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We study asymptotically compact nonautonomous dynamical systems given byabstract cocycles in Banach spaces. Our main assumptions are given by asqueezing property in a quadratic cone field (given by a family of indefinitequadratic Lyapunov-like functionals) and asymptotic compactness. Under suchconditions it is possible to reconstruct foliations as in the theory ofnormally hyperbolic manifolds. Our approach allows to unify many "practical"theories of local and nonlocal invariant manifolds, such asstable/unstable/center manifolds and inertial manifolds, and, moreover, itoften leads to optimal conditions for their existence.  We give applications for semilinear parabolic equations and neutral delayequations, where the Frequency Theorem is used to construct constant conefields.

Inertial manifolds and foliations for asymptotically compact cocycles in Banach spaces