Second-order, three-point, boundary value problems with jumping non-linearities

We consider the non-linear, three-point boundary value problem consisting of the equation (1)- u^? = f (u) + h, a.e. on (0, 1), where h ? L¹ (0, 1), together with the boundary conditions (2)u (0) = 0, u (1) = a u (?), where ?, a ? (0, 1). The function f : R ? R is continuous, and we assume that the following limits exist and are finite: f_{± 8} {colon equals} under(lim, s ? ± 8) frac(f (s), s) . We allow f₈ ? f_{- 8}-such a non-linearity f is said to be jumping. Related to (1) is the equation (3)- u^? = a u⁺ - b u^- + ? u, on (0, 1), where a, b, ? ? R, and u^± (x) = max {± u (x), 0} for x ? [0, 1]. The problem (2) and (3) is 'positively homogeneous' and jumping. Values of ? for which (2) and (3) has a non-trivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1) and (2). The set of half-eigenvalues is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability and non-solvability results for (1) and (2) are obtained from either the half-eigenvalue or the Fucík spectrum approach. © 2007 Elsevier Ltd. All rights reserved.