Newton’s Method under Different Lipschitz Conditions

Abstract

The classical Kantorovich theorem for Newton’s method assumes that the derivative of the involved operator satisfies a Lipschitz condition ∥;F’(x ₀)^-1 [F’(x) -’’(y)] ∥≤ L∥x - y∥ In this communication, we analyse the different modifications of this condition, with a special emphasis in the center-Lipschitz condition: ∥F’(x₀)^-1 [F’(x) - F-(x₀)]∥≤ω(∥x - x₀∥) being ω a positive increasing real function and x₀ the starting point for Newton’s iteration.

In this paper we make a survey of the convergence of Newton’s method in Banach spaces. So, let X, Y be two Banach spaces and let F : X → Y be a Fréchet differentiable operator. Starting from x₀ ∈ X, the well-known Newton’s method is defined by the iterates x_n+1 = x_n - F’(x_n)^-1F(x_n), n = 0, 1, 2, . . . (1) provided that the inverse of the linear operator F’(x_n) is defined at each step.

Research of both authors has been supported by a grant of the Universidad de La Rioja (ref. API-99/B14) and two grants of the DGES (refs. PB98-0198 and PB96- 0120-C03-02).