Two-sided Remotely Almost Periodic Solutions of Ordinary Differential Equations in Banach Spaces
David Cheban
TL;DR
The paper addresses the existence and structure of two-sided remotely almost periodic solutions for nonautonomous ODEs in Banach spaces with a linear part that satisfies an exponential trichotomy. It develops a framework based on hyperbolic linear cocycles, Green's functions, and fixed-point arguments to obtain two-sided RAP solutions for linear and semilinear equations with small nonlinear perturbations, providing explicit integral representations and norm bounds. The results show that, under exponential trichotomy and Lagrange stability, RAP solutions exist and persist under Lipschitz perturbations, with unique two-sided RAP solutions in favorable cases and continuous dependence on perturbations. By extending RAP theory to two-sided time domains and employing a contraction approach, the work yields robust constructions and insights for two-sided remotely almost periodic dynamics in Banach spaces, with concrete representations via Green's functions.
Abstract
The aim of this paper is studying the two-sided remotely almost periodic solutions of ordinary differential equations in Banach spaces of the form $x'=A(t)x+f(t)+F(t,x)$ with two-sided remotely almost periodic coefficients if the linear equation $x'=A(t)x$ satisfies the condition of exponential trichotomy and nonlinearity $F$ is "small".