Stable manifolds for nonautonomous equations without exponential dichotomy (Q818430)

The authors consider linear equations \[ v' = A(t)v \] in Banach spaces with nonuniform exponential dichotomy and show the existence of stable invariant manifolds for sufficiently small perturbations \[ v' = A(t)v + f(t,v) \] corresponding to the negative Lyapunov exponents of the linear equation. The novelty of the authors' approach to the old existence problem of invariant manifolds is the quite general assumption on the linearization admitting only a nonuniform exponential dichomoty, a typical property from a measure-theoretical point of view. The approach is based on Lyapunov regularity. For extensions and related results see also [the authors, Commun. Math. Phys. 259, No.3, 639-677 (2005; Zbl 1098.34041)].