Estimates of majorizing sequences in the Newton–Kantorovich method: A further improvement

Let f :B(x0,R) ⊆ X→Y be an operator, with X and Y Banach spaces, and f be Hölder continuous with exponent θ. The convergence of the sequence of Newton–Kantorovich approximations xn = xn−1 − f (xn−1) −1f (xn−1), n ∈ N, is a classical tool to solve the equation f (x) = 0. The convergence of xn is often reduced to the study of the majorizing sequence rn defined by r0 = 0, r1 = a, rn+1 = rn + bk(rn −rn−1)1+θ (1+ θ)(1−bkrθn ) , n∈ N, with a, b, k parameters related to f and f . In the paper [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton– Kantorovich method, submitted for publication] we proved that, if ξ := aθ bk 1 1+θ θ 1−θ 1−θ θ 1+θ θ , then the following estimates for rn hold rn (bk) −1 θ 1+θ θ 1−θ 1−θ θ 1− 1 (1+θ)n , ∀n ∈ N.In the present paper we give a stronger (at least asymptotically) estimates on rn under a weaker condition on ξ . The techniques employed in the paper are similar to the ones used in [F. Cianciaruso, E. De Pascale, Estimates of majorizing sequences in the Newton–Kantorovich method, submitted for publication]. Finally, we make a comparison with previous results. © 2005 Elsevier Inc. All rights reserved