A Convergence Theorem for a Splitting Method and Its Applications in Geodesic Metric Spaces With Negative Curvature

In this paper, we study a splitting proximal method for minimizing the sum of convex functions defined on metric spaces with negative curvature. Our approach utilizes the resolvent operator and is tailored to the geometry of such spaces. We establish convergence rate theorems for the proposed splitting method by imposing additional conditions on the objective function. Finally, we apply our results to convex optimization problems arising in convex feasibility problems, the centroid problem, and, in particular, the computation of Karcher means.