On "Nonlinear eigenvalue problems"

TL;DR

This paper offers an alternative derivation of the asymptotic behavior of solutions to a nonlinear differential equation, confirming previous results using a new method based on oscillatory behavior analysis.

Contribution

It introduces a novel approach analyzing solutions in different regions to confirm asymptotic behavior of a nonlinear eigenvalue problem.

Findings

Confirmed previous asymptotic results for large initial values

Developed a new method analyzing oscillatory behavior in different regions

Validated the relation between initial conditions and solution maxima

Abstract

The asymptotic behaviour of solutions to $y'(x)=\cos[\pi x y(x)]$ was investigated by Bender, Fring and Komijani \cite{BenderEtAl:2014}. They found, for example, a relation between the initial value $y(0)=a$ and the number of maxima that the solution exhibited. We present an alternative derivation of the asymptotic results that looks at the solutions in the regions $x<y$ and $x>y$, and confirms the behaviour found previously for larger values of $a$. This method uses the small amplitude and high frequency of the oscillatory behaviour in the region $x<y$.