THE COMBINATION OF THE ESTABLISHMENT METHOD AND NEWTONS METHOD FOR SOLVING NONLINEAR DIFFERENTIAL PROBLEMS

An iterative scheme for the non-linear equation phi (z) = 0 in Banach space based on the establishment method is studied. A non-linear evolutionary equation is associated with the initial problem. The corresponding discrete scheme leads to ''almost linear'' problems with an adjustable non-linearity norm, which have been solved numerically by the continuous analogue of Newton's method. The advantage of the proposed scheme over the latter is that it can be widely used in near-degenerate cases (\\phi (z)\\ --> 0). The results are employed to solve a boundary-value problem for a non-linear second-order differential equation. Theorems on the monotone convergence of the iterative scheme as a whole and of the Newtonian iterative scheme at each step of the iterations are proved. Numerical examples are given.