Generalized nonuniform dichotomies and local stable manifolds

We establish the existence of local stable manifolds for semiflows generated by nonlinear perturbations of nonautonomous ordinary linear differential equations in Banach spaces, assuming the existence of a general type of nonuniform dichotomy for the evolution operator that contains the nonuniform exponential and polynomial dichotomies as a very particular case. The family of dichotomies considered allow situations for which the classical Lyapunov exponents are zero. Additionally, we give new examples of application of our stable manifold theorem and study the behavior of the dynamics under perturbations.Comment: 18 pages. New version with minor corrections and an additional theorem and an additional exampl

Interpolation of compact non-linear operators

Let and be two Banach couples and let be a continuous map such that is a Lipschitz compact operator and is a Lipschitz operator. We prove that if is also compact or is continuously embedded in or is continuously embedded in , then is also a compact operator when and . We also investigate the behaviour of the measure of non-compactness under real interpolation and obtain best possible compactness results of Lions&#8211;Peetre type for non-linear operators. A two-sided compactness result for linear operators is also obtained for an arbitrary interpolation method when an approximation hypothesis on the Banach couple is imposed.</p