## 3 Semi-Local Convergence of [PERSON]'s Method
### Main Result  
We now give a semi-local convergence result for [PERSON]'s method from a modification of the convergence conditions required in Theorem 1. Therefore, we consider Conditions (A1), (A2) and the modification of Condition (A3) given by:  
* \(\gamma\leq\frac{1+p}{2+p}\), \(b_{1}\) satisfies (16) and \(B(x_{0},R)\subset\Omega\), where \(R=\left(1+\frac{\delta}{1-a_{1}}\right)\eta\).  
Remember that Condition (3) follows from Condition (A2) for the starting point \(x_{0}\).  
**Theorem 3**.: _Let \(F:\Omega\subseteq X\longrightarrow Y\) be a continuously-differentiable operator defined on a nonempty open convex domain \(\Omega\) of a Banach space \(X\) with values in a Banach space \(Y\). Suppose that Conditions (A1), (A2) and (A3b) are satisfied. Then, [PERSON]'s sequence, given by (1), converges to a solution \(x^{*}\) of the equation \(F(x)=0\), starting at \(x_{0}\), and \(x_{n},x^{*}\in\overline{B(x_{0},R)}\), for all \(n=0,1,2,\ldots\)._  
**Proof.** We begin by proving the following four items for Sequences (1), (4) and (5) and \(n\geq 3\):  
* (I) There exists \(\Gamma_{n-1}=[F^{\prime}(x_{n-1})]^{-1}\) and \(\|\Gamma_{n-1}\|\leq\frac{\|\Gamma_{n-2}\|}{1-b_{n-2}}\),
* (II) \(\|x_{n}-x_{n-1}\|\leq a_{n-2}\|x_{n-1}-x_{n-2}\|\),
* (III) \(K\|\Gamma_{n-1}\|\|x_{n}-x_{n-1}\|^{p}\leq b_{n-1}\),
* (IV) \(x_{n}\in\Omega\).  
Observe that \(x_{1}\in\Omega\), since \(\eta<R\). Moreover, from (6), (7), (8) and (9), it follows that \(x_{2}\in\Omega\). Furthermore, from (11), (12), (13) and (14), we have that Items (I), (II), (III) and (IV) hold for \(n=3\). If we now suppose that Items (I), (II) and (III) are true for some \(n-1\), it follows, by analogy to the case where \(n=3\) and induction, that Items (I), (II) and (III) also hold for \(n\). Notice that \(b_{n}<1\) for all \(n\geq 1\). Now, we prove (IV). Therefore,\[\|x_{n}-x_{0}\| \leq \|x_{n}-x_{n-1}\|+\|x_{n-1}-x_{n-2}\|+\cdots+\|x_{1}-x_{0}\|\] \[\stackrel{{\text{(II)}}}{{\leq}} \left(1+\sum_{i=1}^{n-2}\left(\prod_{j=1}^{i}a_{j}\right)\right)\|x _{2}-x_{1}\|+\|x_{1}-x_{0}\|\] \[< \left(1+\sum_{i=1}^{n-2}a_{1}^{i}\right)\|x_{2}-x_{1}\|+\|x_{1}- x_{0}\|\] \[< \frac{1}{1-a_{1}}\|x_{2}-x_{1}\|+\|x_{1}-x_{0}\|\] \[\leq \left(1+\frac{\delta}{1-a_{1}}\right)\|x_{1}-x_{0}\|\] \[\leq R\]  
and \(x_{n}\in B(x_{0},R)\). As \(B(x_{0},R)\subset\Omega\), then \(x_{n}\in\Omega\) for all \(n\geq 0\). Note that the conditions given in (15) are satisfied for all \(x_{n}\) and \(b_{n-1}\) with \(n\geq 3\).  
Next, we prove that \(\{x_{n}\}\) is a Cauchy sequence. For this, we follow an analogous procedure to the latter. Therefore, for \(m\geq 2\) and \(n\geq 2\), we have:  
\[\|x_{n+m}-x_{n}\| \leq \|x_{n+m}-x_{n+m-1}\|+\|x_{n+m-1}-x_{n+m-2}\|+\cdots+\|x_{n+1}-x_{ n}\|\] \[\stackrel{{\text{(II)}}}{{\leq}} \sum_{i=n-1}^{n+m-2}\left(\prod_{j=1}^{i}a_{j}\right)\|x_{2}-x_{1}\|\] \[\stackrel{{\text{Lemma \ref{lemma:1}}}}{{<}} \left(\sum_{i=n-1}^{n+m-2}a_{1}^{i}\right)\|x_{2}-x_{1}\|\] \[= \left(\sum_{i=0}^{m-1}a_{1}^{n+i-1}\right)\|x_{2}-x_{1}\|\] \[= \frac{1-a_{1}^{m}}{1-a_{1}}\,a_{1}^{n-1}\|x_{2}-x_{1}\|.\]  
Thus, \(\{x_{n}\}\) is a Cauchy sequence.  
After that, we prove that \(x^{*}\) is a solution of equation \(F(x)=0\). As \(\|\Gamma_{n}F(x_{n})\|\to 0\) when \(n\to\infty\), if we take into account that:  
\[\|F(x_{n})\|\leq\|F^{\prime}(x_{n})\|\|\Gamma_{n}F(x_{n})\|\]  
and \(\{\|F^{\prime}(x_{n})\|\}\) is bounded, since:  
\[\|F^{\prime}(x_{n})\|\leq\|F^{\prime}(x_{0})\|+K_{0}\|x_{n}-x_{0}\|^{p}\leq\|F ^{\prime}(x_{0})\|+K_{0}R^{p},\]  
it follows that \(\|F(x_{n})\|\to 0\) when \(n\to\infty\). As a consequence, we obtain \(F(x^{*})=0\) by the continuity of \(F\) in \(\overline{B(x_{0},R)}\). \(\qed\)