Convergence of Newton's method and inverse function theorem in Banach space

Under the hypothesis that the derivative satisfies some kind of weak Lipschitz condition, a proper condition which makes Newton’s method converge, and an exact estimate for the radius of the ball of the inverse function theorem are given in a Banach space. Also, the relevant results on premises of Kantorovich and Smale types are improved in this paper. We continue to discuss the problem of convergence in the Newton method xn+1 = xn − f ′(xn)−1f(xn), n = 0, 1, · · · , (0.1) to solve an operator equation f which maps from some domain D in a real or complex Banach space X to another Banach space Y, f(x) = 0. (0.2) Now we come back to the problem which we bypassed in [1]. We always assume that f ′(x0)−1 exists and f ′(x0)−1f ′ satisfies some kind of Lipschitz condition similar to that of [1] in some open ball B(x0, r) ⊂ D with center x0 and radius r (or some closed ball B(x0, r) ⊂ D) in order to study the convergence of Newton’s method and the domain of the local inverse function of f at x0. 1. The domain of the inverse function The inverse function theorem asserts that there is an inverse function f−1 x0 defined on some open ball B(f(x0), ε) ⊂ Y with the property that f−1 x0 (f(x0)) = x0, f(f−1 x0 (y)) = y, ∀y ∈ B(f(x0), ε), and f−1 x0 is differentiable. Now we study the exact lower bound estimate of the radius of this ball. For this reason, we assume that f has a continuous derivative in the ball B(x0, r), f ′(x0)−1 exists and f ′(x0)−1f ′ satisfies the center Lipschitz condition with the L average, ∥∥f ′(x0)−1f ′(x)− I∥∥ ≤ ∫ ρ(x) 0 L(u)du, ∀x ∈ B(x0, r), (1.1) Received by the editor March 12, 1997 and, in revised form, June 6, 1997. 1991 Mathematics Subject Classification. Primary 65H10. Supported by the China State Major Key Project for Basic Research and the Zhejiang Provincial Natural Science Foundation. c ©1999 American Mathematical Society 169 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use