A modification of the classical Kantorovich conditions for Newton's method (Q5953960)

The paper deals with the Newton method to approximate a locally unique solution of the equation \(F(x)=0\), where \(F\) is a second Fréchet differentiable operator in a Banach space. The author studies a modification of the classical Kantorovich condition \(\|F''(x)\|\leq K \) in an appropriate domain, assuming that \(\|F''(x)\|\leq \omega(\|x\|) \), where \(\omega\) is a continuous and non-decreasing real function.