## 2 Semilocal Convergence  
The following hypotheses shall be used in the semilocal convergence analysis (H):  
1. [label=(H0)]
2. Let \(x_{0}\in\mathbb{C}^{n}\). There exist \(\beta_{1}>0\), \(\beta_{2}>0\), \(\gamma>0\) and \(\mu>0\) such that \[\|H(x_{0})\|\leq\beta_{1},\ \|S(x_{0})\|\leq\beta_{2},\ \|F^{\prime}(x_{0})^{-1}\|\leq \gamma,\ \|F(x_{0})\|\leq\mu.\]
3. There exist \(v_{1}:[0,+\infty)\to\mathbb{R}\), \(v_{2}:[0,+\infty)\to\mathbb{R}\), continuous and nondecreasing functions with \(v_{1}(0)=v_{2}(0)=0\) such that, for each \(x,y\in D\), \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\] \[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\] Define functions w and v by \(w(t)=w_{1}(t)+w_{2}(t)\) and \(v(t)=v_{1}(t)+v_{2}(t)\). Let \(r_{0}=\sup\{t\geq 0:\gamma v(t)<1\}\) and set \[D_{0}=D\cap U(x_{0},r_{0}).\]
4. There exist \(w_{1}:[0,+\infty)\to\mathbb{R}\), \(w_{2}:[0,+\infty)\to\mathbb{R}\), continuous and nondecreasing functions with \(w_{1}(0)=w_{2}(0)=0\) such that, for each \(x,y\in D_{0}\), \[\|H(x)-H(y)\|\leq w_{1}(\|x-y\|),\] \[\|S(x)-S(y)\|\leq w_{2}(\|x-y\|).\]  
We need the following auxiliary results for the semilocal convergence analysis that follows. **Lemma 1**. : _Under the (H) hypotheses, the following items hold for each \(x,y\in D_{0}\):_  
\[\|F^{\prime}(x)-F^{\prime}(y)\|\leq w(\|x-y\|), \tag{3}\]  
\[\|F^{\prime}(x)-F^{\prime}(x_{0})\|\leq v(\|x-y\|), \tag{4}\]  
\[\|F^{\prime}(x)\|\leq v(\|x-y\|)+\beta_{1}+\beta_{2}, \tag{5}\]  
\[\|F^{\prime}(x)-F(y)-F^{\prime}(y)(x-y)\|\leq\int_{0}^{1}w(\|x-y\|\xi)d\xi\|x -y\|, \tag{6}\]  
_and_  
\[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]0  
Proof. : By hypothesis \((H_{3})\) and \(F^{\prime}(x)=H(x)+S(x)\), we have that  
\[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]1 \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]2 \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]3  
and by \((H_{2})\)  
\[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]4 \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]5 \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]6  
which show inequalities (3) and (4), respectively. Then, we get, by \((H_{1})\) and \((H_{3})\),  
\[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]7 \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]8 \[\|H(x)-H(x_{0})\|\leq v_{1}(\|x-x_{0}\|),\]9  
which shows inequality (5). Using \((H_{3})\), we obtain that  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]0 \[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]1  
which shows inequality (6). By \((H_{1})\), \((H_{2})\) and inequality (4), we get, in turn, that for \(x\in D_{0}\) :  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]2  
It follows from inequality (8) and the Banach lemma on invertible operators [4] that \(F^{\prime}(x)^{-1}\) exists so that inequality (7) is satisfied. We shall define some scalar functions and parameters to be used in the semilocal convergence analysis. Let \(t_{0}=0\) and \(s_{0}^{(1)}=(1+\eta)\gamma\mu\). Define scalar sequences \(\{t_{k}\}\), \(\{s_{k}^{(1)}\},\ldots,\{s_{k}^{(m-1)}\}\) by the following schemes:  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]3\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]4  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]5  
Moreover, define functions \(q\) and \(h_{q}\) on the interval \([0,r_{0})\) by  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]6  
and  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]7  
We have that \(h_{q}(0)=\eta-1<0\) and \(h_{q}\to\infty\) as \(t\to r_{0}^{-}\). It follows from the intermediate value theorem that function \(h_{q}\) has zeros in interval \((0,r_{0})\). Denote by \(r_{q}\) the smallest such zero. Then, we have that for each \(t\in[0,r_{0})\)  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]8  
**Lemma 2**. : _Suppose that equation_  
\[\|S(x)-S(x_{0})\|\leq v_{2}(\|x-x_{0}\|).\]9  
_has zeros in interval \((0,r_{q})\). Denote by \(r\) the smallest such zero. Then, sequence \(\{t_{k}\}\), generated by Equation (9) is nondecreasing, bounded from above by \(r_{q}\) and converges to its unique least upper bound \(r^{*}\), which satisfies_  
\[D_{0}=D\cap U(x_{0},r_{0}).\]0  
Proof.