Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 146, pp. 1-21.
Title: Linear second-order problems with Sturm-Liouville-type multi-point
boundary conditions
Author: Bryan P. Rynne (Heriot-Watt Univ., Edinburgh, Scotland)
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.