LOCAL CONVERGENCE FOR SOME THIRD-ORDER ITERATIVE METHODS UNDER WEAK CONDITIONS Newton method;order of convergence;local convergence; The solutions of equations are usually found using iterative methods whose convergence order is determined by Taylor expansions. In particular, the local convergence of the method we study in this paper is shown under hypotheses reaching the third derivative of the operator involved. These hypotheses limit the applicability of the method. In our study we show convergence of the method using only the first derivative. This way we expand the applicability of the method. Numerical examples show the applicability of our results in cases earlier results cannot.

In this paper we consider regularized modified Newton's method for approximately solving the nonlinear ill-posed problem F(x) = y, where the right hand side is replaced by noisy data y^δ ∈ Y with ‖y – y^δ ‖ ≤ δ and F : D(F) ⊂ X → Y is a nonlinear operator between Hilbert spaces X and Y. Under the assumption that Fréchet derivative F′ of F is Lipschitz continuous, a choice of the regularization parameter and a stopping rule based on a majorizing sequence are presented. We prove that under a general source condition on , the error between the regularized approximation and the solution of optimal order.