Further remarks on a problem of Moser and a conjecture of Mawhin. (English) Zbl 0851.34046

Andres, Jan

Topol. Methods Nonlinear Anal. 6, No. 1, 163-174 (1995).

The author deals with the pendulum equation (1) \(x''+ ax'+ b\sin x= p(t)\), where \(a\), \(b\) are real constants and \(p\) is a continuous function such that \(p(t+ T)\equiv p(t)\), \({1\over T} \int^T_0 p(t) dt= 0\). Together with (1) he considers the auxiliary equation \((1^*)\) \(x''+ ax'+ b\sin^*_\delta x= p(t)\), where \[ \sin^*_\delta x= \begin{cases} \sin(x+ \pi)\quad & \text{for}\quad |x|\leq \pi- \delta,\\ -\sin \delta\text{ sgn } x \quad & \text{for} \quad |x|\geq \pi- \delta,\end{cases} \] with \(0< \delta< \pi\). Supposing \(a> 0\), \(b> 0\), he gives sufficient conditions for \((1^*)\) to admit a \(T\)-periodic solution \(x\) such that \(|x(t)|< \pi\). As a consequence he obtains conditions under which the equation (1) has at least two geometrically distinct \(T\)-periodic solutions.
Finally, the anti-periodic case \(p(t+ T)= - p(t)\) is studied. It is shown that (1) admits a \(2T\)-periodic solution provided \[ |a|< {4\over T(T+ 2)} \] is satisfied. As a corollary, the conditions for (1) to have at least two geometrically distinct \(2T\)-periodic solutions are given.

Reviewer: J.Kalas (Brno)

MSC:

34C25	Periodic solutions to ordinary differential equations
34D05	Asymptotic properties of solutions to ordinary differential equations
37G99	Local and nonlocal bifurcation theory for dynamical systems

Keywords:

forced oscillators; problem of Moser; Mawhin’s conjecture; periodic solution; pendulum equation; anti-periodic