Levitan Almost Periodic Solutions of Linear Differential Equations

TL;DR

This paper extends Levitan's theorem by proving that linear differential equations with Levitan almost periodic coefficients have Levitan almost periodic solutions if they possess at least one bounded solution, removing previous separation assumptions.

Contribution

It generalizes Levitan's theorem by removing the separation condition among bounded solutions for existence of almost periodic solutions.

Findings

Proves existence of Levitan almost periodic solutions without separation condition.

Extends results to difference equations.

Analyzes solutions within nonautonomous dynamical systems framework.

Abstract

The known Levitan's Theorem states that the linear differential equation $$ x'=A(t)x+f(t) \ \ \ (*) $$ with Bohr almost periodic coefficients $A(t)$ and $f(t)$ admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations $$ x'=A(t)x\ .\ \ \ (**) $$ In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. In this case, the separation from zero of bounded solutions of equation (**) is not assumed. The analogue of this result for difference equations also is given. We study the problem of existence of Bohr/Levitan almost periodic solutions for equation (*) in the framework of general nonautonomous dynamical systems (cocycles).