# On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space
## 3 Convergence  
The following common set of conditions is sufficient for the convergence of these methods. Suppose:  
\((C_{1})\) There exist a starting point \(x_{0}\in\Omega\) and a parameter \(\xi\geq 0\) such that \(F^{\prime}(x_{0})^{-1}\in L(Y,X)\) and  
\[\|F^{\prime}(x_{0})^{-1}F(x_{0})\|\leq\xi.\]  
\((C_{2})\)\(\|F^{\prime}(x_{0})^{-1}(F^{\prime}(v)-F^{\prime}(x_{0}))\|\leq w_{0}(\|v-x_{0}\|)\) for all \(v\in\Omega\), where the function \(w_{0}:[0,\infty)\to[0,\infty)\) is continuous and nondecreasing. \((C_{3})\) Equation \(w_{0}(t)-1=0\) has a smallest positive solution \(\rho\). Set \(T=[0,\rho)\) and \(\Omega_{0}=B(x_{0},\rho)\cap\Omega\). \((C_{4})\)\(\|F^{\prime}(x_{0})^{-1}(F^{\prime}(v_{2})-F^{\prime}(v_{1})\|\leq w(\|v_{2}-v_{1} \|)\) for all \(v_{2},v_{1}\in\Omega_{0}\), where the function \(w:T\to[0,\infty)\) is continuous and nondecreasing. and  
\((C_{6})\)\(B[x_{0},t_{*}]\subset\Omega\). Next, the semi-local convergence is given first for method (2). **Theorem 1**. : _Suppose that the conditions \((C_{1})\)-\((C_{6})\) hold. Then, the sequence \(\{x_{n}\}\) generated by the Formula (2) is well defined in the ball \(B(x_{0},t_{*})\), stays in the ball \(B(x_{0},t_{*})\) and converges to a limit point \(x_{*}\in B[x_{0},t_{*}]\) satisfying \(t\)\(F(x_{*})=0\). Moreover, the solution \(x_{*}\) relates to the method (2) and the sequence \(\{t_{n}\}\) by_  
\[\|x_{n}-x_{*}\|\leq t_{*}-t_{n}\quad for\ all\quad n=0,1,2,\ldots. \tag{9}\]  
Proof. : The following items shall be shown using mathematical induction on the number \(k\):  
\[\|y_{k}-x_{k}\| \leq s_{k}-t_{k}, \tag{10}\] \[\|z_{k}-y_{k}\| \leq u_{k}-s_{k},\] (11) \[\|x_{k+1}-z_{k}\| \leq t_{k+1}-u_{k}. \tag{12}\]  
Item (10) holds for \(k=0\), since by the condition \((C_{1})\), the definition of the method (2) and the sequence (5)  
\[\|y_{0}-x_{0}\|=|\alpha|\|\|F^{\prime}(x_{0})^{-1}F(x_{0})\|\leq|\alpha|\xi=( s_{0}-t_{0})<t_{*}.\]  
Notice also that the iterates \(y_{0},z_{0}\) and \(x_{1}\) are well defined and \(y_{0}\in B(x_{0},t_{*})\). Then, for \(v\in B(x_{0},t_{*})\), conditions (7), \((C_{1})\) and \((C_{2})\) give  
\[\|F^{\prime}(x_{0})^{-1}(F^{\prime}(v)-F^{\prime}(x_{0}))\|\leq w_{0}(\|v-x_{ 0}\|)<1.\]  
This estimate together with the standard lemma by [PERSON] on linear operator [2] implies that  
\[\|F^{\prime}(v)^{-1}F^{\prime}(x_{0})\|\leq\frac{1}{1-w_{0}(\|v-x_{0}\|)}. \tag{13}\]By replacing the value of \(y_{k}\) given in the first substep in the second substep of the method (2), we have  
\[z_{k}-y_{k} = (\alpha I-\frac{23}{8}I+3A_{k}-\frac{9}{8}A_{k}^{2})F^{\prime}(x_{k })^{-1}F(x_{k}) \tag{14}\] \[= -\frac{1}{8\alpha}[8(\alpha-1)I-6(I-A_{k})-9(I-A_{k})^{2}](y_{k}- x_{k}).\]  
In view of the definition of the sequence (5), condition \((C_{3})\), (13) (for \(v=x_{k}\)) and the identity (14), we obtain the estimate  
\[\|x_{n}-x_{*}\|\leq t_{*}-t_{n}\quad for\ all\quad n=0,1,2,\ldots. \tag{9}\]0 \[\|x_{n}-x_{*}\|\leq t_{*}-t_{n}\quad for\ all\quad n=0,1,2,\ldots. \tag{9}\]1  
where  
\[\tilde{b}_{k}=\left\{ \begin{array}{l}w_{0}(\|x_{k}-x_{0}\|)+w_{0}(\|y_{k}-x_ {0}\|)\\ or\\ w(\|y_{k}-x_{k}\|)\end{array} \right.\]  
and \(\tilde{\gamma}_{k}=\frac{\tilde{b}_{k}}{1-w_{0}(\|x_{k}-x_{0}\|)}\).