  Assuming the existence of a general nonuniform dichotomy for the evolution
operator of a non-autonomous ordinary linear differential equation in a Banach
space, we establish the existence of invariant stable manifolds for the
semiflow generated by sufficiently small nonlinear perturbations of the linear
equation. The family of dichotomies considered satisfies a general growth rate
given by some increasing differentiable function, allows situations for which
the classical Lyapunov exponents are zero, and contains the nonuniform
exponential dichotomies as a very particular case. In addition we also give
explicit examples of linear equations that admit all the possible considered
dichotomies.
