On a generalization of Moret's theorem for inexact Newton-like methods (Q2910012)

The article deals with the Newton-like iterations NEWLINE\[NEWLINEx_{n+1} = x_n - A(x_n)^{-1}F(x_n)\qquad (n = 0,1,\ldots)\tag{1}NEWLINE\]NEWLINE for approximate solution of the nonlinear operator equation \(F(x) = 0\) with a smooth operator \(F\) between Banach spaces \(X\) and \(Y\). The authors formulate some conditions under which the sequence \((x_n)\) is well defined and converges to the solution \(x_*\) of the equation \(F(x) = 0\). Some partial cases are also considered. The arguments of the authors are based on the use the following `non-standard' Lipschitz condition NEWLINE\[NEWLINE\|A(x_0)^{-1}(F'(x) - F'(y))\| \leq \int\limits_{\|x - x_0\|}^{\|x - y\| + \|x - x_0\|} L(u) \, duNEWLINE\]NEWLINE with a positive non-decreasing function \(L(u)\).