# Expanding the Applicability of Some High Order Householder-Like Methods
## 2 Semilocal Convergence
Then, the following assertion holds:_ \[f_{1}(\bar{t}_{n+1})=\left(\frac{1}{2}f_{1}^{\prime\prime}(\bar{ t}_{n})\frac{A_{0}(f_{1}\bar{t}_{n})}{f_{1}^{\prime}(\bar{t}_{n})^{2}}-1 \right)A_{0}(f_{1};\bar{t}_{n})\] \[+f_{1}^{\prime\prime}(\bar{t}_{n})\frac{A_{0}(f_{1};\bar{t}_{n}) }{f_{1}^{\prime}(\bar{t}_{n})}\left(1+\frac{\bar{h}(f_{1};\bar{t}_{n})}{f_{1}^ {\prime}(\bar{t}_{n})}\right)\frac{f_{1}(\bar{t}_{n})}{f_{1}^{\prime}(\bar{t}_ {n})}\] \[+\frac{1}{2}f_{1}^{\prime\prime}(\bar{t}_{n})\left(\frac{\bar{h}( f_{1}(\bar{t}_{n})}{f_{1}^{\prime}(\bar{t}_{n})}\right)^{2}\] \[+\frac{1}{2}f_{1}^{\prime\prime}(\bar{t}_{n})\left(\frac{\bar{h}( f_{1};\bar{t}_{n})f_{1}(\bar{t}_{n})}{f_{1}^{\prime}(\bar{t}_{n})}\right)^{2}\] \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]0 \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]1 _where:_ \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]2 _and_ \(A_{k}(f_{1};-):\mathbb{R}\to\mathbb{R}\) _are real differentiable functions._
2. _If_ \(F\) _has a continuous third-order Frechet-derivative on_ \(D\)_, then the following assertion holds:_ \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]3 \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]4 \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]5 \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]6 \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]7 \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]8 _where:_ \[\alpha_{0}\leq 3-2\sqrt{2} \tag{16}\]9 _where:_ \[\beta\leq\bar{t}^{*}\leq\left(1+\frac{1}{\sqrt{2}}\right)\beta\leq\left(1- \frac{1}{\sqrt{2}}\right)\frac{1}{\delta}\leq\bar{t}^{**}\leq\frac{1}{2\delta}.\]0
2. \(\|\Gamma_{0}F^{(m-1)}(x)\|\leq f_{1}^{(m-1)}(\|x-x_{0}\|),\quad m=3\quad\text {or}\quad m=5\)_._  
Proof. :
1. See Remark 1. 2. Simply replace function \(f\) by \(f_{1}\) in Lemma 4 in [4]. **Lemma 4**. : _Suppose \(F\) satisfies the \(\gamma\)-condition on \(D\). Then, the following items hold:_  
1. \(F\) _satisfies the_ \(\gamma_{0}\)_-condition on_ \(D\) _and the_ \(\delta\)_-_\(\gamma\)_-condition on_ \(D_{0}\)_;_
2. \(\gamma_{0}\leq\gamma\)_;_
3. \(\delta\leq\gamma\)_;_  
_Moreover, if \(\alpha\leq 3-2\sqrt{2}\), then \(\bar{\alpha}_{0}=b_{0}\gamma\leq 3-2\sqrt{2}\), \(\alpha_{0}\leq 3-2\sqrt{2}\),_  
\[\beta\leq\bar{t}^{*}\leq\left(1+\frac{1}{\sqrt{2}}\right)\beta\leq\left(1- \frac{1}{\sqrt{2}}\right)\frac{1}{\delta}\leq\bar{t}^{**}\leq\frac{1}{2\delta}.\]1  
_and:_  
\[\beta\leq\bar{t}^{*}\leq\left(1+\frac{1}{\sqrt{2}}\right)\beta\leq\left(1- \frac{1}{\sqrt{2}}\right)\frac{1}{\delta}\leq\bar{t}^{**}\leq\frac{1}{2\delta}.\]2  
_Furthermore, (15) supposes that \(\gamma_{0}\leq\delta\) and from (18); we have:_  
\[\beta\leq\bar{t}^{*}\leq\left(1+\frac{1}{\sqrt{2}}\right)\beta\leq\left(1- \frac{1}{\sqrt{2}}\right)\frac{1}{\delta}\leq\bar{t}^{**}\leq\frac{1}{2\delta}.\]3  
Proof. : The assertions hold from Lemma 3, (18), the definition of functions \(f_{0},f_{1},f\) and the definition of \(\gamma_{0}\), \(\gamma\) and \(\delta\)-condition. **Theorem 2**. : _Suppose that the operator \(F\) satisfies the conditions of Lemma 1, 2, 3, (9), (18), and \(A_{0}(f_{1};t)>=0\) hold. Then, sequences \(\{x_{n}\}\), \(\{y_{n}\}\), generated by the method (7) are well defined in \(U(x_{0},\overline{t}^{*})\), remain in \(\overline{U}(x_{0},\overline{t}^{*})\) for each \(n=0,1,2,\ldots\), and converge to a solution \(\overline{x}^{*}\in\overline{U}(x_{0},\overline{t}^{*})\) of equation \(F(x)=0\). The limit point \(\overline{x}^{*}\) is the only solution of equation \(F(x)=0\) in \(U(x_{0},R_{0})\). Moreover, the following estimates are true for each \(n=0,1,2,\ldots\):_  
\[ \begin{array}{l}\|y_{n}-x_{n}\|\leq\bar{s}_{n}-\bar{t}_{n},\\ \|x_{n+1}-y_{n}\|\leq\bar{t}_{n+1}-\bar{s}_{n},\\ \|y_{n}-x^{*}\|\leq\overline{t}^{*}-\bar{s}_{n},\\ \|x_{n}-x^{*}\|\leq\overline{t}^{*}-t_{n},\end{array}  \tag{19}\]  
_where \(\overline{t}^{*}=\lim_{n}\bar{t}_{n}\)._  
Proof. : We shall show the estimates (19). Using mathematical induction as a consequence of the following recurrence relations:  
\[(M_{k}^{1})\quad\quad x_{k}\in\overline{U}(x_{0},\bar{t}_{k});\]\[