Asymptotic Integration of a Linear Fourth Order Differential Equation of Poincar\'e Type

TL;DR

This paper investigates the asymptotic behavior of a perturbed fourth order linear differential equation of Poincaré type, transforming it into a Riccati equation and establishing solution existence and asymptotics under specific hypotheses.

Contribution

It introduces a novel approach using variable change and fixed point theorem to analyze the asymptotics of fourth order equations with perturbations, extending existing methods.

Findings

Existence of a unique solution to the Riccati equation.

Asymptotic formulas for solutions of the fourth order differential equation.

Conditions ensuring the fundamental system of solutions exists.

Abstract

This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable satisfies a third order differential equation of Riccati type. We assume three hypothesis. The first is the following: all roots of the characteristic polynomial associated to the fourth order linear equation has distinct real part. The other two hypothesis are related with the behavior of the perturbation functions. Under this general hypothesis we obtain four main results. The first two results are related with the application of fixed point theorem to prove that the Riccati equation has a unique solution. The next result concerns with the asymptotic behavior of the solutions of the Riccati equation. The fourth main theorem is introduced to establish the existence of a fundamental system of solutions and to precise formulas for the asymptotic behavior of the linear fourth order differential equation.