Universality of Newton's method

Convergence of the classical Newton’s method and its DSM version for solving operator equations $F(u)=h$ is proved without any smoothness assumptions on $F^{\mathrm{\prime }}(u)$. It is proved that every solvable equation $F(u)=f$ can be solved by Newton’s method if the initial approximation is sufficiently close to the solution and $\mathrm{\mid }\mathrm{\mid }[F^{\mathrm{\prime }}(y){]}^{-1}\mathrm{\mid }\mathrm{\mid }\le m$, where $m>0$ is a constant.