Periodic solutions for second order differential equations with discontinuous restoring forces. (Q1419717)

The authors study the boundary value problem \[ u''+g(u)=h(t),\qquad u(0)=u(T),\;u'(0)=u'(T),\tag{1} \] where \(h\) is continuous in \([0,T]\) and \(g\) is continuous in \({\mathbb R}\setminus\{0\}\) with a jump discontinuity at \(u=0\). They use an approximation procedure in which \(g\) is replaced by a sequence of continuous functions that fill the gap between \(g(0-)\) and \(g(0+)\) and the method of the lower and upper solutions. If moreover \(g(x)>0\) for all \(x\), \(g(-\infty)=g(+\infty)=0\), the limits \(\lim_{x\to\pm0}g(x)>0\) exist and are finite and the mean value of \(h\) is positive and sufficiently small, the authors prove the existence of at least two generalized solutions of \((1)\). If the restoring term is unbounded and becomes larger at \(+\infty\) than at \(-\infty\), the authors investigate the existence of a generalized solution of (1).