Second order, multi-point problems with variable coefficients

In this paper, we consider the eigenvalue problem consisting of the equation − u ″ = λ r u , on ( − 1 , 1 ) , where r ∈ C 1 [ − 1 , 1 ] , r > 0 and λ ∈ R , together with the multi-point boundary conditions u ( ± 1 ) = ∑ i = 1 m ± α i ± u ( η i ± ) , where m ± ⩾ 1 are integers, and, for i = 1 , … , m ± , α i ± ∈ R , η i ± ∈ [ − 1 , 1 ] , with η i + ≠ 1 , η i − ≠ − 1 . We show that if the coefficients α i ± ∈ R are sufficiently small (depending on r ), then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients α i ± are not sufficiently small, then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case ( r ≡ 1 ), but the variable coefficient case has not been considered previously (apart from the existence of ‘principal’ eigenvalues). Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for the existence of general solutions and also of nodal solutions—these results rely on the spectral properties of the linear problem.