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Right-definite multiparameter Sturm-Liouville problems with eigenparameter-dependent boundary conditions. (English) Zbl 1022.34009
The authors study the distribution of eigenvalues of the multiparameter Sturm-Liouville system defined by \[ -y_{j}^{\prime \prime }+q_{j}y_{j}=\left( \sum_{j=1}^{n}\lambda _{k}r_{jk}\right) y_{j},\quad j=1,\dots n, \] with eigenparameter-dependent boundary conditions defined by \[ ( a_{j0}\lambda _{j}+b_{j0})y_j(0)= (c_{j0}\lambda _{j}+d_{j0}) y_j'(0), \] where \(q_{i}\), \(r_{jk}\in C^{1}[0,1]\) and \(r_{jk}\) satisfy the Minkowsky condition \(\int_{0}^{1}r_{jk}(x)|y(x)|^{2} dx\leq 0\) for \(j\neq k\) while \(\sum_{k=1}^{n} \int_{0}^{1}r_{jk}(x)|y(x)|^{2} dx>0\) for \(j=1, 2,\dots , n\). This gives rise to a right-definite regular Sturm-Liouville system whose spectrum, made of the vectors \((\lambda_{1}, \lambda _{2}, \dots , \lambda_{n})\in \mathbb{R}^{n}\), is discrete. Using the oscillation properties of the eigenfunctions, the intersection of hypersurfaces of eigenvalues are examined in the cases when either one or both boundary conditions depend on the eigenvalue parameter.
MSC:
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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