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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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