Invariant Manifolds for Abstract Functional Differential Equations and Related Volterra Difference Equations in a Banach Space

Abstract

For abstract functional differential equations (FDE) and Volterra difference equations (VDE) in a Banach space, the local existence and smoothness of invariant manifolds, such as stable/unstable manifolds, center-stable/center-unstable manifolds and center manifolds, are established by means of the variation of constants formula in the phase space in [18] and [12]. Also, it is shown that in a neighborhood of the zero solution the behavior of solutions of FDE (resp. VDE) is described, in some sense, by a certain ordinary differential equation (resp. first order difference equation) in a finite dimensional space. As a corollaly, the principle of linearized stability is derived.