Some existence results for singular boundary value problems (Q1261635)

The topological transversality theorem is used to establish existence of positive solutions to the differential equation \(y''+f(t,y)=0\) subject either to the boundary conditions \(y(0)=y'(1)=0\) or to \(y(0)=y(1)=0\), \(f\) is allowed to be singular at \(y=0\), \(t=0\) or \(t=1\). Besides the authors use a priori bounds. Their particular treatments are based on exploiting the observation that if \(f(t,y)\) is a decreasing function of \(y\), then close upper bounds on \(y''\), and hence on \(y'\), will follow from close lower bounds on \(y\). Consequently, the results typically allow somewhat stronger singularities than do many studies of such singular problems. In particular, they do not impose the overly restrictive condition that \(f\) be integrable in \(t\) for fixed \(y>0\).