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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.

MSC:

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