Journal of Inequalities and Applications 
Volume 3 (1999), Issue 1, Pages 25-43
doi:10.1155/S1025583499000028

Inequalities among eigenvalues of Sturm–Liouville problems

M. S. P. Eastham,1 Q. Kong,2 H. Wu,2 and A. Zettl2

 1Computer Science Department, Cardiff University of Wales, Cardiff CF2 3XF, UK
2Department of Mathematics, Northern Illinois University, DeKalb 60115, IL, USA
Received 15 June 1997; Revised 9 October 1997

Abstract

There are well-known inequalities among the eigenvalues of Sturm–Liouville problems with periodic, semi-periodic, Dirichlet and Neumann boundary conditions. In this paper, for an arbitrary coupled self-adjoint boundary condition, we identify two separated boundary conditions corresponding to the Dirichlet and Neumann conditions in the classical case, and establish analogous inequalities. It is also well-known that the lowest periodic eigenvalue is simple; here we prove a similar result for the general case. Moreover, we show that the algebraic and geometric multiplicities of the eigenvalues of self-adjoint regular Sturm–Liouville problems with coupled boundary conditions are the same. An important step in our approach is to obtain a representation of the fundamental solutions for sufficiently negative values of the spectral parameter. Our approach yields the existence and boundedness from below of the eigenvalues of arbitrary self-adjoint regular Sturm–Liouville problems without using operator theory.