A modification of the classical Kantorovich conditions for Newtonʹs method

In the classical Kantorovich theorem on Newtonʹs method it is assumed that the second Fréchet derivative of the involved operator satisfies the condition ||F″(x)||⩽K in an appropiate domain. In this paper we study a modification of this condition, assuming that ||F″(x)||⩽ω(||x||), where ω is a continuous and non-decreasing real function.