Shadowing for infinite dimensional dynamics and exponential trichotomies. Issue 3 (June 2021)

• Record Type: Journal Article
• Authors: Backes, Lucas; Dragičević, Davor
• Is Part Of: Proceedings. Volume 151:Issue 3(2021)
• Page Start: 863
• Page End: 884
• Publication Date: 2021-06
• Subjects: Shadowing, -- Nonautonomus systems, -- Exponential trichotomies, -- Nonlinear perturbations, -- Hyers–Ulam stability; 37C50, -- 34D09, -- 34D10; Mathematics -- Periodicals; 510
• DOI: 10.1017/prm.2020.42
• Languages: English
• ISSNs: 0308-2105

Abstract

Abstract : 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.