α-theory for Newton-Moser method

In this work we consider a variant of Newton’s method to approximate the solution of a nonlinear equation F (x) = 0, (1) where F is an operator defined between two Banach spaces X and Y. In fact, we study the so called Newton-Moser method [4], that is defined by the iterative scheme{ xn+1 = xn −BnF (xn), n ≥ 0, Bn+1 = 2Bn −BnF ′(xn+1)Bn, n ≥ 0, (2) where x0 is a given point in X and B0 is a given linear operator from Y to X. The method exhibits several attractive features. First, it avoids the calculus of inverse operators that appears in Newton’s method, xn+1 = xn−F ′(xn)−1F (xn), n ≥ 0. So it is not necessary to solve a linear equation at each iteration. Second, it has quadratic convergence, the same as Newton’s method. Third, in addition to solve the nonlinear equation (1), the method produces successive approximations {Bn} to the value of F ′(x∗)−1, being x ∗ a solution of (1). We analyze the convergence of Newton-Moser method (2) under estimations at one point. This theory, introduced by Smale ([3], [5]), is an alternative to Kantorovich theory [2] to study the semilocal convergence of iterative processes to solve nonlinear equations. Roughly speaking, this theory establishes conditions on the initial value x0 for the convergence of the sequence {xn} to the solution x ∗ of equation (1). Actually, we assume i) ‖F ′(x0)−1F (x0) ‖ ≤ β, ii) 1 k