EXISTENCE AND GENERALIZED QUASILINEARIZATION METHODS FOR SINGULAR NONLINEAR DIFFERENTIAL EQUATIONS

We consider the existence problem for the differential equation lu = F(u), where l is a formally self-adjoint singular second order differential expression and F is nonlinear. Under certain assumptions on l and F we develop an existence theorem. If the problem has upper and lower solutions these assumptions can be relaxed. A generalized quasilinearization method is then developed for this problem and we obtain a monotonic sequence of approximate solutions converging to a solution of the problem. If F is monotone then the convergence is quadratic.