Estimates of majorizing sequences in the Newton-Kantorovich method

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 $$ r_0=0, \quad r_1 =a , \quad r_{n+1}=r_n +\frac{bk(r_n-r_{n-1})^{1+\theta}}{(1+\theta)(1-bkr_n ^{\theta})} \,, $$ with $a, b, k$ parameters related to $f$ and $f'$.\\ We extend an estimate for $r_n$, known in the Lipschitz case, to the H\"{o}lder case. The proof requires the introduction of a multiplicative factor in the sequence estimating $r_n$, estimates of the ratio $\displaystyle{\frac{r_{n+1}}{r_n}}$ and the use of two parallel induction processes on the sequences $r_n$ and $\displaystyle{\frac{r_{n+1}}{r_n}}$.\\ In the last part we make a comparison with our previous results.