Two-sided Remotely Almost Periodic Solutions of Ordinary Differential Equations in Banach Spaces

TL;DR

This paper investigates the existence of two-sided remotely almost periodic solutions for certain differential equations in Banach spaces, focusing on equations with almost periodic coefficients and small nonlinearities under exponential trichotomy conditions.

Contribution

It establishes conditions for the existence of two-sided remotely almost periodic solutions in Banach spaces for differential equations with almost periodic coefficients.

Findings

Existence of solutions under exponential trichotomy.

Conditions for small nonlinearities to ensure solutions.

Extension of almost periodic solution theory to Banach spaces.

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".