Multiplicity of periodic solutions for Duffing equations under nonuniform non-resonance conditions

Wang, Chengwen

doi:10.1090/S0002-9939-98-04520-1

Multiplicity of periodic solutions for Duffing equations under nonuniform non-resonance conditions
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Proc. Amer. Math. Soc. 126 (1998), 1725-1732 Request permission

Abstract:

This paper is devoted to the study of multiple $2 \pi$-periodic solutions for Duffing equations \begin{equation*} x'' +cx’ + g(t,x) = s(1+h(t)) \end{equation*} under the condition of nonuniform non-resonance related to the positive asymptotic behavior of $g(t,x)x ^{-1}$ at the first two eigenvalues $0$ and $1$ of the periodic BVP on $[0,2 \pi ]$ for the linear operator $L = - x''$, and the condition on the negative asymptotic behavior of $g(t,x)$ at infinity. The techniques we use are degree theory and the upper and lower solution method.

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