Convergence and Uniqueness Properties of Gauss-Newton's Method

K e y w o r d s G a u s s N e w t o n ' s method, Lipschitz conditions with L average, Convergence ball, Uniqueness bali. 1. I N T R O D U C T I O N Finding the solution of a nonlinear operator equation f(x) = 0 (1.1) in Banach space X is a very general subject which is widely used in both theoretical and applied areas of mathematics, where f is a nonlinear operator from some domain D in a real or complex Banach space X to another Y. When f is Freehet differentiable, the most important method to find the approximation solution is Newton's method which is defined by / --1 x ~ + l = x ~ f (x~) f(x~), n = 0 , 1 , . . . , for some initial value x0 E D. The well-known works, due to Kantorovich [1] and Kantorovich and Akilov [2], on the convergence of Newton's method, under the hypothesis that f/i is bounded on D *The research of this author is supported by the National Natural Science Foundation of China (Grant No. 10271025). tThe research of this author is supported by the research Grant RG026/00-01S/JXQ/FST Kom University of Macau. 0898-1221/04/$ see front matter @ 2004 Elsevier Ltd. Ali rights reserved. doi:10.1016/j.camwa.2002.12.014 Typeset by A A//8-~X