Resumen
We consider the Sturm-Liouville boundary value problem where pC1([0, ]), qC0([0, ]), with p (x)>0, x[0, ], c2i0+c2i1>0, i=0, 1, hL2(0, ), and f: [0, ]× is a Carathéodory function. We assume that the rate of growth of f (x, ) is at most linear as ||, but the asymptotic behaviour may be different as ±, 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 Fuík 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 Fuík spectrum conditions.