Existence and exact multiplicity of positive periodic solutions to forced non-autonomous Duffing type differential equations

Abstract

The paper studies the existence, exact multiplicity, and a structure of the set of positive solutions to the periodic problem $$ u''=p(t)u+q(t,u)u+f(t);\quad u(0)=u(\omega),\ u'(0)=u'(\omega), $$ where $p,f\in L([0,\omega])$ and $q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R}$ is Carathéodory function. The general results obtained are applied to the forced non-autonomous Duffing equation $$ u''=p(t)u+h(t)|u|^{\lambda}\operatorname{sgn} u+f(t), $$ with $\lambda>1$ and a~non-negative $h\in L([0,\omega])$. We allow the coefficient $p$ and the forcing term $f$ to change their signs.

Keywords

• positive periodic solution
• second-order differential equation
• duffing equation
• existence
• uniqueness
• multiplicity