Existence and uniqueness of Remotely Almost Periodic solutions of differential equations and applications

Abstract

We establish existence and uniqueness of remotely almost periodic (RAP) solutions for nonlinear ordinary differential systems $x' = A(t)x + f(t,x) + g_{\nu}(t,x).$ Assuming that the linear equation $x' = A(t)x$ admits an exponential dichotomy and that the associated Green kernel is exponentially bi-remotely almost periodic, we derive sufficient conditions guaranteeing a unique RAP solution of the perturbed system for $\nu$ in a suitable range. As an application, we obtain RAP solutions for a nonautonomous Brusselator model.