## 3 Semilocal Convergence  
Let \(\left(X,\left(E,K,\left\|\cdot\right\|\right),\left/\cdot\right/\right)\) and \(Y\) be generalized Banach spaces, \(D\subset X\) an open subset, \(G:D\to Y\) a continuous operator and \(A\left(\cdot\right):D\to L\left(X,Y\right)\). A zero of operator \(G\) is to be determined by a Newton-type method starting at a point \(x_{0}\in D\). The results are presented for an operator \(F=JG\), where \(J\in L\left(Y,X\right)\). The iterating elements are determined through a fixed point problem:  
\[x_{n+1}=x_{n}+y_{n},\text{ }A\left(x_{n}\right)y_{n}+F\left(x_{n} \right)=0 \tag{3.1}\] \[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]  
Let \(U\left(x_{0},r\right)\) stand for the ball defined by  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]for some \(r\in K\). Next, we present the semilocal convergence analysis of Newton-type method Equation (3.1) using the preceding notation. **Theorem 3.1**. : _Let \(F:D\subset X\), \(A\left(\cdot\right):D\to L\left(X,Y\right)\) and \(x_{0}\in D\) be as defined previously. Suppose:_  
_(\(H_{1}\)) There exists an operator \(M\in B\left(I-A\left(x\right)\right)\) for each \(x\in D\)._  
_(\(H_{2}\)) There exists an operator \(N\in L_{+}\left(E,E\right)\) satisfying for each \(x,y\in D\)_  
\[/F\left(y\right)-F\left(x\right)-A\left(x\right)\left(y-x\right)/\leq N\,/y-x/\]  
_(\(H_{3}\)) There exists a solution \(r\in K\) of_  
\[R_{0}\left(t\right):=\left(M+N\right)t+/F\left(x_{0}\right)/\leq t\]  
_(\(H_{4}\)) \(U\left(x_{0},r\right)\subseteq D\)._  
_(\(H_{5}\)) \(\left(M+N\right)^{k}r\to 0\) as \(k\rightarrow\infty\)._  
_Then, the following hold:_  
_(\(C_{1}\)) The sequence \(\left\{x_{n}\right\}\) defined by_  
\[x_{n+1}=x_{n}+T_{n}^{\infty}\left(0\right)\text{, }\text{ }T_{n}\left(y\right):= \left(I-A\left(x_{n}\right)\right)y-F\left(x_{n}\right) \tag{3.2}\]  
_is well-defined, remains in \(U\left(x_{0},r\right)\) for each \(n=0,1,2,...\) and converges to the unique zero of operator \(F\) in \(U\left(x_{0},r\right)\)._  
_(\(C_{2}\)) An a priori bound is given by the null-sequence \(\left\{r_{n}\right\}\) defined by \(r_{0}:=r\) and for each \(n=1,2,...\)_  
\[r_{n}=P_{n}^{\infty}\left(0\right)\text{, }\text{ }P_{n}\left(t\right)=Mt+Nr_{n-1}\]  
_(\(C_{3}\)) An a posteriori bound is given by the sequence \(\left\{s_{n}\right\}\) defined by_  
\[s_{n}=R_{n}^{\infty}\left(0\right)\text{, }\text{ }R_{n}\left(t\right)=\left(M+N \right)t+Na_{n-1}\]  
\[b_{n}:=/x_{n}-x_{0}/\leq r-r_{n}\leq r\]  
_where_  
\[a_{n-1}=/x_{n}-x_{n-1}/\text{ \ for each }n=1,2,...\]  
Proof. : Let us define for each \(n\in\mathbb{N}\) the statement:  
(I\({}_{n}\)) \(x_{n}\in X\) and \(r_{n}\in K\) are well-defined and satisfy  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]0  
We use induction to show (I\({}_{n}\)). The statement (I\({}_{1}\)) is true: By Lemma 2.4, (H\({}_{3}\)) and (H\({}_{5}\)), there exists \(q\leq r\) such that:  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]1  
Hence, by Lemma 2.5\(x_{1}\) is well-defined and we have \(a_{0}\leq q\). Then, we get the estimate  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]2  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]3  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]4It follows with Lemma 2.4 that \(r_{1}\) is well-defined and  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]5  
Suppose that (I\({}_{j}\)) is true for each \(j=1,2,...,n\). We need to show the existence of \(x_{n+1}\) and obtain a bound \(q\) for \(a_{n}\).