STABLE MANIFOLDS FOR NON-AUTONOMOUS EQUATIONS WITH NON-UNIFORM POLYNOMIAL DICHOTOMIES

We establish the existence of stable manifolds for semiflows defined in Banach spaces by non-autonomous ordinary differential equations v′ = A(t)v + f(t, v) assuming that the non-autonomous linear equation v′ = A(t)v admits a type of non-uniform dichotomy that we call non-uniform polynomial dichotomy. We consider two families of perturbations f and obtain for one of them global C¹ stable manifolds and for the other local Lipschitz stable manifolds. In both cases we obtain the polynomial decay of the flow on the stable manifold and in the first case we also obtain the polynomial decay of its derivative. Additionally, we study how the manifolds obtained vary with the perturbations considered and present examples of linear non-autonomous differential equation admitting a non-uniform polynomial dichotomy that is not a uniform polynomial dichotomy.