# Local Convergence of an Efficient Multipoint Iterative Method in Banach Space
## 1 Introduction  
We study local criteria for obtaining a unique solution \(u_{*}\) of the nonlinear model  
\[F(u)=0, \tag{1}\]  
for Banach space valued mappings with \(F:\Omega\subset B\to B\), where \(F\) is differentiable in the sense of Frechet [1, 2]. For a good survey of literature on local and semilocal convergence criteria of iterative methods see [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13].  
The most popular numerical method for approximating a solution \(u_{*}\) of Equation (1) is the quadratically convergent [PERSON]'s method, which is expressed as  
\[u_{n+1}=u_{n}-F^{\prime}(u_{n})^{-1}F(u_{n}),\ \ \text{for each}\ n=0,1,2,\ldots\]  
In quest of efficient higher order method, a number of improved, multipoint [PERSON]'s or Newton-like iterative schemes have been proposed in literature; see, for example [3, 5, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19] and references cited therein.  
In particular, [PERSON] et al. [16] have recently developed an eighth order method for solving \(F(u)=0\) using a derivative-free composite scheme. The method is of order eight using only divided differences, derivatives up to the order nine and Taylor expansions in the special case when \(B=\mathbb{R}^{i}\) and \(Q(u)=\left(f_{1}^{m}(u),f_{2}^{m}(u),\cdots,f_{i}^{m}(u)\right)\), \(m\geq 2\), where \(Q:B\to B\), \(Q(u_{*})=F(u_{*})=0\). We study this method in the more general setting of a Banach space setting:\[y_{n} = u_{n}-[u_{n}+Q(u_{n}),u_{n};F]^{-1}F(u_{n}),\] \[z_{n} = y_{n}-\bigg{[}\frac{13}{4}I-A_{n}\bigg{(}\frac{7}{2}I-\frac{5}{4}A_{ n}\bigg{)}\bigg{]}[u_{n}+Q(u_{n}),u_{n};F]^{-1}F(y_{n}),\] \[u_{n+1} = z_{n}-\bigg{[}\frac{7}{2}I-A_{n}\bigg{(}4I-\frac{3}{2}A_{n}\bigg{)} \bigg{]}[u_{n}+Q(u_{n}),u_{n};F]^{-1}F(z_{n}), \tag{2}\]  
where \(u_{0}\in\Omega\) is an initial point, \(A_{n}=[u_{n}+Q(u_{n}),u_{n};F]^{-1}[y_{n}+Q(y_{n}),y_{n};F]\), \([.,.;F]:\Omega\times\Omega\to L(B,B)\) is divided difference of order one with  
\[[u,y;F](u-y)=F(u)-F(y)\]  
for each \(u,y\in\Omega\) with \(u\
eq y\) and \([u,u;F]=F^{\prime}(u)\) for each \(u\in\Omega\), if \(F\) is differentiable at \(u\). Here \(L(B,B)\) is the set of bounded linear operators from \(B\) into \(B\).  
The benefits of using method (2) over others in the literature have been well explained in [16]. Then to avoid repetitions, we refer the reader to [16]. But there are drawbacks when it comes to using method (2) limiting its applicability. These are: The existence of the ninth derivative is needed to show the order of convergence; the upper bounds on \(\|u_{n}-u_{*}\|\) or results on the uniqueness of the solution are not given; the initial point is a shot in the dark; the method is restricted only on the \(i\)-dimensional Euclidean space and higher order derivatives do not appear on the method. Notice that the method cannot even guarantee convergence, if we consider the scalar function \(\varphi\) on \(\Omega=[-\frac{1}{2},\frac{3}{2}]\) given as  
\[\varphi(x)=\left\{\begin{array}{ll}x^{3}\ln x^{2}+x^{5}-x^{4},\ x\
eq 0\\ 0,\quad x=0.\end{array}\right.\]  
Then, clearly \(\varphi^{\prime\prime\prime}(x)\) is unbounded on \(\Omega\). Hence, there is no assurance that \(\lim_{n\to\infty}u_{n}=u_{*}\) under the conditions in [16]. The novelty of this work is that we deal with all these drawbacks using only conditions on the divided difference of order one which actually used in Equation (2). Hence, we extend its applicability and for operators valued on Banach space.