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

Abstract

We consider the linear eigenvalue problem \tag{1} -u" = λu, \quad \text{on $(-1,1)$}, where $λ\in \mathbb{R}$, together with the general multi-point boundary conditions \tag{2} α_0^\pm u(\pm 1) + β_0^\pm u'(\pm 1) = \sum^{m^\pm}_{i=1} α^\pm_i u(η^\pm_i) + \sum_{i=1}^{m^\pm} β^\pm_i u'(η^\pm_i). We also suppose that: α_0^\pm \ge 0, \quad α_0^\pm + |β_0^\pm| > 0, \tag{3} \pm β_0^\pm \ge 0, \tag{4} (\frac{\sum_{i=1}^{m^\pm} |α_i^\pm|}{α_0^\pm})^2 + (\frac{\sum_{i=1}^{m^\pm} |β_i^\pm|}{β_0^\pm})^2 < 1, \tag{5} with the convention that if any denominator in (5) is zero then the corresponding numerator must also be zero, and the corresponding fraction is omitted from (5) (by (3), at least one denominator is nonzero in each condition). In this paper we show that the basic spectral properties of this problem are similar to those of the standard Sturm-Liouville problem with separated boundary conditions. Similar multi-point problems have been considered before under more restrictive hypotheses. For instance, the cases where $β_i^\pm = 0$, or $α_i^\pm = 0$, $i = 0,..., m^\pm$ (such conditions have been termed Dirichlet-type or Neumann-type respectively), or the case of a single-point condition at one end point and a Dirichlet-type or Neumann-type multi-point condition at the other end. Different oscillation counting methods have been used in each of these cases, and the results here unify and extend all these previous results to the above general Sturm-Liouville-type boundary conditions.