Some invariant manifolds for functional difference equations with infinite delay

For nonlinear autonomous functional difference equations, the existence of the local stable manifolds, together with the local unstable manifolds and the local center-unstable manifolds, is shown. As an application, the principle of linearization which states that stabilities and instabilities for the zero solution of nonlinear equations can be derived from those of the zero solution of the linearized equations under some circumstances is established for functional difference equations.