  We consider semidifferentiable (possibly nonsmooth) maps, acting on a subset
of a Banach space, that are nonexpansive either in the norm of the space or in
the Hilbert's or Thompson's metric inherited from a convex cone. We show that
the global uniqueness of the fixed point of the map, as well as the geometric
convergence of every orbit to this fixed point, can be inferred from the
semidifferential of the map at this point. In particular, we show that the
geometric convergence rate of the orbits to the fixed point can be bounded in
terms of Bonsall's nonlinear spectral radius of the semidifferential. We derive
similar results concerning the uniqueness of the eigenline and the geometric
convergence of the orbits to it, in the case of positively homogeneous maps
acting on the interior of a cone, or of additively homogeneous maps acting on
an AM-space with unit. This is motivated in particular by the analysis of
dynamic programming operators (Shapley operators) of zero-sum stochastic games.
