Shadowing for infinite dimensional dynamics and exponential trichotomies

Abstract

Let $(A_m{)}_{m\in \mathbb{Z}}$ be a sequence of bounded linear maps acting on an arbitrary Banach space $X$ and admitting an exponential trichotomy and let $f_m:X\to X$ be a Lispchitz map for every $m\in \mathbb{Z}$. We prove that whenever the Lipschitz constants of $f_m$, $m\in \mathbb{Z}$, are uniformly small, the nonautonomous dynamics given by $x_{m+1}=A_m{x}_m+f_m(x_m)$, $m\in \mathbb{Z}$, has various types of shadowing. Moreover, if $X$ is finite dimensional and each $A_m$ is invertible we prove that a converse result is also true. Furthermore, we get similar results for one-sided and continuous time dynamics. As applications of our results we study the Hyers-Ulam stability for certain difference equations and we obtain a very general version of the Grobman-Hartman's theorem for nonautonomous dynamics.

Revised version. Accepted for publication in Proceedings of the Royal Society of Edinburgh Section A: Mathematics

References in corpus (1)

• Lipschitz shadowing implies structural stability

Cited by in corpus (6)

• A general approach to nonautonomous shadowing for nonlinear dynamics
• Conditional Lipschitz shadowing for ordinary differential equations
• A generalized Grobman-Hartman theorem for nonautonomous dynamics
• Shadowing, Hyers--Ulam stability and hyperbolicity for nonautonomous linear delay differential equations
• Shadowing and hyperbolicity for linear delay difference equations
• Parameterized shadowing for nonautonomous dynamics