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)=∑k=1mαky(ηk), will be studied, where q∈L1([0,1], ), α=(αk)∈ m, 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.