# On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space
## 3 Convergence
Method (16) also gives  
\[\leq \|F^{\prime}(x_{k})^{-1}(F^{\prime}(x_{k})-F^{\prime}(x_{0}))\|+ \|F^{\prime}(x_{k})^{-1}(F^{\prime}(y_{k})-F^{\prime}(x_{0}))\|\]4 \[\leq \|F^{\prime}(x_{k})^{-1}(F^{\prime}(x_{k})-F^{\prime}(x_{0}))\|+ \|F^{\prime}(x_{k})^{-1}(F^{\prime}(y_{k})-F^{\prime}(x_{0}))\|\]5  
leading to  
\[\leq \|F^{\prime}(x_{k})^{-1}(F^{\prime}(x_{k})-F^{\prime}(x_{0}))\|+ \|F^{\prime}(x_{k})^{-1}(F^{\prime}(y_{k})-F^{\prime}(x_{0}))\|\]6 \[\leq \|F^{\prime}(x_{k})^{-1}(F^{\prime}(x_{k})-F^{\prime}(x_{0}))\|+ \|F^{\prime}(x_{k})^{-1}(F^{\prime}(y_{k})-F^{\prime}(x_{0}))\|\]7 \[\leq \|F^{\prime}(x_{k})^{-1}(F^{\prime}(x_{k})-F^{\prime}(x_{0}))\|+ \|F^{\prime}(x_{k})^{-1}(F^{\prime}(y_{k})-F^{\prime}(x_{0}))\|\]8 \[\leq \|F^{\prime}(x_{k})^{-1}(F^{\prime}(x_{k})-F^{\prime}(x_{0}))\|+ \|F^{\prime}(x_{k})^{-1}(F^{\prime}(y_{k})-F^{\prime}(x_{0}))\|\]9  
since  
\[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]0 \[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]1  
On the other hand, we can write  
\[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]2 \[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]3  
and  
\[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]4 \[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]5 \[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]6 \[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]7 \[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]8  
Then, by the first substep of the method (2)  
\[\leq \frac{w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_{0}\|)}{1-w_{0}(\|x_ {k}-x_{0}\|)}=\tilde{\gamma}_{k}\]9 \[\leq \frac{w_{0}(t_{k})+w_{0}(s_{k})}{1-w_{0}(t_{k})}=\gamma_{k},\]0and \(\|y_{k+1}-x_{0}\|\leq\|y_{k+1}-x_{k+1}\|+\|x_{k+1}-x_{0}\|\leq s_{k+1}-t_{k+1}+t_{ k+1}=s_{k+1}<t_{*}\). It follows that the item (10) hold for \(k+1\) replacing \(k\) and \(y_{k+1}\in B(x_{0},t_{*})\). Then, the induction for the items (10)-(12) is completed. The condition \((C_{5})\) implies that the sequence \(\{t_{k}\}\) is Cauchy as convergent. Consequently, the sequence \(\{x^{\prime}_{k}\}\) is also Cauchy by estimates (10)-(12), and as such it is convergent to some limit point \(x_{*}\in B[x_{0},t_{*}]\). Furthermore, the continuity of the operator \(F\) and (18) imply \(F(x_{*})=0\) if \(k\to\infty\). Let \(m\geq 0\). Then, by (10)-(12) the following can be written in turn  
\[\leq \frac{w_{0}(t_{k})+w_{0}(s_{k})}{1-w_{0}(t_{k})}=\gamma_{k},\]1 \[\leq \frac{w_{0}(t_{k})+w_{0}(s_{k})}{1-w_{0}(t_{k})}=\gamma_{k},\]2 \[\leq \frac{w_{0}(t_{k})+w_{0}(s_{k})}{1-w_{0}(t_{k})}=\gamma_{k},\]3  
By letting \(m\to\infty\) in the estimate (21), the item (9) follows. **Remark 1**. : _The parameter \(\rho\) can replace the limit point \(t_{*}\) in the condition \((C_{6})\) or \(\tau\) in the condition (8)._  
The next result discusses the location and the uniqueness of a solution for the equation \(F(x)=0\). **Proposition 1**. : _Suppose: \((i)\) The exists a solution \(\bar{x}\in B(x_{0},\rho_{1})\) of the equation \(F(x)=0\) for some parameter \(\rho_{1}>0\)._  
\((ii)\) _The condition \((C_{2})\) hold._  
\((iii)\) _For \(\rho_{2}\geq\rho_{1}\)_  
\[\leq \frac{w_{0}(t_{k})+w_{0}(s_{k})}{1-w_{0}(t_{k})}=\gamma_{k},\]4  
_Set \(\Omega_{1}=B(x_{0},\rho_{2})\cap\Omega\)._  
_Then, the equation \(F(x)=0\) is uniquely solved by \(\bar{x}\) in the region \(\Omega_{1}\)._  
Proof. : Let \(M=\int_{0}^{1}F^{\prime}(\bar{x}+\theta(\bar{y}-\bar{x}))d\theta\) for some \(\bar{y}\in\Omega_{1}\) with \(F(\bar{y})=0\).