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Subharmonic solutions for some second-order differential equations with singularities. (English) Zbl 0787.34035

A scalar equation of the form \(u''+g(u)=e(t)(\equiv e(t+T))\) is considered, where \(g\) is a continuous function defined on an open interval \((A,B)\subset R'\), with some singular behavior at the boundary. The authors study the following two main cases:
(a) The interval \((A,B)\) is unbounded and \(g\) is sublinear at infinity. In this case, via critical point theory, they prove the existence of a sequence of subharmonics whose amplitudes and minimal periods tend to infinity;
(b) The interval \((A,B)\) is bounded and the periodic forcing term \(e(t)\) has minimal period \(T>0\). In this case, using the generalized Poincaré- Birkhoff fixed-point theorem, they prove that for any positive integer \(m\), there are infinitely many periodic solutions having \(mT\) as minimal period.
Applications are given to the dynamics of a charged particle moving on a line over which one has placed some electric charges of the same sign.

MSC:

34C25 Periodic solutions to ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
70F99 Dynamics of a system of particles, including celestial mechanics
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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