Application of two parameter eigencurves to Sturm-Liouville problems with eigenparameter-dependent boundary conditions

Oscillation, comparison and asymptotic theory for the Sturm-Liouville problem -(p(x)y'(x))'+q(x)y(x)=lambda r(x)y(x), 0 less than or equal to x less than or equal to 1, with 1/p,q,r is an element of L(1)( 0,1 ),p,r>0, are studied subject to eigenvalue-dependent boundary conditions (a(j) lambda+b(j))y(j)=(c(f)lambda+d(j))(py')(j), j=0,1. This continues previous work on cases with (-1)(j)sigma(j) less than or equal to 0 where sigma(f)=a(f)d(f)-b(j)c(j). We now consider the remaining sign conditions for sigma(f), exploiting the interplay between the graph of cot theta(-)(lambda,1), for a modified Prufer angle theta(-), and the eigencurves of a related two-parameter problem.