Estimates of Majorizing Sequences in the Newton-Kantorovich Method: A further Improvement

Let $f:B(x_0,R)\subseteq X \rightarrow Y$ be an operator, with $X$ and $Y$ Banach spaces, and $f'$ be H\"{o}lder continuous with exponent $\theta$. The convergence of the sequence of Newton-Kantorovich approximations $$ x_n=x_{n-1} -f'(x_{n-1})^{-1}f(x_{n-1}) \,, \quad n \in N $$ 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 $$ r_0=0, \quad r_1 =a , \quad r_{n+1}=r_n +\frac{bk(r_n-r_{n-1})^{1+\theta}}{(1+\theta)(1-bkr_n ^{\theta})} \,, \quad n \in N $$ with $a, b, k$ parameters related to $f$ and $f'$.\\ In the paper \cite{CDP} we proved that, if $\displaystyle{\xi:=a^{\theta}bk \,\le\frac{1}{\Big(1+\theta^{\frac{\theta}{1-\theta}}\Big)^{1-\theta}} \,\Bigg(\frac{\theta}{1+\theta}\Bigg)^{\theta}}$, then the following estimates for $r_n$ hold $$ r_n\le\frac{(bk)^{-\frac{1}{\theta}}}{\Big(1+\theta^{\frac{\theta}{1-\theta}}\Big)^{\frac{1-\theta}{\theta}}} \,\Bigg(1-\frac{1}{(1+\theta)^n}\Bigg)\,, \quad \forall \, n \in N\,. $$ In the present paper we give a stronger (at least asymptotically) estimates on $r_n$ under a weaker condition on $\xi$. The techniques employed in the paper are similar to the ones used in \cite{CDP}.\\ Finally, we make a comparison with previous results.\\