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A note on nonlinear oscillations at resonance. (English) Zbl 0648.34040

Existence of \(2\pi\)-periodic solutions of \(\ddot x+g(x)=p(t)\) is proved under various conditions. Those conditions relate the class of functions to which p is supposed to belong to the behavior (as \(| x| \to +\infty)\) of functions \(\alpha^{\pm}: {\mathbb{R}}\to {\mathbb{R}}\) satisfying a nonresonance condition of the form \(m^ 2+\alpha \quad -(x)\leq g(x)/x\leq (m+1)^ 2-\alpha \quad +(x)\) for larg \(| x|\) and some integer \(m\geq 1\). The relation of the theorem of this paper to other results in the field is discussed at great length.
Reviewer: B.Aulbach

MSC:

34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
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