Half eigenvalues and the Fučík spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equationequation(1)−u″=f(u)+h,a.e. on (−1,1), where h∈L1(−1,1)h∈L1(−1,1), together with the multi-point, Dirichlet-type boundary conditionsequation(2)u(±1)=∑i=1m±αi±u(ηi±), where m±⩾1m±⩾1 are integers, α±=(α1±,…,αm±)∈[0,1)m±, η±∈m±(−1,1)η±∈(−1,1)m±, and we suppose that∑i=1m±αi±<1. We also suppose that f:R→Rf:R→R is continuous, and00λ,a,b>0, and u±(x)=max{±u(x),0}u±(x)=max{±u(x),0} for x∈[−1,1]x∈[−1,1]. The problem  and  is ‘positively-homogeneous’ and jumping. Regarding a, b   as fixed, values of λ=λ(a,b)λ=λ(a,b) for which  and  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  and . The set of half-eigenvalues is closely related to the ‘Fučík spectrum’ of the problem, which we briefly describe. Equivalent solvability and non-solvability results for  and  are obtained from either the half-eigenvalue or the Fučík spectrum approach.