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Shadowing for infinite dimensional dynamics and exponential trichotomies
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 3 / June 2021
- Published online by Cambridge University Press:
- 24 June 2020, pp. 863-884
- Print publication:
- June 2021
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- Article
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Let
$(A_m)_{m \in {\mathop Z}}$ 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 {\mathop Z} $. We prove that whenever the Lipschitz constants of 
$f_m$, 
$m \in {\mathop Z} $, are uniformly small, the nonautonomous dynamics given by 
$x_{m+1}=A_mx_m+f_m(x_m)$, 
$m\in {\mathop 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.
