Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions (Q1943260)

This paper deals with the second-order equation \[ u''+f(t,u)=0, \] where \(f\) is \(T\)-periodic in \(t\). The authors prove the existence of infinitely many sub-harmonic solutions, and they also obtain precise information on the minimal periods and the number of zeros of these solutions. The main assumptions are the existence of ordered upper and lower solutions, \(\beta (t)>\alpha (t)\), together with a \(T\)-periodic solution \(u^* (t)\) lying between \(\alpha\) and \(\beta\) and satisfying the condition \[ \int_0^T \frac{\partial f}{\partial u} (t,u^* (t))\,dt >0. \] This elegant result can be applied to many known equations, in particular to the forced pendulum equation.