## 3 Semilocal Convergence
To achieve this, notice that:  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]6  
Then, it follows from Lemma 2.4 that there exists \(q\leq r_{n}\) such that  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]7  
By (I\({}_{j}\)) it follows that  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]8  
Hence, \(x_{n}\in U\left(x_{0},r\right)\subset D\) and by (H\({}_{1}\)) \(M\) is a bound for \(I-A\left(x_{n}\right)\)  
We can write by (H\({}_{2}\)) that  
\[\Leftrightarrow y_{n}=T\left(y_{n}\right)=\left(I-A\left(x_{n} \right)\right)y_{n}-F\left(x_{n}\right)\]9  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]0  
It follows from Equations (3.3) and (3.4) that  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]1  
By Lemma 2.5, \(x_{n+1}\) is well-defined and \(a_{n}\leq q\leq r_{n}\). In view of the definition of \(r_{n+1}\) we have that  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]2  
so that by Lemma 2.4, \(r_{n+1}\) is well-defined and  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]3  
which proves (I\({}_{n+1}\)). The induction for (I\({}_{n}\)) is complete. Let \(m\geq n\), then we obtain in turn that  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]4  
Moreover, we get inductively the estimate  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]5  
It follows from (H\({}_{5}\)) that \(\left\{r_{n}\right\}\) is a null-sequence. Hence, \(\left\{x_{n}\right\}\) is a complete sequence in a Banach space \(X\) by Equation (3.5) and as such it converges to some \(x^{*}\in X\). By letting \(m\rightarrow\infty\) in Equation (3.5) we deduce that \(x^{*}\in U\left(x_{n},r_{n}\right)\). Furthermore, Equation (3.4) shows that \(x^{*}\) is a zero of \(F\). Hence, (C\({}_{1}\)) and (C\({}_{2}\)) are proved. In view of the estimate  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]6  
the a priori bound of (C\({}_{3}\)) is well-defined by Lemma 2.4. That is, \(s_{n}\) is smaller in general than \(r_{n}\). The conditions of Theorem 3.1 are satisfied for \(x_{n}\) replacing \(x_{0}\). A solution of the inequality of (C\({}_{2}\)) is given by \(s_{n}\)--see Equation (3.4). It follows from Equation (3.5) that the conditions of Theorem 3.1 are easily verified. Then, it follows from (C\({}_{1}\)) that \(x^{*}\in U\left(x_{n},s_{n}\right)\), which proves (C\({}_{3}\)). In general, the a posterior estimate is of interest. Then, condition (H\({}_{5}\)) can be avoided as follows:  
**Proposition 3.2**. : _Suppose: condition (H\({}_{1}\)) of Theorem 3.1 is true._  
_(H\({}_{3}^{\prime}\)) There exists \(s\in K\), \(\theta\in\left(0,1\right)\) such that_  
\[U\left(x_{0},r\right)=\left\{x\in X:/x-x_{0}/\leq r\right\}\]7  
_(H\({}_{4}^{\prime}\)) \(U\left(x_{0},s\right)\subset D\)._  
_Then, there exists \(r\leq s\) satisfying the conditions of Theorem 3.1. Moreover, the zero \(x^{\ast}\) of \(F\) is unique in \(U\left(x_{0},s\right)\)._  
**Remark 3.3**. : _(i) Notice that by Lemma 2.4\(R_{n}^{\infty}\left(0\right)\) is the smallest solution of \(R_{n}\left(s\right)\leq s\). Hence any solution of this inequality yields on the upper estimate for \(R_{n}^{\infty}\left(0\right)\). Similar inequalities appear in (H\({}_{2}\)) and (H\({}_{2}^{\prime}\))._  
_(ii) The weak assumptions of Theorem 3.1 do not imply the existence of \(A\left(x_{n}\right)^{-1}\). In practice, the computation of \(T_{n}^{\infty}\left(0\right)\) as a solution of a linear equation is of no problem, and the computation of the expensive or impossible to compute in general \(A\left(x_{n}\right)^{-1}\) is not needed._  
_(iii) We can use the following result for the computation of the a posteriori estimates. The proof can be found in [14, Lemma 4.2] by simply exchanging the definitions of \(R\)._  
**Lemma 3.4**. : _Suppose that the conditions of Theorem 3.1 are satisfied.