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 equation(1)−u″=f(u)+h,a.e. on (0,1), where h∈L1(0,1)h∈L1(0,1), together with the boundary conditions equation(2)u(0)=0,u(1)=αu(η), where η,α∈(0,1)η,α∈(0,1). The function f:R→Rf:R→R is continuous, and we assume that the following limits exist and are finite: f±∞≔lims→±∞f(s)s. We allow f∞≠f−∞f∞≠f−∞—such a non-linearity ff is said to be jumping.Related to (1) is the equation equation(3)−u″=au+−bu−+λu,on (0,1), where a,b,λ∈Ra,b,λ∈R, and u±(x)=max{±u(x),0}u±(x)=max{±u(x),0} for x∈[0,1]x∈[0,1]. The problem (2) and (3) is ‘positively homogeneous’ and jumping. Values of λλ for which (2) and (3) has a non-trivial solution uu will be called half-eigenvalues  , while the corresponding solutions uu 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 ‘Fučík spectrum’ of the problem, and equivalent solvability and non-solvability results for (1) and (2) are obtained from either the half-eigenvalue or the Fučík spectrum approach.