Oscillatory solutions of some perturbed second order differential equations

TL;DR

This paper investigates oscillatory solutions that decay to zero in certain perturbed second order differential equations, highlighting a new type of behavior where small perturbations cause oscillations independent of the original equation's coefficients.

Contribution

It introduces a novel class of perturbed equations exhibiting oscillatory decay, with perturbations that are arbitrarily small and independent of the unperturbed coefficients.

Findings

Oscillatory solutions decay to zero as s approaches infinity.

Small perturbations can induce oscillations regardless of original coefficients.

Reveals a new pathology in the theory of perturbed oscillations.

Abstract

We discuss the occurrence of oscillatory solutions which decay to 0 as $s\to+\infty$ for a class of perturbed second order ordinary differential equations. As opposed to other results in the recent literature, the perturbation is as small as desired in terms of its improper integrals and it is independent of the coefficients of the non-oscillatory unperturbed equation. This class of equations reveals thus a new pathology in the theory of perturbed oscillations.