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

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. For more information see https://ejde.math.txstate.edu/Volumes/2023/65/abstr.html