Electron. J. Diff. Equ., Vol. 2012 (2012), No. 146, pp. 1-21.

Linear second-order problems with Sturm-Liouville-type multi-point boundary conditions

Bryan P. Rynne

We consider the eigenvalue problem for the equation $-u'' = \lambda u$ on $(-1,1)$, together with general Sturm-Liouville-type, multi-point boundary conditions at $\pm 1$. We show that the basic spectral properties of this problem are similar to those of the standard Sturm-Liouville problem with separated boundary conditions. In particular, for each integer $k \ge 0$ there exists a unique, simple eigenvalue $\lambda_k$ whose eigenfunctions have 'oscillation count' equal to k. Similar multi-point problems have been considered before for Dirichlet-type or Neumann-type multi-point boundary conditions, or a mixture of these. Different oscillation counting methods have been used in each of these cases. A new oscillation counting method is used here which unifies and extends all the results for these special case to the general Sturm-Liouville-type boundary conditions.

Submitted October 28, 2011. Published August 21, 2012.
Math Subject Classifications: 34B05, 34B10, 34B24, 34B25.
Key Words: Second order ordinary differential equations; multi-point boundary conditions; Sturm-Liouville problems.

Show me the PDF file (361 KB), TEX file, and other files for this article.

Bryan P. Rynne
Department of Mathematics and the
Maxwell Institute for Mathematical Sciences
Heriot-Watt University
Edinburgh EH14 4AS, Scotland
email: bryan@ma.hw.ac.uk

Return to the EJDE web page