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

Abstract:
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.

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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

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