A new semilocal convergence theorem for Newtonʹs method involving twice Fréchet-differentiability at only one point

In this study, we provide a new semilocal convergence theorem for Newtonʹs method. It is assumed that an operator is twice continuously Fréchet-differentiable at only one point. Finally, a simple numerical example is given to show that our results can apply, but earlier similar ones fail.