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, π),c00u(0)+c01u′(0)=0, c10u(π)+c11u′(π)=0,where p C1([0, π]), q C0([0, π]), with p(x)>0, x [0, π], c2i0+c2i1>0, i=0, 1, h L2(0, π), and f: [0, π]× → is a Carathéodory function. We assume that the rate of growth of f(x, ξ) is at most linear as ξ→∞, but the asymptotic behaviour may be different as ξ→±∞, 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 Fu ík 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 Fu ík spectrum conditions.