We prove comparison, uniqueness and existence results for viscosity solutions to a wide class of fully nonlinear second order partial differential equations F (x, u, du, d 2 u) = 0 defined on a finite-dimensional Riemannian manifold M. Finest results (with hypothesis that require the function F to be degenerate elliptic, that is nonincreasing in the second order derivative variable, and uniformly continuous with respect to the variable x) are obtained under the assumption that M has nonnegative sectional curvature, while, if one additionally requires F to depend on d 2 u in a uniformly continuous manner, then comparison results are established with no restrictive assumptions on curvature.

In this paper we establish several results which allow to find fixed points and zeros of set-valued mappings on Riemannian manifolds. In order to prove these results we make use of subdifferential calculus. We also give some useful applications.