Non-resonance conditions for semilinear Sturm-Liouville problems with jumping non-linearities

We consider the Sturm Liouville boundary value problem - (p(x)u'(x))' + q(x)u(x)=f(x, u(x))+h(x), N is an element of (0,pi), cc(00)u(0)+c(01)u'(0)+0, c(10)u(pi) + c(11)u'(pi )0, where p is an element of C-1([0, pi]), q is an element of C-0([0, pi]), wit h p(x) > 0, N is an element of [0, pi], c(10)(2)+c(11)(2) > 0, i = 0.1. h i s an element of L-2(0, pi), and f: [0, pi] X R --> R is a Caratheodory func tion. We assume that the rate of growth of f(x, zeta) is at most linear as \zeta\ --> alpha, but the asymptotic behaviour may be different as zeta --> +/- alpha, so the non-linearity is termed "jumping". Conditions for existe nce of solutions of this problem are usually expressed in terms of "non-res onance" with respect to the standard Fucik spectrum. In this paper we give conditions for both existence and non-existence of solutions in terms of a slightly different idea of the spectrum. These conditions extend the usual Fucik spectrum conditions. (C) 2001 Academic Press.