Periodic solutions to superlinear indefinite planar systems: a topological degree approach

TL;DR

This paper proves the existence of positive periodic solutions in a superlinear indefinite planar system using topological degree theory, generalizing previous results on Butler's problem.

Contribution

It introduces a new approach based on coincidence degree theory to establish positive periodic solutions for a class of superlinear indefinite systems, extending prior work.

Findings

Existence of positive T-periodic solutions depending on parameter λ

Generalization of Butler's problem to broader differential systems

Unified framework for previous results on positive solutions

Abstract

We deal with a planar differential system of the form \begin{equation*} \begin{cases} \, u' = h(t,v), \\ \, v' = - \lambda a(t) g(u), \end{cases} \end{equation*} where $h$ is $T$-periodic in the first variable and strictly increasing in the second variable, $\lambda>0$, $a$ is a sign-changing $T$-periodic weight function and $g$ is superlinear. Based on the coincidence degree theory, in dependence of $\lambda$, we prove the existence of $T$-periodic solutions $(u,v)$ such that $u(t)>0$ for all $t\in\mathbb{R}$. Our results generalize and unify previous contributions about Butler's problem on positive periodic solutions for second-order differential equations (involving linear or $\phi$-Laplacian-type differential operators).