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Bifurcation of positive periodic solutions to non-autonomous undamped Duffing equations. (English) Zbl 1464.34041

Summary: We study a bifurcation of positive solutions to the parameter-dependent periodic problem \[ u'' = p(t) u- h(t) |u|^\lambda \mathrm{sgn}\, u+ \mu f(t); \quad u(0) = u(\omega),\, u'(0) =u'(\omega), \] where \(\lambda >1\), \(p\), \(h\), \(f \in L([0,\omega])\), \(\mu \in \mathbb{R}\) is a parameter. Both the coefficient \(p\) and the forcing term \(f\) my change their signs, \(h\geq 0\) a.e. on \([0,\omega]\). We provide sharp conditions on the existence and multiplicity as well as non-existence of positive solutions to the given problem depending on the choice of the parameter \(\mu\).

MSC:

34B08 Parameter dependent boundary value problems for ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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