Proximal calculus on Riemannian manifolds, with applications to fixed point theory

2004/03/26 by Daniel Azagra, Juan Ferrera, Azagra, Daniel +1
Computer Science · Mathematics · #47H10 #49J52 #58C30 #58E30 #Differential Geometry (math.DG) #FOS: Mathematics #Fixed Point Theorems Analysis #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Optimization and Variational Analysis #math.DG #math.OC #msc:47H10 #msc:49J52 #msc:58C30 #msc:58E30

paper · pdf · doi:10.48550/arxiv.math/0403465

27 pages

arxiv created 2004/03/26 · openalex publication_date 2004/03/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold M. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset C of M; 2) solvability and implicit function theorems for nonsmooth functions on M; 3) conditions on the existence of a circumcenter for three different points of M; and especially 4) fixed point theorems for expansive and nonexpansive mappings and certain perturbations of such mappings defined on M.

Citations

• Nonsmooth analysis and Hamilton-Jacobi equations on Riemannian manifolds

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