Eigencurves for two-parameter self-adjoint ordinary differential equations of even order (Q1121420)

Previous results of the authors [Proc. R. Soc. Edinb., Sect. A 106, 39-51 (1987; Zbl 0642.47003)] are sharpened in the special case of the differential equation \(\sum^{n}_{j=0}(p_ jy^{(j)})^{(j)}=(\lambda f-\mu r)y,\) where \(r>0\), with certain self- adjoint boundary conditions. Necessary and sufficient conditions are established for the existence of asymptotes to eigencurves and equations for these asymptotes are derived. The existence of line segments in the eigencurves is also investigated.