## 2 Local Convergence  
As we indicated above, the local convergence results for iterative methods require conditions on the operator \(F\) and the solution \(x^{*}\) of Equation (1). Note that a local result provides what we call ball of convergence and denote by \(B(x^{*},\widetilde{R})\). From the value \(\widetilde{R}\), the ball of convergence gives information about the accessibility of the solution \(x^{*}\). An interesting local result for [PERSON]'s method Equation (2) is given in [7] by [PERSON] and [PERSON], where the following conditions are required:  
1. Let \(x^{*}\) be a solution of Equation (1) and exist \(r>0\), so that \(B(x^{*},r)\subset\Omega\) and the operator \([F^{\prime}(x^{*})]^{-1}\) exists with \(\|[F^{\prime}(x^{*})]^{-1}\|\leq\beta\),
2. \(\|F^{\prime}(x)-F^{\prime}(y)\|\leq K\|x-y\|\), \(K\geq 0\), for all \(x,y\in\Omega\). Now, our aim is to establish the local convergence study for family Equation (3), with cubical convergence (see [6]), from conditions \((C_{1})\) and \((C_{2})\). So, we obtain a local convergence result in the same initial conditions that for the Newton method, which is a second order method. Firstly, we provide two technical lemmas, where we obtain some results about the operator \(F\) and the sequences \(\{x_{n}\}\), \(\{y_{n}\}\) and \(\{z_{n}\}\), which define family Equation (3). **Lemma 1**. : _Let us suppose that conditions \((C_{1})\)-\((C_{2})\) are satisfied. Let us assume that exists \(S\), a positive real number, with \(S\leq r\). If \(\beta KS<1\), then, for all \(x\in B(x^{*},S)\), the operator \([F^{\prime}(x)]^{-1}\) exists and_  
\[\|[F^{\prime}(x)]^{-1}\|\leq\frac{\beta}{1-\beta KS}\]  
Proof. : Taking into account the Banach lemma on invertible operators and conditions \((C_{1})\)-\((C_{2})\), the proof follows. **Lemma 2**. : _Fixed \(n\in\mathbb{N}\), we assume that \(x_{n},z_{n}\in B(x^{*},r)\) and there exists \(\Gamma_{n}\). Then, we obtain the following decompositions_  
Proof. : Firstly, fixed \(p\in(0,1]\), we define two auxiliary scalar functions:  
\[f(t)=1-p+p\frac{t}{2}\ \ \text{and}\ \ g(t)=\frac{t}{2p^{2}}\left(f(t)^{2}+2f(t)+1-p ^{2}\right)\]  
with \(t\in\mathbb{R}\) and \(p\in(0,1]\), which are strictly increasing in \(\mathbb{R}_{+}\). Secondly, let \(\widetilde{R}<\min\{r,R\}\), where \(R=\frac{\zeta_{p}}{(1+\zeta_{p})\beta K}\) and \(\zeta_{p}\) is the positive real root of the equation \(g(t)-1=0\). As \(x_{0}\in B(x^{*},\widetilde{R})\), with \(\widetilde{R}<r\) and \(\beta K\widetilde{R}<\beta KR<1\), then by Lemma 1 the operator \(\Gamma_{0}=[F^{\prime}(x_{0})]^{-1}\) exists and is such that  
\[\|\Gamma_{0}\|<\frac{\beta}{1-\beta K\widetilde{R}}\]  
So, \(y_{0}\) is well-defined and, from Lemma 2\((i_{0})\), we obtain  
\[\|y_{0}-x^{*}\|<\frac{\delta}{2}\|x_{0}-x^{*}\|\]  
where \(\delta=\frac{\beta K\widetilde{R}}{1-\beta K\widetilde{R}}\). As \(\widetilde{R}<R\), then \(\delta<\frac{\beta KR}{1-\beta K\widetilde{R}}=\zeta_{p}\). On the other hand, taking into account that \(g(2)=\frac{4-p^{2}}{p^{2}}>1\) and \(g(\zeta_{p})=1\), it follows that \(\frac{\delta}{2}<1\) and then \(\|y_{0}-x^{*}\|<\frac{\delta}{2}\|x_{0}-x^{*}\|<\|x_{0}-x^{*}\|\). Therefore \(y_{0}\in B(x^{*}\widetilde{R})\). Moreover, \(z_{0}\) is well-defined and \(z_{0}\in B(x^{*},\widetilde{R})\), since, from item \((ii_{0})\) of Lemma 2, it follows  
\[\|z_{0}-x^{*}\|<\left(1-p+p\frac{\delta}{2}\right)\|x_{0}-x^{*}\|<\|x_{0}-x^{*}\|\]  
as a consequence of \(f(\delta)<1\).