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), 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], c²_i0+c²_i1>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.