Poincar\'e-Perron problem for high order differential equations in the class of almost periodic type functions

TL;DR

This paper extends classical approximation results for high order linear differential equations to the class of almost periodic functions, providing explicit solutions, conditions for fundamental systems, and demonstrating the existence of almost periodic solutions.

Contribution

It introduces explicit solution formulas and sufficient conditions for solutions of high order differential equations in almost periodic function classes, extending previous second order results.

Findings

Derived explicit solution formulas for high order equations.

Established conditions for the existence of fundamental solution systems.

Proved the existence of almost periodic and asymptotically almost periodic solutions.

Abstract

We address the Poincar\'e-Perron's classical problem of approximation for high order linear differential equations in the class of almost periodic type functions, extending the results for a second order linear differential equation in [23]. We obtain explicit formulae for solutions of these equations, for any fixed order $n\ge 3$, by studying a Riccati type equation associated with the logarithmic derivative of a solution. Moreover, we provide sufficient conditions to ensure the existence of a fundamental system of solutions. The fixed point Banach argument allows us to find almost periodic and asymptotically almost periodic solutions to this Riccati type equation. A decomposition property of the perturbations induces a decomposition on the Riccati type equation and its solutions. In particular, by using this decomposition we obtain asymptotically almost periodic and also $p$-almost periodic solutions to the Riccati type equation. We illustrate our results for a third order linear differential equation.