Parameter Dependence of Stable Invariant Manifolds for Delay Differential Equations under -Dichotomies

We obtain the existence of stable invariant manifolds for the nonlinear equation provided that the linear delay equation admits a nonuniform -dichotomy and is a sufficiently small Lipschitz perturbation. We show that the stable invariant manifolds are dependent on parameter . Namely, the stable invariant manifolds are Lipschitz in the parameter . In addition, we also show that nonuniform -contraction persists under sufficiently small nonlinear perturbations. 1. Introduction The theory of exponential dichotomy is attracting much attention in recent years. This is mostly because the concept of exponential dichotomy plays an important role in the obtention of invariant manifolds of differential equations. The notion of exponential dichotomy can be traced back to Perron in [1]. Since then, Sacker and Sell [2–5] investigated sufficient conditions for the existence of an exponential dichotomy, also in the infinite-dimensional setting. In [6], Barreira and Valls discussed the much weaker notion of nonuniform exponential dichotomy. This theory of nonuniform hyperbolicity was introduced in the seventies by Barreira and Pesin [7]. For background material on exponential dichotomy for differential equations, the books [8–10] may be consulted. We note that the study of robustness in the case of exponential behavior has a long history. For the early work we can refer to Perron [1] and Massera and Sch？ffer [11]. For more recent works we refer to [12–15]. In particular, in [12], Barreira and Valls studied the existence of stable invariant manifold for any sufficiently small nonlinear perturbation of the linear equation. In [13], the authors discussed the parameter dependence of stable manifolds under nonuniform exponential dichotomy. Recently, general stable and unstable behaviors with growth rates given by two increasing functions are exhibited by Bento and Silva [16]. The results in this work rely on the notion of nonuniform -dichotomy, which includes the traditional exponential dichotomy and the polynomial dichotomy [17–19] and the -nonuniform exponential dichotomy [20–22]. Since the nonuniform -dichotomy is more general than nonuniform dichotomy before, so it is an emerging field drawing attention from both theoretical and applied disciplines. For example, in [23], the authors showed the robustness of nonuniform -dichotomy provided that and were two differentiable functions. Authors of [24] showed the existence of invariant manifolds for sufficiently small Lipschitz perturbations of a linear equation with nonuniform -dichotomy. In the implementation of differential