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.