## 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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