Multiple solutions of two point boundary value problems with jumping nonlinearities (Q1065226)

This paper is concerned with solutions of the ordinary differential equation \[ (1)\quad y''+g(y)=s \sin x+h(x),\quad y(0)=y(\pi)=0 \] where \(\lim_{\xi \to -\infty}g(\xi)/\xi =a\), \(\lim_{\xi \to +\infty}g(\xi)/\xi =b\). The main theorem is that if \(n^ 2<a<(n+1)^ 2\), \((n+k)^ 2<b<(n+k+1)^ 2\), and the problem \(y''+by^+-ay^-=0\), \(y(0)=y(\pi)=0\) has no non-trivial solution, then the total number of solutions of (1) for s large positive and s large negative is at least \(2(n-k)+2\). The method for calculating the precise number for s positive and the precise number for s negative is implicit in the paper. Whether these numbers are exact remains an open question.