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 Holder 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−bkrnθ),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:=aθbk⩽1(1+θθ1−θ)1−θ(θ1+θ)θ, then the following estimates for rn hold rn⩽(bk)−1θ(1+θθ1−θ)1−θθ(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.