# Expanding the Applicability of Some High Order Householder-Like Methods
## 2 Semilocal Convergence  
We consider Equation (1), where \(F\) is a nonlinear operator defined in a non-empty open convex subset \(D\) of a Banach space \(B_{1}\) with values in another Banach space \(B_{2}\). We need the definition of the \(\gamma\)-condition given in [7]. **Definition 1**. : _Suppose that \(F:D\to B_{2}\) is five-times Frechet-differentiable. Let \(\beta>0\) and \(x_{0}\in D\). We say that \(F\) satisfies the \(\gamma\)-condition, if \(\Gamma_{0}=F^{{}^{\prime}}(x_{0})^{-1}\in\mathcal{L}(B_{2},B_{1})\) and for each \(x\in D\):_  
\[\|\Gamma_{0}F(x_{0})\| \leq \beta,\] \[\|\Gamma_{0}F^{{}^{\prime\prime}}(x_{0})\| \leq \frac{2\gamma}{(1-\gamma\|x-x_{0}\|)^{2}},\] \[\|\Gamma_{0}F^{{}^{\prime\prime}}(x)\| \leq \frac{6\gamma^{2}}{(1-\gamma\|x-x_{0}\|)^{4}}=f^{{}^{\prime \prime}}(\|x-x_{0}\|),\] \[\|\Gamma_{0}F^{(iv)}(x_{0})\| \leq 24\gamma^{3},\] \[\|\Gamma_{0}F^{(v)}(x)\| \leq \frac{120\gamma^{4}}{(1-\gamma\|x-x_{0}\|)^{6}}=f^{(v)}(\|x-x_{0} \|),\]  
_where:_  
\[f(t)=\beta-t+\frac{\gamma t^{2}}{1-\gamma t}. \tag{8}\]  
Then, the following semilocal convergence result for the method defined in (7) under \(\gamma\)-type conditions was given in [4], Theorem 1:  
**Theorem 1**. : _Assume that the operator \(F\) satisfies for each \(x\in D\):_  
\[\|[F^{\prime}(x)]^{-1}A_{k}(F;x)\|\leq k^{p},\,\forall k\geq 2,\ldots,p>0\qquad \text{and}\qquad\|[F^{\prime}(x)]^{-1}F(x)\|<1, \tag{9}\]  
_the \(\gamma\)-condition and hypotheses:_  
\[ \begin{split}&\alpha\leq\beta\gamma\leq 3-2\sqrt{2},\\ & A_{0}(f,t)\geq 0,\quad\left(\sum_{k=2}^{\infty}\frac{A_{k}(f;t)}{f ^{\prime}(t)}\left[\frac{f(t)}{f^{\prime}(t)}\right]^{n-1}\right)\geq 0,\end{split}  \tag{10}\]  
_with \(f\) given in (30) and \(\overline{U}(x_{0},R)\subseteq D\), where \(R=(1-\frac{1}{\sqrt{2}})\frac{1}{\gamma}\). Then, sequences \(\{x_{n}\}\), \(\{y_{n}\}\) generated by the method (7) are well defined in \(U(x_{0},R)\), remain in \(U(x_{0},t^{*})\) for each \(n=0,1,2\ldots\) and converge to a solution \(x^{*}\in U(x_{0},t^{*})\) of equation \(F(x)=0\), which is unique in \(U(x_{0},R)\). Moreover, the following estimates hold true for each \(n=0,1,2\ldots\):_  
\[ \begin{array}{lll}\|y_{n}-x_{n}\|&\leq&s_{n}-t_{n},\\ \|x_{n+1}-y_{n}\|&\leq&t_{n+1}-s_{n},\\ \|y_{n}-x^{*}\|&\leq&t^{*}-s_{n},\\ \|x_{n}-x^{*}\|&\leq&t^{*}-t_{n},\end{array}  \tag{11}\]  
_where \(t_{0}=0\), \(s_{0}=\beta\),_  
\[s_{n} = t_{n}-\frac{f(t_{n})}{f^{\prime}(t_{n})},\qquad n\geq 0,\] \[t_{n+1} = s_{n}-\frac{A_{0}(f;t_{n})}{f^{\prime}(t_{n})}-\left(\sum_{k=2}^ {\infty}\frac{A_{k}(f;t_{n})}{f^{\prime}(t_{n})}\left[\frac{f(t_{n})}{f^{ \prime}(t_{n})}\right]^{k-1}\right)\frac{f(t_{n})}{f^{\prime}(t_{n})}.\]  
_and:_  
\[t^{*} = \frac{1+\alpha-\sqrt{(1+\alpha)^{2}-8\alpha}}{4\gamma}.\]  
Next, we show how to achieve the advantages of our approach as stated in the introduction of this study. Definition 2: Let \(F:D\to B_{2}\) be a Frechet-differentiable operator. Operator \(F\) satisfies the center \(\gamma_{0}\)-Lipschitz condition at \(x_{0}\), if for each \(x\in D\):  
\[\|\Gamma_{0}(F^{\prime}(x)-F^{\prime}(x_{0}))\| \leq \frac{1}{(1-\gamma_{0}\|x-x_{0}\|)^{2}}-1.\]  
Remark 1: Let \(D_{0}:=U(x_{0},(1-\frac{1}{\sqrt{2}})\frac{1}{\gamma_{0}})\cap D\). Then, we have that:  
\[\|\Gamma_{0}(F^{\prime}(x)-F^{\prime}(x_{0}))\|\leq\frac{1}{(1- \gamma_{0}\|x-x_{0}\|)^{2}}-1<1.\]  
In view of the Banach lemma on invertible operators [8], \(\Gamma_{0}=F^{{}^{\prime}}(x_{0})^{-1}\in\mathcal{L}(B_{2},B_{1})\) and:  
\[\|[F^{\prime}(x)]^{-1}F^{\prime}(x_{0})\|\leq\left(2-\frac{1}{(1 -\gamma_{0}\|x-x_{0}\|)^{2}}\right)^{-1}. \tag{12}\]  
The corresponding result using the \(\gamma\)-condition given in [7] is:  
\[\|[F^{\prime}(x)]^{-1}F^{\prime}(x_{0})\|\leq\left(2-\frac{1}{(1 -\gamma\|x-x_{0}\|)^{2}}\right)^{-1}. \tag{13}\]  
for each \(x\in D\). However, if \(D_{0}\subseteq D\), then \(\gamma_{0}\leq\gamma\), so if \(\gamma_{0}<\gamma\), the estimate (12) is more precise than (13), leading to more precise majorizing sequences, which in turn lead to the advantages already stated.