# Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative
## 2 Local Convergence: One Dimensional Case  
In this section, we define some scalar functions and parameters to study the local convergence of method Equation (2). Let \(K_{0}>0,\ K_{1}>0,\ K>0,L_{0}>0,\ L>0,\ M\geq 1\), \(\lambda>1\) be given constants. Let us also assume \(g_{2}:\left[0,\ \frac{1}{L_{0}}\right)\rightarrow\mathbb{R}\), be a nondecreasing and continuous function. Define functions \(g_{1}\) and \(p\) in the interval \(\left[0,\ \frac{1}{L_{0}}\right)\) by  
\[g_{1}(t)=\frac{Lt}{2(1-L_{0}t)},p(t)=(K_{0}g_{2}(t)t^{\lambda-1}+K_{1})t\]  
and parameter \(r_{1}\) by  
\[r_{1}=\frac{2}{2L_{0}+L}.\]  
Notice that the function \(g_{1}\) is a monotonically increasing function on the interval \(\left[0,\ 1\right)\). We have \(g_{1}(r_{1})=1\) and for each \(t\in\left[0,\ r_{1}\right):0\leq g_{1}(t)<1\). Further, define function \(h_{2}:\left[0,\ \frac{1}{L_{0}}\right)\rightarrow\mathbb{R}\) by \(h_{2}(t)=g_{2}(t)t^{\lambda-1}-1\). Suppose that  
\[ \begin{split}& g_{2}(t)t^{\lambda-1}<1,\ \ \text{for each}\ \ \left[0,\ r_{1}\right),\\ & h_{2}(l)>0,\\ &\text{as }t\to l<r_{1}\text{ for some }l>0.\end{split}  \tag{3}\]  
Then, we have \(h_{2}(0)=-1<0\). By Equation (3) and the intermediate value theorem, function \(h_{2}\) has zeros in the interval \(\left(0,\ l\right)\). Further, let \(r_{2}\) be the smallest such zero. Further, 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 \(\left[0,\ r_{q}\right)\) by  
\[g_{3}(t)=\left(1+\frac{\left(K+(L_{0}+K_{1})t+K_{0}g_{1}(t)t\right)M}{\left(1 -L_{0}t\right)\left(1-q(t)\right)}\right)g_{2}(t)t^{\lambda-1}\]  
and  
\[h_{3}(t)=g_{3}(t)-1.\]  
Then, we get \(h_{3}(0)=-1\) 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 \(\left(0,\ r_{q}^{-}\right)\). 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 \(t\in\left[0,r\right)\)  
\[0\leq g_{1}(t), \tag{6}\]\[0\leq p(t), \tag{7}\]  
\[0\leq q(t)<1, \tag{8}\]  
\[0\leq g_{2}(t)<1 \tag{9}\]  
and  
\[0\leq g_{3}(t)<1. \tag{10}\]  
\(U(\gamma,\ s)\) and \(\bar{U}(\gamma,\ s)\) stand, respectively for the open and closed balls in \(\mathbb{X}\) with center \(\gamma\in\mathbb{X}\) and radius \(s>0\). Next, we present the local convergence analysis of scheme Equation (2) using the preceding notations. **Theorem 1**. : _Let us consider \(F:\mathbb{D}\subset\mathbb{X}\rightarrow\mathbb{Y}\) be a Frechet differentiable operator. Let us also assume \([\cdot,\cdot\ ;F]:\mathbb{D}^{2}\to L(\mathbb{X})\) be a divided difference of order one. Suppose there exist \(x^{*}\in\mathbb{D}\), a neighborhood \(\mathbb{D}_{0}\) of \(\mathbb{D}\), \(L_{0}>0,\ \lambda>1\) such that Equation (3) holds and for each \(x\in\mathbb{D}_{0}\)_  
\[F(x^{*})=0,\quad F^{\prime}(x)^{-1}\in L(\mathbb{Y},\ \mathbb{X}) \tag{11}\]  
\[h_{3}(t)=g_{3}(t)-1.\]0  
_and_  
\[h_{3}(t)=g_{3}(t)-1.\]1  
_where \(z(x)=\phi_{4}\left(x,\ x-F^{\prime}(x)^{-1}F(x)\right)\)._  
_Moreover, suppose that there exist \(K_{0}>0,\ K_{1}>0,\ K>0,\ L>0\) and \(M\geq 1\) such that for each \(x,\ y\in U\left(x^{*},\ \overline{L_{0}}\right)\cap\mathbb{D}_{0}\)_  
\[h_{3}(t)=g_{3}(t)-1.\]2  
\[h_{3}(t)=g_{3}(t)-1.\]3  
\[h_{3}(t)=g_{3}(t)-1.\]4  
\[h_{3}(t)=g_{3}(t)-1.\]5  
_and_  
\[h_{3}(t)=g_{3}(t)-1.\]6  
_where the radius of convergence \(r\) is defined by Equation (4). Then, the sequence \(\{x_{n}\}\) generated by method Equation (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 \(x^{*}\).