## 1 Introduction  
It is often necessary in many scientific and engineering problems to find a root of a nonlinear equation. To solve this equation, we can use iterative methods. To give sufficient generality to the problem of approximating a solution of a nonlinear equation by iterative methods, we are concerned, in this work, with the problem of approximating a locally-unique solution \(x^{*}\) of the equation \(F(x)=0\), where \(F\) is a nonlinear operator defined on a nonempty open convex subset \(\Omega\) of a Banach space \(X\) with values in a Banach space \(Y\), so that many scientific and engineering problems can be written as a nonlinear equation in Banach spaces.  
[PERSON]'s method is probably the best-known iterative method, and it is well known that it converges quadratically. [PERSON]'s method for solving scalar equations was extended to nonlinear equations in Banach spaces by [PERSON] in 1948 [1] in the following way:  
\[x_{0}\text{ given in }\Omega,\quad x_{n}=x_{n-1}-[F^{\prime}(x_{n-1})]^{-1}F(x_{ n-1}),\quad n\in\mathbb{N}, \tag{1}\]where \(F^{\prime}\) is the Frechet derivative of the operator \(F\). For this reason, many authors refer to it as the [PERSON] method.  
[PERSON] proved the semi-local convergence of [PERSON]'s method under the hypothesis that the operator involved \(F\) is twice differentiable Frechet with bounded second derivative:  
There exists a constant \(C\geq 0\), such that \(\|F^{\prime\prime}(x)\|\leq C\) for \(x\in\Omega\).  
It is well known that this condition can be replaced by a Lipschitz condition on the first derivative \(F^{\prime}\) of the operator involved [2]:  
There exists a constant \(L\geq 0\), such that \(\|F^{\prime}(x)-F^{\prime}(y)\|\leq L\|x-y\|\) for \(x,y\in\Omega\).  
In view of the applications, numerous papers have recently appeared where the convergence of the method is proven under different conditions for the derivatives of the operator \(F\). A known variant is the study of the convergence of the method under a Holder condition on the first derivative [3; 4; 5]:  
There exist two constants \(K\geq 0\) and \(p\in(0,1]\) such that \(\|F^{\prime}(x)-F^{\prime}(y)\|\leq K\|x-y\|^{p}\) for \(x,y\in\Omega\).  
In order to improve the accessibility of [PERSON]'s method under the last condition, we can use different strategies that change the condition. In this work, we use a similar one as that used by [PERSON] in [6] for the Lipschitz condition on the first derivative \(F^{\prime}\), which consists of noticing that, as a consequence of the last condition being satisfied in \(\Omega\), we have, for the starting point \(x_{0}\), that:  
There exist two constants \(K_{0}\geq 0\) and \(p\in(0,1]\), such that \(\|F^{\prime}(x)-F^{\prime}(x_{0})\|\leq K_{0}\|x-x_{0}\|^{p}\) for \(x\in\Omega\),  
with \(K_{0}\leq K\). We then say that \(F^{\prime}\) is center Holder in \(x_{0}\).  
In this paper, we focus our attention on the analysis of the semi-local convergence of Sequence (1), which is based on demanding the last condition to the initial approximation \(x_{0}\) and provides the so-called domain of parameters corresponding to the conditions required for the initial approximation that guarantee the convergence of Sequence (1) to the solution \(x^{*}\).  
In this work, we carry out an analysis of the domain of parameters for [PERSON]'s method under the last two conditions for \(F^{\prime}\) and use a technique based on recurrence relations. As a consequence, we improve the domains of parameters associated with the semi-local convergence result given for [PERSON]'s method by [PERSON] in [4].  
We prove in this paper that center conditions on the first derivative of the operator involved in the solution of nonlinear equations play an important role in the semi-local convergence of [PERSON]'s method, since we can improve the accessibility of [PERSON]'s method from them.  
Throughout the paper, we denote \(\overline{B(x,\varrho)}=\{y\in X;\|y\ -\ x\|\ \ \leq\ \varrho\}\) and \(B(x,\varrho)=\{y\in X;\|y-x\|<\varrho\}\).