

In this paper, we define a resolvent of a proper lower semicontinuous convex function in a complete geodesic space with negative curvature and we show its well-definedness and fundamental properties. We further show a fixed point theorem for hyperbolically nonspreading mappings, which can be applied to the resolvent defined in this paper. We also apply these results to convex optimization problems in complete geodesic spaces with negative curvature.