Structure of Eigenvalues of Multi-Point Boundary Value Problems

The structure of eigenvalues of −y q x y λy, y 0 0, and y 1 ∑m k 1 αky ηk , will be studied, where q ∈ L1 0, 1 ,R , α αk ∈ R, and 0 < η1 < · · · < ηm < 1. Due to the nonsymmetry of the problem, this equation may admit complex eigenvalues. In this paper, a complete structure of all complex eigenvalues of this equation will be obtained. In particular, it is proved that this equation has always a sequence of real eigenvalues tending to ∞. Moreover, there exists some constant Aq > 0 depending on q, such that when α satisfies ‖α‖ ≤ Aq, all eigenvalues of this equation are necessarily real.