Estimates of Majorizing Sequences in the Newton–Kantorovich Method

Abstract

Let f:B(x ₀,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

is a classical tool to solve the equation f(x) = 0. The convergence of x _n is often reduced to the study of the majorizing sequence r _n defined by

with a, b, k parameters related to f and f′. We extend an estimate for r _n , known in the Lipschitz case, to the Hölder case. The proof requires the introduction of a multiplicative factor in the sequence estimating r _n , estimates of the ratio , and the use of two parallel induction processes on the sequences r _n and . In the last section, we make a comparison with our previous results.

Keywords:

• Estimates of majorizing sequences
• Hölder continuous derivative
• Newton–Kantorovich approximations

AMS Subject Classification:

• 65J15
• 47J25

ACKNOWLEDGMENT

We wish to thank Giuseppe Marino for a conversation about the proof of Lemma 2.1.