We construct stable invariant manifolds for semiflows generated by the nonlinear impulsive differential equation with parameters x＇= A（t）x ＋ f（t, x, λ）, t≠τi and x（τ＋i） = Bix（τi） ＋ gi（x（τi）, λ）, i ∈ N in Banach spaces, assuming that the linear impulsive differential equation x＇= A（t）x, t≠τi and x（τ＋i） = Bix（τi）, i ∈ N admits a nonuniform （μ, ν）-dichotomy. It is shown that the stable invariant manifolds are Lipschitz continuous in the parameter λ and the initial values provided that the nonlinear perturbations f, g are sufficiently small Lipschitz perturbations.
In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.
