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arXiv · 2211.06070

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

Guglielmo Feltrin·Juan Carlos Sampedro·Fabio Zanolin
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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).

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Guglielmo Feltrin, Juan Carlos Sampedro, Fabio Zanolin. 2022-11-11. Periodic solutions to superlinear indefinite planar systems: a topological degree approach. https://arxiv.org/abs/2211.06070

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