Convergence of iterations of Euler family under weak condition

The iteration maps of Euler family for finding zeros of an operator f in Banach spaces is defined as the partial sum of Taylor expansion of the local inverse fz-1 of f at z. The unified convergence theorem is established for the iterations of Euler family under the assumption that α≤3 - 2 , while the strong condition that f is analytic in Smale’s criterion a is replaced by the weak condition that f is of finite order derivative.