Positive solutions and conjugate points for multipoint boundary value problems (Q1183230)

Let \(Ly=y^{(n)}+a_ 1(x)y^{(n-1)}+\cdots+a_ ny\) be disconjugate on \([\alpha,\infty)\) and \(T_ by=0\) be homogeneous \(k\)-point conjugate \(BC\)'s. Under sign conditions on \(p\in C[\alpha,\infty]\), the authors first characterize the existence of a conjugate point of \(Ly=py\), \(T_ by=0\) by the existence of a nontrivial solution that lies in a cone. This is done by proving first certain sign properties of Green's functions, needed to use some eigenvalue results for positive maps. The result is then applied to establish the existence of solutions for a class of nonlinear problems \(Ly=f(x,y)\), \(T_ b y=0\). The results extend some earlier ones by Nehari, Schmitt-Smith, Hankerson-Henderson, etc.