Electron. J. Differential Equations, Vol. 2023 (2023), No. 65, pp. 1-23.

Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term

Jiri Sremr

We study the existence, exact multiplicity, and structure of the set of positive solutions to the periodic problem $$ u''=p(t)u+h(t)|u|^{\lambda}\operatorname{sgn} u+\mu f(t);\quad u(0)=u(\omega),\; u'(0)=u'(\omega), $$ where \(\mu\in \mathbb{R}\) is a parameter. We assume that \(p,h,f\in L([0,\omega])\), \(\lambda>1\), and the function \(h\) is non-negative. The results obtained extend the results known in the existing literature. We do not require that the Green's function of the corresponding linear problem be positive and we allow the forcing term \(f\) to change its sign.

Submitted November 22, 2022. Published October 5, 2023.
Math Subject Classifications: 34B08, 34C23, 34C25, 34B18.
Key Words: Periodic solution; second-order differential equation; existence; Duffing equation; multiplicity; bifurcation; positive solution.
DOI: 10.58997/ejde.2023.65

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Jiri Sremr
Institute of Mathematics
Faculty of Mechanical Engineering
Brno University of Technology
Technicka 2, 616 69 Brno, Czech Republic
email: sremr@fme.vutbr.cz

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