Positive subharmonic solutions to nonlinear ODEs with indefinite weight

TL;DR

This paper proves the existence of positive subharmonic solutions for a class of nonlinear indefinite ODEs with sign-changing weights, extending previous results and using a combination of coincidence degree theory and Poincaré-Birkhoff theorem.

Contribution

It establishes the existence of positive subharmonic solutions for indefinite equations with mean value conditions, generalizing earlier work and applying advanced topological methods.

Findings

Existence of positive subharmonic solutions for large k

Application of coincidence degree and Poincaré-Birkhoff theorems

Extension to a broader class of indefinite equations

Abstract

We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition $\int_{0}^{T} a(t)~\!dt < 0$, has positive subharmonic solutions of order $k$ for any large integer $k$, thus providing a further contribution to a problem raised by G. J. Butler in its pioneering paper (JDE, 1976). The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincar\'e-Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it).