Bryan P. Rynne
Abstract:
We consider the eigenvalue problem for the equation
on
,
together with general Sturm-Liouville-type, multi-point boundary
conditions at
.
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
there exists a unique, simple
eigenvalue
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