A new semilocal convergence theorem for Newton's method involving twice Fréchet-differen\-tiabil\-ity at only one point (Q557695)

This paper is concerned with the problem of approximating a locally unique solution of the equation \( F(x)=0\) , where \(F\) is defined an open convex subset of a Banach space \(X\) with values in a Banach space \(Y\). It is assumed that an operator \(F\) is twice continuously Fréchet-differentiability at only one point. The author provide a new semilocal convergence theorem for Newton's method. Error estimates are derived. A simple numerical example is given.