Shadowing for infinite dimensional dynamics and exponential trichotomies

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics Vol. 151; no. 3; pp. 863 - 884
• Main Authors: Backes, Lucas, Dragičević, Davor
• Format: Journal Article
• Language: English
• Published: Edinburgh, UK Royal Society of Edinburgh Scotland Foundation 01.06.2021; Cambridge University Press
• Subjects: Banach spaces; Difference equations; Dynamics; Exponential trichotomies; 37C50; Nonautonomus systems; 34D10; 34D09; Shadowing; Nonlinear perturbations; Hyers–Ulam stability
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/prm.2020.42

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.