Multiparameter SturmLiouville Problems with Eigenparameter Dependent Boundary Conditions

A system of ordinary differential equations, $ -y_j ^n + q_j y_j = \sum_{k=1}^{n} \lambda _k r_{jk}) y_j, j = 1, \cdot \cdot \cdot$ ,n, with real valued and continuous coefficient functions $q_j , r_{jk}$ is studied on [0, 1] subject to boundary conditions $\frac {y_j ^'(0)}{y_j(0)}\ = cot \beta _j$ and $b_j y_j(1) - d_j y_j^'(1)$ = $e^T_j \lambda(c_j y^'_j(1) - a_j y_j(1))$ (0.2) for j = 1, \cdot \cdot \cdot ,n. Here $E^T$ = $[e_1, e_2 \cdot \cdot \cdot e_n]$ is an arbitrary n \times n matrix of real numbers and $ \omega _j = a_j d_j - b_j c_j $ \neq 0. A point $\lambda = [ \lambda_1 \cdot \cdot \cdot \lambda_n]^T \epsilon C^n$, satisfying (0.1) and (0.2) is called an eigenvalue of the system.Results are given on the existence and location of the eigenvalues and completeness and oscillation of the eigenfunctions