## 2 Local Convergence  
In this section, we present the local convergence analysis of Method (3). Let \(L_{0}>0\), \(L>0\), \(M\geq 1\), \(L_{1}>0\), \(\beta\in\mathbb{S}\) and \(L_{2}>0\). It is convenient for the local convergence analysis that follows to introduce some functions and parameters. Define functions \(g_{1}\), \(p\) and \(h_{p}\) on the interval \([0,\frac{1}{L_{0}})\) by:  
\[g_{1}(t) =\frac{Lt}{2(1-L_{0}t)},\] \[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\] \[h_{p}(t) =p(t)-1,\]  
and parameter \(r_{1}\) by:  
\[r_{1}=\frac{2}{2L_{0}+L}.\]  
We have that \(h_{p}(0)=-1<0\) and \(h_{p}(t)\rightarrow\infty\) as \(t\rightarrow\frac{1^{-}}{L_{0}}\). It follows from the intermediate value theorem that function \(h_{p}\) has zeros in the interval \(\left(0,\ \frac{1}{L_{0}}\right)\). Denote by \(r_{p}\) the smallest such zero. Moreover, define functions \(g_{2}\) and \(h_{2}\) on the interval \([0,\ r_{p})\) by:  
\[g_{2}(t)=\left(1+\frac{M^{2}(1+|\beta|g_{1}(t))}{(1-p(t))(1-L_{0}t)}\right)g_{ 1}(t)\]  
and:  
\[h_{2}(t)=g_{2}(t)-1.\]  
Then, we get \(h_{2}(0)=-1<0\) and \(h_{2}(t)\rightarrow\infty\) as \(t\to r_{p}^{-}\). Denote by \(r_{2}\) the smallest zero of function \(h_{2}\) on the interval \((0,\ r_{p})\). Furthermore, define functions \(q\) and \(h_{q}\) on the interval \([0,\ r_{p}]\) by:  
\[q(t)=[4L_{1}+(3L_{1}+L_{2})\big{(}g_{1}(t)+g_{2}(t)\big{)}]t,\]and:  
\[h_{q}(t)=q(t)-1.\]  
We have that \(h_{q}(0)=-1<0\) and \(h_{q}(t)\rightarrow\infty\) as \(t\to r_{p}^{-}\). Denote by \(r_{q}\) the smallest zero of function \(h_{q}\) on the interval \((0,\ r_{q})\). Finally, define functions \(g_{3}\) and \(h_{3}\) on the interval \([0,\ r_{q})\) by:  
\[g_{3}(t)=\left(1+\frac{M}{1-q(t)}\right)g_{2}(t),\]  
and:  
\[h_{3}(t)=g_{3}(t)-1.\]  
We get that \(h_{3}(0)=-1<0\) and \(h_{3}(t)\rightarrow\infty\) as \(t\to r_{q}^{-}\). Denote by \(r_{3}\) the smallest zero of function \(h_{3}\) on the interval \((0,\ r_{q})\). Set:  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]0  
Then, we have that:  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]1  
and for each \(t\in[0,\ r)\):  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]2  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]3  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]4  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]5  
and:  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]6  
Let \(U(\gamma,\ \rho)\), \(\bar{U}(\gamma,\ \rho)\) stand, respectively, for the open and closed balls in \(\mathbb{S}\), with center \(\gamma\in\mathbb{S}\) and of radius \(\rho>0\). Next, we present the local convergence analysis of Method (3) using the preceding notation. **Theorem 1**. : _Let \(F:\mathbb{D}\subset\mathbb{S}\rightarrow\mathbb{S}\) be a differentiable function. Let \([.,\. ;\ F]:\mathbb{D}\times\mathbb{D}\to L(\mathbb{S})\) be a divided difference of order one. Suppose that there exist \(x^{*}\in\mathbb{D}\), \(L_{0}>0\), \(L>0\), \(M\geq 1\), \(L_{1}\geq 0,\ L_{2}\geq 0\), \(\beta\in\mathbb{S}\), such that for all \(x,\ y\in\mathbb{D}\):_  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]7  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]8  
\[p(t) =\frac{1}{2}\left(L_{0}t+2M|\beta-2|g_{1}(t)\right),\]9  
\[h_{p}(t) =p(t)-1,\]0  
\[h_{p}(t) =p(t)-1,\]1  
\[h_{p}(t) =p(t)-1,\]2  
_and:_  
\[h_{p}(t) =p(t)-1,\]3_where the radius \(r\) is defined by Equation (4). Then, the sequence \(\{x_{n}\}\) generated for \(x_{0}\in U(x^{*},\ r)-\{x^{*}\}\) by Method (3) is well defined, remains in \(U(x^{*},\ r)\) for each \(n=0,1,2,\ldots\) and converges to \(x^{*}\). Moreover, the following estimates hold:_  
\[h_{p}(t) =p(t)-1,\]4  
\[h_{p}(t) =p(t)-1,\]5  
_and:_  
\[h_{p}(t) =p(t)-1,\]6  
_where the \"\(g\)\" functions are defined previously. Furthermore, for \(T\in[r,\frac{2}{L_{0}})\), the limit point \(x^{*}\) is the only solution of equation \(F(x)=0\) in \(\bar{U}(x^{*},\ T)\cap\mathbb{D}\)._  
**Proof.