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