For a given Sturm-Liouville equation whose leading coefficient function changes sign, we establish inequalities among the eigenvalues for any coupled self-adjoint boundary condition and those for two corresponding separated self-adjoint boundary conditions. By a recent result of Binding and Volkmer, the eigenvalues (unbounded from both below and above) for a separated self-adjoint boundary condition can be numbered in terms of the Pr\"ufer angle; and our inequalities can then be used to index the eigenvalues for any coupled self-adjoint boundary condition. Under this indexing scheme, we determine the discontinuities of each eigenvalue as a function on the space of such Sturm-Liouville problems, and its range as a function on the space of self-adjoint boundary conditions. We also relate this indexing scheme to the number of zeros of eigenfunctions. In addition, we characterize the discontinuities of each eigenvalue under a different indexing scheme.

The nonlinear Sturm-Liouville equation $$ -(py')' + qy = \lambda(1 - f)ry \text{ on } [0,1] $$ is considered subject to the boundary conditions $$ (a_j\lambda + b_j) y(j) = (c_j\lambda + d_j) (py') (j), \quad j = 0,1. $$ Here $a_0 = 0 = c_0$ and $p,r > 0$ and $q$ are functions depending on the independent 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 linearized eigenvalues, and for which $y$ has prescribed oscillation count, and on completeness of the $y$ in an appropriate sense.