Newton's method under different Lipchitz conditions.

Resumen

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 xn+1 = xn - F’(xn)-1F(xn), n = 0, 1, 2, . . . (1) provided that the inverse of the linear operator F’(xn) is defined at each step