## 4 Accessibility of [PERSON]'s Method  
The accessibility of an iterative method is analyzed from the set of possible starting points that guarantee the convergence of the iterative method when it starts at them. As we have indicated, the set of starting points that guarantees the convergence of the iterative method is related to the domain of parameters associated with a result of semi-local convergence of the iterative method. Next, we study the domain of parameters associated with Theorem 1 and compare it with that associated with Theorem 3. To guarantee the convergence of [PERSON]'s method from Theorem 3, the following three conditions must be satisfied:  
\[\gamma\leq\frac{1+p}{2+p},\qquad b_{1}<\frac{1+p}{2+p}\qquad\text{and}\qquad b _{1}+\frac{b_{1}^{p}}{(1+p)^{p}(1-b_{1})^{p}}<1, \tag{17}\]  
where \(p\in(0,1]\). From the auxiliary function given in (2), the third condition of (17) can be written as:  
\[\phi(b_{1};p)>0. \tag{18}\]  
Observe that \(\phi(x;p)\) is a non-increasing and convex function and \(\phi(0;p)=(1+p)^{p}>0\) and \(\phi\left(\frac{1}{2};p\right)\leq 0\), for all \(p\in(0,1]\). Besides, if \(p=1\), the unique zero of \(\phi(x;1)\) in the interval \(\left(0,\frac{1}{2}\right]\) is \(\frac{1}{2}\). If we now denote, for a fixed \(p\in(0,1]\), the unique zero of \(\phi(x;p)\) in \(\left(0,\frac{1}{2}\right]\) by \(\xi(p)\) and demand \(b_{1}<\xi(p)\), then Condition (18) holds. Moreover, since \(\xi(p)\leq\frac{1}{2}\), the second condition of (17) is satisfied. Now, as:  
\[b_{1}=\frac{\delta^{p}b_{0}}{1-\gamma}=\frac{\gamma^{p}b_{0}}{(1+p)^{p}(1- \gamma)^{1+p}},\]  
the second and third conditions of (17) are satisfied, provided that:  
\[b_{0}<\xi(p)\frac{(1+p)^{p}(1-\gamma)^{1+p}}{\gamma^{p}}.\]  
After that, we write the first condition of (17) as:  
\[\mathpzc{h}\leq\frac{1+p}{\varepsilon(2+p)} \tag{19}\]  
and take into account that the second and third conditions of (17) are satisfied if:  
\[\mathpzc{h}<\xi(p)\frac{(1+p)^{p}(1-\varepsilon\mathpzc{h})^{1+p}}{ \varepsilon^{p}\mathpzc{h}^{p}}, \tag{20}\]  
since \(b_{0}=\mathpzc{h}\) and \(\gamma=\varepsilon\mathpzc{h}\). In addition, Condition (20) is equivalent to:  
\[\varpi(\mathpzc{h})=\varepsilon^{p}\mathpzc{h}^{1+p}-\xi\left(1+p\right)^{p} (1-\varepsilon\mathpzc{h})^{1+p}<0.\]  
Furthermore, \(\varpi^{\prime}(\mathpzc{h})\geq 0\), so that \(\varpi(\mathpzc{h})\) is a nondecreasing function for all \(\mathpzc{h}\geq 0\), and \(\varpi(0)\leq 0\) and \(\varpi\left(\frac{1+p}{\varepsilon(2+p)}\right)>0\). Therefore, Condition (19) is satisfied if Condition (20) holds, and we can then give the following result. **Corollary 4**. : _Let \(F:\Omega\subseteq X\longrightarrow Y\) be a continuously-differentiable operator defined on a nonempty open convex domain \(\Omega\) of a Banach space \(X\) with values in a Banach space \(Y\). Suppose that (A1)-(A2) are satisfied. If Condition (20), where \(\xi(p)\) is the unique zero of function (2) in the interval \(\left(0,\frac{1}{2}\right]\), is satisfied and \(B(x_{0},R)\subset\Omega\), where \(R=\left(1+\frac{\upsilon}{1-\
u}\right)\eta\), \(\upsilon=\frac{\mathpzc{h}}{(1+p)(1-\varepsilon\mathpzc{h})}\), \(\
u=\frac{\vartheta}{(1+p)(1-\vartheta)}\) and \(\vartheta=\frac{\mathpzc{h}\varepsilon^{p}}{1-\varepsilon\mathpzc{h}}\), then [PERSON]'s sequence, given by (1), converges to a solution \(x^{*}\) of the equation \(F(x)=0\), starting at \(x_{0}\), and \(x_{n},x^{*}\in\overline{B(x_{0},R)}\), for all \(n=0,1,2,\ldots\)._From the last result, we can define the following domains of parameters:  
\[\mathcal{D}_{\mathrm{C}}^{\varepsilon}=\left\{(x,y)\in\mathbb{R}^{2}:\;xy\leq \xi(p)\,\frac{(1+p)^{p}(1-\varepsilon xy)^{1+p}}{\varepsilon^{p}x^{p}y^{p}} \right\},\]  
where \(\varepsilon=\frac{K_{0}}{K}\in[0,1]\), \(p\in(0,1]\) and \(\xi(p)\) is the unique zero of Function (2) in the interval \(\left(0,\frac{1}{2}\right]\). Next, we compare the conditions required for the semi-local convergence of [PERSON]'s method in Theorem 1 and Corollary 4.