Two-Parameter Uniformaly Elliptic Sturm–Liovviville Problems With Eigenparameter-Dependent Boundary Conditions

We consider the two-parameter Sturm–Liouville system $-y_1" + q_1y_1 = (\lambda r_{11} + \mu r_{12})y_1\quad\text{on }[0,1], $ with the boundary conditions $\frac{y_1'(0)}{y_1(0)} = \cot\alpha_1\quad\text{and}\quad\frac{y_1'(1)}{y_1(1)} = \frac{a_1\lambda + b_1}{c_1\lambda + d_1}, $ and $-y_2"+q_2y_2 = (\lambda r_{21} + \mu r_{22})y_2\quad\text{on}[0,1], $ with the boundary conditions $\frac{y_2'(0)}{y_2(0)} = \cot\alpha_2\quad\text{and}\quad\frac{y_2'(1)}{y_2(1)} = \frac{a_2\mu + b_2}{c_2\mu+d_2}, $ subject to the uniform-left-definite and uniform-ellipticity conditions; where $q_{i}$ and $r_{ij}$ are continuous real valued functions on $[0,1]$, the angle $\alpha_{i}$ is in $[0,\pi)$ and $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ are real numbers with $\delta_{i} = a_{i}d_{i}-b_{i}c_{i}>0$ and $c_{i}\neq 0$ for $i,j = 1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues