Newton method under weak Lipschitz continuous derivative in Banach spaces.
Let \(f: X \to Y\) be a Fréchet differentiable nonlinear operator from a Banach space \(X\) into a Banach space \(Y\). The Newton method for solving the equation \(f(x) = 0\) is described by the iterative formula \(x^{n+1} = x^n - (f'(x^n))^{-1} f(x^n),\) \(n>0\), provided that \((f'(x^n))^{-1}\) exists in each step. \textit{A. Ostrovski} [Comment. Math. Helv. 9, 79--103 (1937; Zbl 0015.36403)] and \textit{L. Kantorovich} [On Newton's method, Trudy. Math. Inst. Steklov 28, 104--144 (1949)] used majorant function techniques to determine whether the iterative sequence defined by this method, starting from some pivot value \(x^0\) converges and at the next step get a theoretical error estimates which reflects the convergent order two of the sequence of iterations. In this paper the results of Ostrovski and Kantorovich are generalized to weaker conditions for \(f\) and \(f''\). A relaxed Kantorovich-type condition to guarantee the convergence is obtained. It is also demonstrated, that the conditions of Kantorovich and {those of \textit{S. Smale} [Proc. Symp. Honor G. S. Young, Laramie/Wyo. 1985, 185--196 (1986; Zbl 0613.65058)] are special cases of the developed ones. The relations between the new condition and the old ones are described.}
scientific article; zbMATH DE number 1978068