A new angle on Sturm-Liouville problems
In this note Sturm Liouville problems \(- (py')' + qy = \lambda ry\), where \(p > 0\), \(r > 0\), \({1 \over p}, q,r \in L_1 ([0,1], \mathbb{R})\), with boundary conditions \(y(0) \cos \beta_0 = (py') (0) \sin \beta_0\), \(0 \leq \beta_0 < \pi\), \((a \lambda + b) y(0) = (c \lambda + d) (py') (0)\), where \(0 \neq (a,b,c,d) \in \mathbb{R}^4\), are considered. For the three cases (I) \(c = 0 \neq d\), \(e \geq 0\), (II) \(c \neq 0\), \(e > 0\), (III) \(c \neq 0\), \(e < 0\), a Sturm theory is presented. The results are oscillation theorems, asymptotics for the eigenvalues and monotonicity (continuity) of the eigenvalues with respect to a parameter \(t\) upon which \(- \beta_0\), \(- bd,p\) and \(q\) depend monotonically (continuously).
scientific article; zbMATH DE number 796906