Multiparameter Sturm-Liouville problems with eigenparameter dependent boundary conditions

Abstract

A system of ordinary differential equations,− y″j + qjyj = ∑k = 1nλkrjkyj, j = 1,…,n, 0.1 with real valued and continuous coefficient functions q_j, r_jk is studied on [0, 1] subject to boundary conditionsy′j0yj0 = cot Βj and bjyj1 − djy′j1 = eTjλcjy′j1 − ajyj1 0.2for j = 1,…,n. Here E^T = [e₁1 e₂ ··· e_n] is an arbitrary n × n matrix of real numbers and ω_j = a_jd_j − b_jc_j ≠ 0. A point λ = [λ₁ ··· λ_n]^T ∈ Cⁿ 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.