Advances in Difference Equations
Volume 2010 (2010), Article ID 381932, 24 pages
doi:10.1155/2010/381932
Research Article

Structure of Eigenvalues of Multi-Point Boundary Value Problems

Jie Gao,1 Dongmei Sun,2 and Meirong Zhang3

1School of Mathematics and Information Sciences, Weifang University, Weifang, Shandong 261061, China
2College of Applied Science and Technology, Hainan University, Haikou, Hainan 571101, China
3Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received 7 January 2010; Revised 19 March 2010; Accepted 29 March 2010

Academic Editor: Gaston M. N'Guérékata

Copyright © 2010 Jie Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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],ℝ), α=(αk)∈ℝm, 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.