A convergence analysis for Newton-like methods in Banach space under weak hypotheses and applications (Q2719894)

The reviewed article contains some results about convergence of Newton-like methods \(x_{n+1}=x_n-{A(x_n)}^{-1} F(x_n)\) for the equation \(F(x)=0\) with Frechet-differentiable nonlinear operator \(F\) satisfying Hölder continuity conditions the form NEWLINE\[NEWLINE\begin{aligned}\|A(x_0)^{-1} (A(x)-A(x_0))\|&\leq C_1 \|x-x_0 \|+C_0\\ \|{A(x_0)}^{-1} (F'(x+t(y-x))-A(x))\|&\leq C_2 (t\|x-y\|)^p +C_3\|x-x_0\|^p +C_4 \end{aligned}NEWLINE\]NEWLINE for all \(x,y\in U(x_0,R)\), \(t\in[0,1].\) Computations for two examples are given when the usual Newton method does not work.