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

Abstract

We consider the two-parameter Sturm–Liouville system –y "₁+q₁y₁=(λr₁₁+μr₁₂)y₁ [0,1], with the boundary conditions y'₁(0)/y₁(0)=cotΑ₁ and y'₁(1)/y₁(1)=a₁λ+b₁/c₁λ+d₁ and –y" ₂+q₂y₂=(λr₂₁+μr₂₂)y₂ on [0,1] with the boundary conditions y'₂(0)/y₂(0)=cotΑ₂ and y'₂(1)/y₂(1)=a₂μ+b₂/c₂μ+d₂ subject to the uniform-left-definite and uniform-ellipticity conditions; where and q_i and r_ij are continuous real valued functions on [0,1], the angle Α_i is in [0, π) and a_i, b_i, c_i, d_i, and are real numbers with δ _i = a_id_i - b_ic_i > 0 and c_i ≠0 for i,j = 1,2. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.