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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.

(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.

Reviewer: Ding Tongren (Beijing)

### 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 |