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 monotonie sequence of approximate solutions converging to a solution of the problem. If F is monotone then the convergence is quadratic.