Spectral problems for non-linear Sturm-Liouville equations with eigenparameter dependent boundary conditions

The nonlinear Sturm-Liouville equation -(py')' + qy = lambda(1 - f)ry on [0, 1] is considered subject to the boundary conditions (a(j)lambda + b(j))y(j) = (c(j)lambda + d(j))(py')(j), j=0,1. Here a(0) = 0 = c(0) and p, r > 0 and q are functions depending on the inde pendent variable x alone, while f depends on x, y and y'. Results are given on existence and location of sets of (lambda, y) bifurcating from the line arized eigenvalues, and for which y has prescribed oscillation count, and o n completeness of the y in an appropriate sense.