Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 65, pp. 1-23.

Title: Parameter-dependent periodic problems for non-autonomous Duffing equations with sign-changing forcing term
       
Author: Jiri Sremr (Brno Univ. of Technology, Czech Republic)
         
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 05, 2023.
Math Subject Classifications: 34B08, 34C23, 34C25, 34B18.
Key Words: Periodic solution; second-order differential equation; existence; Duffing equation; 
  multiplicity; bifurcation; positive solution.