Existence and exact multiplicity of positive periodic solutions to forced non-autonomous Duffing type differential equations. (English) Zbl 07444219

Summary: 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 \[ u''=p(t)u+h(t)\vert u\vert ^{\lambda}\operatorname{sgn} u+f(t), \] 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.


34C25 Periodic solutions to ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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