We determine a solution x of the equation F(x) = 0. where X and Y are Banach spaces, D ⊆ X a convex set and F : D → Y is a Fréchet-differentiable operator. In particular, we expand the applicability of the Newton’s method defined by xn + 1 = xn −[F′(xn)]−1F(xn), for each n = 0,1,2, …, (2.1) by considering weaker sufficient convergence criteria than in earlier studies [13]. We will denote along the chapter.