Second order, multi-point problems with variable coefficients Original Research Article

In this paper, we consider the eigenvalue problem consisting of the equation View the MathML source−u″=λru,on (−1,1), Turn MathJax on where View the MathML sourcer∈C1[−1,1],r>0 and λ∈Rλ∈R, together with the multi-point boundary conditions View the MathML sourceu(±1)=∑i=1m±αi±u(ηi±), Turn MathJax on where m±⩾1m±⩾1 are integers, and, for i=1,…,m±i=1,…,m±, View the MathML sourceαi±∈R, View the MathML sourceηi±∈[−1,1], with View the MathML sourceηi+≠1, View the MathML sourceηi−≠−1. We show that if the coefficients View the MathML sourceαi±∈R are sufficiently small (depending on rr), then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients View the MathML sourceαi± 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≡1r≡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.