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Subharmonic oscillations of forced pendulum-type equations. (English) Zbl 0708.34028

The authors consider the differential equation \(\ddot x+g(x(t))=f(t)\) where f: \({\mathbb{R}}\to {\mathbb{R}}\) is a continuous periodic function with a minimal period \(T>0\) with \(\int^{T}_{0}f(t)dt=0\) and g: \({\mathbb{R}}\to {\mathbb{R}}\) is continuous with the function \(G(x)=\int^{x}_{0}g(s)ds\) being \(2\pi\)-periodic. They prove that if the T-periodic solutions are isolated and if every such solution having Morse index equal to zero is nondegenerate, then there exists an integer \(k_ 0\geq 2\) such that for every prime integer \(k\geq k_ 0\) there is a periodic solution of this equation with minimal period kT. Furthermore they show that under certain additional conditions two such subharmonic solutions exist, which are geometrically distinct.
Reviewer: L.Janos

MSC:

34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
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