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On the periodic problem for the equation \(x''(t)+g(x(t))=f(t)\). (English) Zbl 0579.34031
An existence result for periodic solutions of second order equations of the type \(x''+g(x)=f\) is proved under the following assumptions: \(f=f(t)\) is a \(2\pi\)-periodic function, \(g=g(x)\) is a monotone continuous function such that \(| g(x)| \leq \gamma | x|\) \(+C,x\in {\mathbb{R}}\), \(C=const>0\), \(0\leq \gamma <1\) and \(g(-\infty)<(2\pi)^{- 1}\int^{2\pi}_{0}f(x)dx<g(+\infty).\) The proof is based on a technique introduced by Brezis and Nirenberg.

MSC:
34C25 Periodic solutions to ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
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