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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
##### Keywords:
second order equations; monotone continuous function