Convergence of Newton’s method and uniqueness of the solution of equations in Banach space

where f is an operator mapping from some domain D in a real or complex Banach space X to another Y. Ostrowski and Kantorovich once gave a popular formulation which does not presume the existence of the solution (see Wang, 1999). However, it implies the existence of the solution because their hypothesis can result in the existence of x∗ = lim xn . Therefore, the existence of the solution is a very natural hypothesis. The advantage of such a hypothesis is that it can make us clearly see how big the radius of convergence ball is in the study. For x ∈ X and a positive number r , let B(x, r) denote an open ball with radius r and centre x and let B(x, r) denote its closure. For example, under the hypothesis that f ′(x∗)−1 f ′ satisfies the Lipschitz condition ‖ f ′(x∗)−1( f ′(x) − f ′(x ′))‖ L‖x − x ′‖, ∀x ′, x ∈ B(x∗, r), (1.3)