Two-parameter uniformly elliptic Sturm-Liouville problems with eigenparameter-dependent boundary conditions

We consider the two-parameter Sturm-Liouville system -y(1)'' + q(1)y(1) = (lambda r(11) + mu r(12))y(1) on [0, 1], with the boundary conditions y(1)'(0)/y(1)(0) = cot alpha(1) and y(1)'(1)/y(1)(1) = a(1)lambda + b(1)/c(1)lambda + d(1), and -y(2)'' + q(2)y(2) = (lambda r(21) + mu r(22))y(2) on [0, 1], with the boundary conditions y(2)'(0)/y(2)(0) = cot alpha(2) and y(2)'(1)/y(2)(1) = 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) not equal 0 for i, j = 1, 2. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.