## 2 Local Convergence: One Dimensional Case  
In this section, we define some scalar functions and parameters to study the local convergence of Scheme (2). Let \(K_{0}>0,\ K_{1}>0,\ K>0,L_{0}>0,\ L>0,\ M\geq 1\), \(\lambda\geq 1\), be given constants. Let us also assume \(g_{2}:\left[0,\ \frac{1}{L_{0}}\right)\rightarrow\mathbb{R}\), be nondecreasing and continuous function. Further, define function \(h_{2}:\left[0,\ \frac{1}{L_{0}}\right)\rightarrow\mathbb{R}\) and \(h_{2}(t)=g_{2}(t)t^{\lambda-1}-1\). Suppose that  
\[ \begin{array}{l}g_{2}(t)t^{\lambda-1}<1,\ \ \mbox{for each}\ \ \left[0,\ \frac{1}{L_{0}}\right),\\ h_{2}(t)\rightarrow\mbox{a positive number or}\ +\infty,\\ \mbox{as }t\to l<\frac{1}{L_{0}}\mbox{ for some }l>0.\end{array}  \tag{3}\]  
Then, we have \(h_{2}(0)=-1<0\). By Equation (3) and the intermediate value theorem, function \(h_{2}\) has zeros in the interval \((0,\ l)\). Further, let \(r_{2}\) be the smallest such zero. Moreover, define functions \(g_{1},\ p\) and \(h_{p}\) in the interval \(\left[0,\ \frac{1}{L_{0}}\right)\) by \[g_{1}(t)=\frac{Lt}{2(1-L_{0}t)},\] \[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\] \[h_{p}(t)=p(t)-1\] and parameter \(r_{1}\) by \[r_{1}=\frac{2}{2L_{0}+L}.\]  
We have \(g_{1}(r_{1})=1\) and for each \(t\in[0,\ r_{1}):0\leq g_{1}(t)<1\). We also get \(h_{p}(0)=-1\) and \(h_{p}(t)\rightarrow+\infty\) as \(t\rightarrow\frac{1}{L_{0}}\). Denote by \(r_{p}\) the smallest zero of function \(h_{p}\) on the interval \(\left(0,\ \frac{1}{L_{0}}\right)\). Furthermore, define functions \(q\) and \(h_{q}\) on the interval \(\left[0,\ \frac{1}{L_{0}}\right)\) by \(q(t)=p(t)+2\left[K_{0}g_{2}(t)t^{\lambda}+K_{1}g_{1}(t)t\right]\) and \(h_{q}(t)=q(t)-1\). Using \(h_{q}(0)=-1<0\) and Equation (3), we deduce that function \(h_{q}\) has a smallest zero denoted by \(r_{q}\). Finally define functions \(g_{3}\) and \(h_{3}\) on the interval \([0,\ \min\{r_{p},\ r_{q}\})\) by  
\[g_{3}(t)=\left(1+\frac{KM}{\left(1-p(t)\right)\left(1-q(t)\right)}\right)g_{2 }(t)t^{\lambda}\]  
and  
\(h_{3}=g_{3}(t)-1\). Then, we get \(h_{3}(0)=-1\) and \(h_{3}(t)\rightarrow+\infty\) as \(t\rightarrow\min\{r_{p},\ r_{q}\}\). Denote by \(r_{3}\) the smallest zero of function \(h_{3}\) on the interval \((0,\ \min\{r_{p},\ r_{q}\})\). Define  
\[r=\min\{r_{1},\ r_{2},\ r_{3}\}. \tag{4}\]  
Then, we have that  
\[0<r\leq r_{1}<\frac{1}{L_{0}} \tag{5}\]  
and for each \(t\in[0,r)\)  
\[0\leq g_{1}(t)<1, \tag{6}\]  
\[0\leq p(t)<1, \tag{7}\]  
\[0\leq q(t)<1, \tag{8}\]  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]0  
and  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]1  
\(U(\gamma,\ s)\) and \(\bar{U}(\gamma,\ s)\) stand, respectively for the open and closed balls in \(X\) with center \(\gamma\in X\) and radius \(s>0\). Next, we present the local convergence analysis of Scheme (2) using the preceding notations. **Theorem 1**. : _Let us consider \(F:D\subset X\to Y\) be a Frechet differentiable operator. Let us also assume \([\cdot,\cdot\ ;F]:D^{2}\to L(X)\) be a divided difference of order one. Suppose that there exist \(x^{*}\in D,\ L_{0}>0,\ \lambda\geq 1\) such that Equation (3) holds and for each \(x\in D\)_  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]2  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]3  
_and_  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]4  
_Moreover, suppose that there exist \(K_{0}>0,\ K_{1}>0,\ K>0,\ L>0\) and \(M\geq 1\) such that for each \(x,\ y\in U\left(x^{*},\ \frac{1}{L_{0}}\right)\cap D\)_  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]5  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]6  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]7  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]8  
_and_  
\[p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t,\]9  
_where the radius of convergence \(r\) is defined by Equation (4) and \(z(x)=\phi_{4}\left(x,\ x-F^{\prime}(x)^{-1}F(x)\right)\). Then, the sequence \(\{x_{n}\}\) generated by Scheme (2) for \(x_{0}\in U(x^{*},\ r)-\{x^{*}\}\) is well defined, remains in \(U(x^{*},\ r)\) for each \(n=0,1,2,\ldots\) and converges to \(x^{*}\).