Let \(D\subset {\mathbb{R}}^ m\) and \(F: D\to {\mathbb{R}}^ m\) be a mapping. The author studies the approximate solution of the equation \(F(x)=0\) by means of the iterative method for \(n=0,1,2,....:\) (*) \(x_{n+1}:=x_ n+s_ n\in {\mathbb{R}}^ m\) with \(s_ n\) from \(F'(x_ n)s_ n=-F(x_ n)+r_ n\) for some sequence \(\{r_ n\}\subset R^ m\). He gives an affine invariant condition involving \(r_ n\) which ensures the local convergence of (*) to a solution of \(F(x)=0\). Moreover he deduces a radius of convergence result for (*) which is shown to be sharp for both Newton's method and the general difference Newton-like method. The results are applied to the latter two methods and the general Newton-like method in which the iterates are perturbed by the presense of rounding errors. No numerical example.