Approximation Methods for Nonlinear Problems with Application to Two-Point Boundary Value Problems

General nonlinear problems in the abstract form $F(\chi ) = 0$ and corresponding families of approximating problems in the form ${F_h}({\chi _h}) = 0$ are considered (in an appropriate Banach space setting). The relation between "isolation" and "stability" of solutions is briefly studied. The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution. Error estimates are obtained and Newton’s method is shown to converge quadratically. These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear two-point boundary value problems.