We consider the Sturm-Liouville boundary value problem-(p(x)u'(x))'+q(x)u(x)=f(x, u(x))+h(x), x?(0, p),c00u(0)+c01u'(0)=0, c10u(p)+c11u'(p)=0,where p?C1([0, p]), q?C0([0, p]), with p(x)>0, x?[0, p], c2i0+c2i1>0, i=0, 1, h?L2(0, p), and f:[0, p]×R?R is a Carathéodory function. We assume that the rate of growth of f(x, ?) is at most linear as ??8, but the asymptotic behaviour may be different as ??±8, so the non-linearity is termed "jumping." Conditions for existence of solutions of this problem are usually expressed in terms of "non-resonance" with respect to the standard Fucík 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 Fucík spectrum conditions. © 2001 Academic Press.