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

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
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