Approximation to the Sturm-Liouville Problem with a Discontinuous Nonlinearity

We consider a continuous approximation to the Sturm-Liouville problem with a nonlinearity discontinuous in the phase variable. The approximating problem is obtained from the original one by small perturbations of the spectral parameter and by approximating the nonlinearity by Caratheodory functions. The variational method is used to prove the theorem on the proximity of solutions of the approximating and original problems. The resulting theorem is applied to the one-dimensional Gol'dshtik and Lavrent'ev models of separated flows.