## 2 Semilocal Convergence Using Centered Hypotheses
* A bound for \(||\Gamma_{1}^{-1}([x_{1},x_{0};F]-\Phi_{0})||\) From \[x_{n+1} = x_{n}+F^{{}^{\prime}}(x_{n})^{-1}(F(x_{n})-F(y_{n})) \tag{9}\]6 \[x_{n+1} = x_{n}+F^{{}^{\prime}}(x_{n})^{-1}(F(x_{n})-F(y_{n})) \tag{9}\]7 \[x_{n+1} = x_{n}+F^{{}^{\prime}}(x_{n})^{-1}(F(x_{n})-F(y_{n})) \tag{9}\]8 we obtain \[x_{n+1} = x_{n}+F^{{}^{\prime}}(x_{n})^{-1}(F(x_{n})-F(y_{n})) \tag{9}\]9 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]0 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]1 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]2 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]3
* A bound for \(||\Phi_{1}^{-1}\Gamma_{1}||\) First, note that \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]4 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]5 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]6 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]7 Besides, we have \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]8 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](x_{n}-y_{n}))\]9 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]0Thus, we get  
\[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]1 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]2 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]3  
and  
\[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]4 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]5  
Finally  
\[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]6 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]7 \[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]8  
and therefore  
\[= x_{n}+F^{{}^{\prime}}(x_{n})^{-1}([x_{n},y_{n};F](-F^{{}^{\prime }}(x_{n})^{-1}F(x_{n})))\]9 \[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]0  
On the other hand, the relation  
\[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]1  
is equivalent to  
\[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]2  
By definition \(\omega_{0}(x,y)\leq\omega(x,y)\) then  
\[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]3  
Moreover  
\[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]4  
since  
\[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]5Therefore, \[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]6 \[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]7 \[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]8 \[= x_{n}-F^{{}^{\prime}}(x_{n})^{-1}[x_{n},y_{n};F]F^{{}^{\prime}}( x_{n})^{-1}F(x_{n})\]9 \[\Gamma_{n} = F^{{}^{\prime}}(x_{n})\]0 Then, using \((12)\) and \(M<1\), we obtain \[\Gamma_{n} = F^{{}^{\prime}}(x_{n})\]1 Thus, \(x_{2}\in B(x_{0},R)\). By using the same arguments together with an induction strategy we can prove the following facts:  
* \(||x_{n}-x_{0}||\leq\sum_{k=0}^{n-1}M^{k}\eta<R,\) that is, \(x_{n}\in B(x_{0},R)\)
* From the estimate \[\Gamma_{n} = F^{{}^{\prime}}(x_{n})\]2 we conclude that \(\{x_{n}\}\) is a Cauchy sequence, and hence it converges to some \(x^{*}\in B(x_{0},R)\). * Since \[\Gamma_{n} = F^{{}^{\prime}}(x_{n})\]3 and \(||x_{n}-x_{n-1}||\to 0\) when \(n\to+\infty\), we obtain that \(F(x^{*})=0\). Let us remark that, by \((10)\), \(||\Gamma_{n}||\leq||\Gamma_{0}||+\omega_{0}(R,R)\).