# Expanding the Applicability of Some High Order Householder-Like Methods
## 2 Semilocal Convergence
\begin{array}{ll}(M_{k}^{2})&\|F^{\prime}(x_{k})^{-1}F^{\prime}(x_{0})\|\leq- \frac{1}{f_{0}^{\prime}(t_{k})}\leq-\frac{1}{f_{1}^{\prime}(t_{k})};\\ (M_{k}^{3})&\|y_{k}-x_{k}\|\leq\tilde{s}_{k}-\tilde{t}_{k};\\ (M_{k}^{4})&y_{k}\in\bar{U}(x_{0},\xi_{k});\\ (M_{k}^{5})&\|x_{k+1}-x_{k}\|\leq\tilde{t}_{k+1}-\tilde{s}_{k}.\end{array} \]  
Items (\(M_{0}^{i}\)), \(i=1,2,3,4,5\) are true by the initial conditions. Suppose (\(M_{k}^{i}\)) are true for \(k=0,1,\ldots,n\). Then, we shall show that they hold for \(n+1\). We have in turn that:  
\[ \begin{array}{lll}\|x_{k+1}-x_{0}\|&\leq&\|x_{k+1}-y_{k}\|+\|y_{k}-x_{k}\|+\| x_{k}-x_{0}\|\\ &\leq&(\tilde{t}_{k+1}-\tilde{s}_{k})+(\tilde{s}_{k}-\tilde{t}_{k+1})+(\tilde{ t}_{k+1}-\tilde{t}_{0})=\tilde{t}_{k+1},\end{array} \]  
so (\(M_{k+1}^{1}\)) is true. By Lemma 4 and condition \(\gamma_{0}\leq\delta\), we get that \(\Gamma_{k+1}\in\mathcal{L}(Y,X)\) and:  
\[\|\Gamma_{k+1}F^{\prime}(x_{0})\|\leq-\frac{1}{f_{0}^{\prime}(\|x_{k+1}-x_{0} \|}\leq-\frac{1}{f_{0}^{\prime}(\tilde{t}_{k+1})}\leq-\frac{1}{f_{1}^{\prime} (\tilde{t}_{k+1})},\]  
so (\(M_{k+1}^{2}\)) is true. By Lemma 2, we obtain in turn that \(\Gamma_{k+1}\in\mathcal{L}(Y,X)\) and:  
\[ \begin{array}{lll}\|\Gamma_{0}F^{\prime}(x_{k+1})\|\leq f_{1}^{\prime}( \tilde{t}_{k+1}),\end{array} \]  
so:  
\[ \begin{array}{lll}\|y_{k+1}-x_{k+1}\|&=&\|-\Gamma_{k+1}F(x_{k+1})\|\leq\| \Gamma_{k+1}F^{\prime}(x_{0})\|\|\Gamma_{0}F(x_{k+1})\|\\ &\leq&-\frac{f_{1}^{\prime}(\tilde{t}_{k+1})}{f_{1}(\tilde{t}_{k+1})}=\tilde{ s}_{k+1}-\tilde{t}_{k+1},\end{array} \]  
\[ \begin{array}{lll}\|y_{k+1}-x_{0}\|&\leq&\|y_{k+1}-x_{k}\|+\|x_{k}-y_{k-1}\| +\|y_{k-1}-x_{k-1}\|+\|x_{k-1}-x_{0}\|\\ &\leq&(\tilde{s}_{k+1}-\tilde{t}_{k})+(\tilde{t}_{k}-\tilde{s}_{k-1})+(\tilde{ s}_{k-1}-\tilde{t}_{k-1})+(\tilde{t}_{k-1}-\tilde{t}_{0})=\tilde{s}_{k+1},\end{array} \]  
and:  
\[\|x_{k+1}-y_{k}\|\leq\|\frac{1}{2}L_{F}(x_{k})\sum_{k=3}^{\infty}\Gamma_{k}A_{ k}(F;x_{k})(\Gamma_{k}F(x_{k}))^{k-1}\|\|y_{k}-x_{k}\|\leq\tilde{t}_{k+1}- \tilde{t}_{k},\]  
so (\(M_{k+1}^{4}\)) and (\(M_{k+1}^{5}\)) are also true, which completes the induction. **Remark 2**. : _We have so far weakened the sufficient semilocal convergence conditions of the method (7) (see (17)) and have also extended the uniqueness of the solution ball from \(U(x_{0},R)\) to \(U(x_{0},R_{0})\), since for \(\gamma_{0}<\gamma\), we have that \(R<R_{0}\). It is worth noticing that these advantages are obtained under the same computational cost, since in practice, the computation of constant \(\gamma\) requires the computation of constants \(\gamma_{0}\) and \(\delta\) as special cases._  
Next, it follows by a simple inductive argument and the definition of the majorizing sequences that the error bounds on the distances \(\|x_{n+1}-x_{n}\|\), \(\|x_{n+1}-y_{n}\|\), \(\|x_{n}-y_{n}\|\), \(\|y_{n+1}-y_{n}\|\), \(\|x_{n}-x^{*}\|\) and \(\|y_{n}-x^{*}\|\) can be improved. In particular, we have:  
**Proposition 1**. : _Under the hypotheses of Theorem 2 and Lemma 4, further suppose that for each \(u,u_{1},u_{2},v,v_{1},v_{2}\in[0,R]\), the following hold:_  
\[ \begin{array}{lll}u-\frac{f_{1}(u)}{f_{1}^{\prime}(u)}&\leq&v-\frac{f(v)}{f^{ \prime}(v)},\quad\text{for}\quad u\leq v,\\ -\frac{f_{1}(u)}{f_{1}^{\prime}(u)}&\leq&-\frac{f(v)}{f^{\prime}(v)},\quad \text{for}\quad u\leq v,\\ u_{1}-g_{1}(u_{2})&\leq&v_{1}-g(u_{2}),\quad\text{for}\quad u_{1}\leq v_{1},u_{2 }\leq v_{2},u_{1}\leq u_{2},v_{1}\leq v_{2},\\ g_{1}(u)&\leq&g(v),\quad\text{for}\quad u\leq v,\end{array}