## 2 Local Convergence
** We shall show estimate Equations (18)-(20) using mathematical induction. By hypothesis \(x_{0}\in U(x^{*},r)-\{x^{*}\}\), Equations (4) and (12), we get:  
\[h_{p}(t) =p(t)-1,\]7  
It follows from the Equation (21) and the Banach lemma on invertible operators [2,3,14] that \(F^{\prime}(x_{0})\
eq 0\) and:  
\[h_{p}(t) =p(t)-1,\]8  
Hence, \(y_{0}\) is well defined by the first sub-step of Method (3) for \(n=0\). Then, we have by Equations (4), (5), (11), (13) and (22) that:  
\[h_{p}(t) =p(t)-1,\]9 \[r_{1}=\frac{2}{2L_{0}+L}.\]0 \[r_{1}=\frac{2}{2L_{0}+L}.\]1  
which shows Equation (18) for \(n=0\) and \(y_{0}\in U(x^{*},\ r)\). We can write by Equation (11) that:  
\[r_{1}=\frac{2}{2L_{0}+L}.\]2  
Notice that \(|x^{*}+\theta(x_{0}-x^{*})-x_{0}^{*}|=\theta|x_{0}-x^{*}|<r\); hence, \(x^{*}+\theta(x_{o}-x^{*})\in U(x^{*},\ r)\). Then, by Equations (14) and (24), we obtain that:  
\[r_{1}=\frac{2}{2L_{0}+L}.\]3  
We also get that:  
\[r_{1}=\frac{2}{2L_{0}+L}.\]4 \[r_{1}=\frac{2}{2L_{0}+L}.\]5  
It follows that \(x_{1}\) is well defined by the third sub-step of Method (2) for \(n=0\). Then, it follows from Equations (4), (10), (22), (25) (\(for\ x_{0}=z_{0}\)), (29) and (32) that:  
\[ \begin{split}|x_{1}-x^{*}|&\leq|z_{0}-x^{*}|+|A_{0} ^{-1}F^{\prime}(x^{*})||F^{\prime}(x^{*})^{-1}F(z_{0})|\\ &\leq|z_{0}-x^{*}|+\frac{M|z_{0}-x^{*}|}{1-q(|x_{0}-x^{*}|)}\\ &\leq\left(1+\frac{M}{1-q(|x_{0}-x^{*}|)}\right)|z_{0}-x^{*}|\\ &\leq\left(1+\frac{M}{1-q(|x_{0}-x^{*}|)}\right)g_{2}(|x_{0}-x^{ *}|)|x_{0}-x^{*}|\\ &\leq g_{3}(|x_{0}-x^{*}|)|x_{0}-x^{*}|\\ &<|x_{0}-x^{*}|<r,\end{split}  \tag{33}\]  
which shows Equation (20) for \(n=0\) and \(x_{1}\in U(x^{*},\ \ r)\). By simply replacing \(x_{0},\ y_{0},\ z_{0},\ x_{1}\) by \(x_{k},\ y_{k},\ z_{k},\ x_{k+1}\) in the preceding estimates, we arrive at Equations (18)-(20). Using the estimates \(\|x_{k+1}-x^{*}\|<\|x_{k}-x^{*}\|<r\), we deduce that \(\lim\limits_{k\to\infty}x_{k}=x^{*}\) and \(x_{k+1}\in U(x^{*},\ r)\). Finally, to show the uniqueness part, let \(Q=\int_{0}^{1}F^{\prime}(y^{*}+\theta(x^{*}-y^{*}))d\theta\) for some \(y^{*}\in\bar{U}(x^{*},T)\) with \(F(y^{*})=0\). Using Equation (12), we get that:  
\[ \begin{split}&\|F^{\prime}(x^{*})^{-1}(Q-F^{\prime}(x^{*}))\|\leq \|\int_{0}^{1}L_{0}|y^{*}+\theta(x^{*}-y^{*})-x^{*}\|d\theta\\ &\leq\int_{0}^{1}(1-t)\|y^{*}-x^{*}\|d\theta\leq\frac{L_{0}}{2}T <1.\end{split}  \tag{34}\]  
It follows from Equation (34) that \(Q\) is invertible. Then, in view of the identity \(0=F(x^{*})-F(y^{*})=Q(x^{*}-y^{*})\), we conclude that \(x^{*}=y^{*}\). **Remark 1**. :
1. _In view of Equation (_12_) and the estimate:_ \[ \begin{split}|F^{\prime}(x^{*})^{-1}F^{\prime}(x)|& = |F^{\prime}(x^{*})^{-1}(F^{\prime}(x)-F^{\prime}(x^{*}))+I|\\ &\leq 1+|F^{\prime}(x^{*})^{-1}(F^{\prime}(x)-F^{\prime}(x^{*}))|\\ &\leq 1+L_{0}|x_{0}-x^{*}|\end{split} \] _condition Equation (_14_) can be dropped, and_ \(M\) _can be replaced by:_ \[M(t)=1+L_{0}t\] _or by_ \(M(t)=M=2\)_, since_ \(t\in[0,\frac{1}{L_{0}})\)_._
2. _The results obtained here can be used for operators_ \(F\) _satisfying the autonomous differential equation_ _[_2,3_]_ _of the form:_ \[F^{\prime}(x)=P(F(x)),\] _where_ \(P\) _is a known continuous operator. Since_ \(F^{\prime}(x^{*})=P(F(x^{*}))=P(0)\)_, we can apply the results without actually knowing the solution_ \(x^{*}\)_. Let, as an example,_ \(F(x)=e^{x}-1\)_. Then, we can choose_ \(P(x)=x+1\)_
3. _The radius_ \(r_{1}\) _was shown in_ _[_2,3_]_ _to be the convergence radius for [PERSON]'s method Equation (_2_) under conditions Equations (_11_) and (_13_). It follows from Equation (_4_) and the definition of_ \(r_{1}\) _that the convergence radius_ \(r\) _of Method (_3_) cannot be larger than the convergence radius_ \(r_{1}\) _of the second order [PERSON]'s method (_2_).