A generalized quasilinearization method for second-order nonlinear differential equations with nonlinear boundary conditions

We consider the differential equation ℓ(u)=F(u)ℓ(u)=F(u), where ℓℓ is a formally self-adjoint second-order differential expression and F   is nonlinear, with nonlinear boundary conditions. Under appropriate assumptions on ℓ,Fℓ,F and the boundary conditions, existence of solutions is established using the method of lower and upper solutions. A generalized quasilinearization method is then developed for this problem and we obtain two monotonic sequences of approximate solutions converging quadratically to a solution of the equation.