Exponential trichotomies and continuity of invariant manifolds

Abstract

In this work, we consider the invariant manifolds for the family of equationsẋ = Ax + f (ε, x), where A the is generator of a strongly continuous semigroup of linear operators in a Banach space X and f (ε, •) : X → X is continuous. The existence of stable (unstable) and center-stable (center-unstable) manifolds for a large class of these equations has been proved in [2]. We prove here that, if A admits a exponential trichotomy and f satisfies some suitable regularity hypotheses, then those manifolds are continuous with respect to the parameter ε.

Normal Hyperbolicity and Continuity of Global Attractors for a Nonlocal Evolution Equations

“…In the following, using results of [19] we show that the local unstable sets are actually Lipschitz manifolds in a sufficiently small neighborhood and vary continuously with . More precisely, we have the following.…”

“…Therefore, the conditions of Theorems 2.5 and 3.3 from [19] are satisfied and we obtain the existence of locally invariant sets for (71) near the origin, given as graphics of…”

“…Observe that now the "linear part" of (71) does not depend on the parameter , as required by Theorems 2.5 and 3.3 from [19]. Note that…”

“…Moreover, in [8] it is assumed that the zero is a simple eigenvalue of the Frechét derivative of (8) which implies in normal hyperbolicity of curves of equilibria, and in this paper this property is also proven (see Propositions 12 and 14). To prove our results, we use some results given in [18] on the permanence of normally hyperbolic invariant manifolds and one result given in [19] concerning the continuity properties of the local unstable manifolds of the (nonnecessarily isolated) equilibria with respect to the parameter , together with some results of [20] regarding the limiting behavior of the trajectories. This paper is organized as follows.…”

Lower Semicontinuity of Global Attractors for a Class of Evolution Equations of Neural Fields Type in a Bounded Domain

“…To overcome this difficulty we have to replace the hypothesis of hyperbolicity by normal hyperbolicity of curves of equilibria. We then used results of [4] on the permanence of normally hyperbolic invariant manifolds and use one result of [30] of continuity properties of the local unstable manifolds of the curves of equilibria. Finally, in "A Concrete Example" section, we illustrate our results with a concrete example, which satisfies all hypotheses (H1)-(H4).…”