Summary: Nonlinear Analysis 67 (2007) 154174
Applications of proximal calculus to fixed point theory
on Riemannian manifolds
Daniel Azagra, Juan Ferrera
Departamento de An´alisis Matem´atico, Facultad de Matem´aticas, Universidad Complutense, 28040 Madrid, Spain
Received 15 December 2005; accepted 28 April 2006
We prove a general form of a fixed point theorem for mappings from a Riemannian manifold into itself
which are obtained as perturbations of a given mapping by means of general operations which in particular
include the cases of sum (when a Lie group structure is given on the manifold) and composition. In order
to prove our main result we develop a theory of proximal calculus in the setting of Riemannian manifolds.
c 2006 Elsevier Ltd. All rights reserved.
MSC: 49J52; 58E30; 58C30; 47H10
Keywords: Proximal subdifferential; Riemannian manifolds; Fixed point theory
1. Introduction and tools
The proximal subdifferential of lower semicontinuous real-valued functions is a very powerful
tool that has been extensively studied and used in problems of optimization, control theory,
differential inclusions, Lyapunov Theory, stabilization, and HamiltonJacobi equations; see 
and the references therein.