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Periodic solutions of the equation \(\ddot x+g(x)=E\,\cos \,t+\sigma h(t)\dot x\). (English) Zbl 0647.34034

In an early work W. S. Loud [Proc. US-Japan Sem. Differential Functional Equations Minneapolis, Minnesota 1967, 199-224 (1967; Zbl 0162.123)] analyzed the following equation (1) \(\ddot x+g(x)=E \cos t.\) In the present paper the authors prove existence of \(2\pi\)-periodic non- odd-harmonic solutions when (1) is perturbed by a small damping term, \(\sigma h(t)\dot x\), where \(\sigma\) is a real number h is a \(2\pi\)- periodic continuous function satisfying some generic conditions. The autonomous perturbation, \(h(t)=1\), does not satisfy these conditions, and is treated separately. The analysis relies essentially on the Lyapunov- Schmidt reduction.
Reviewer: A.Boucherif

MSC:

34C25 Periodic solutions to ordinary differential equations

Citations:

Zbl 0162.123
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