# On the Semi-Local Convergence of Two Competing Sixth Order Methods for Equations in Banach Space
## 1 Introduction  
Let us consider a Frechet derivable operator \(F:\Omega\subseteq X\to Y\), where \(X\), \(Y\) are Banach spaces and \(\Omega(\
eq\emptyset)\) is a convex and open set. In computational sciences and other related fields, equations of the type  
\[F(x)=0, \tag{1}\]  
are regularly used to address numerous complicated problems. It is important to realize that obtaining the solutions to these equations is a challenging problem. The solutions are only being found analytically in a limited number of cases. Therefore, iterative procedures are often developed to solve these equations. However, it is a difficult task to create an effective iterative strategy for dealing with Equation (1). The popular [PERSON]'s method is widely used to solve this equation. In order to increase the convergence order modifications of methods such as Chebyshev's, Jarratt's, etc. have been developed. Various higher order iterative ways computing solution of (1) have been provided in [1, 2, 3]. These methods are based on Newton-like methods [2, 3, 4, 5, 6, 7, 8, 9, 10]. In [11], two cubically convergent iterative procedures are designed by [PERSON] and [PERSON]. Another third-order convergent method based on the evaluations of two \(F\), one \(F^{\prime}\), and one inversion of the matrix is presented by [PERSON] and [PERSON] [5]. In addition, [PERSON] and [PERSON] [5]also suggested methods having convergence order four. [PERSON] et al. [12] composed two weighted-Newton steps to generate an efficient fourth-order weighted Newton method for nonlinear systems. In addition, fourth and sixth-order convergent iterative algorithms are developed by [PERSON] and [PERSON] [13] to solve nonlinear systems. The main objective of this article is to extend the application of the sixth convergence order methods that we have selected from [13; 14], respectively. These methods are:  
\[y_{n} = x_{n}-\alpha F^{\prime}(x_{n})^{-1}F(x_{n})\] \[z_{n} = x_{n}-\bigg{(}\frac{23}{8}I-3F^{\prime}(x_{n})^{-1}F^{\prime}(y_ {n})+\frac{9}{8}\Big{(}F^{\prime}(x_{n})^{-1}F^{\prime}(y_{n})\Big{)}^{2} \bigg{)}F^{\prime}(x_{n})^{-1}F(x_{n}) \tag{2}\] \[x_{n+1} = z_{n}-\bigg{(}\frac{5}{2}I-\frac{3}{2}F^{\prime}(x_{n})^{-1}F^{ \prime}(y_{n})\bigg{)}F^{\prime}(x_{n})^{-1}F(z_{n})\]  
and  
\[y_{n} = x_{n}-\alpha F^{\prime}(x_{n})^{-1}F(x_{n})\] \[z_{n} = x_{n}-\bigg{(}I+\frac{21}{8}F^{\prime}(x_{n})^{-1}F^{\prime}(y_ {n})-\frac{9}{2}(F^{\prime}(x_{n})^{-1}F^{\prime}(y_{n}))^{2}\] \[+\frac{15}{8}(F^{\prime}(x_{n})^{-1}F^{\prime}(y_{n}))^{3}\bigg{)} F^{\prime}(x_{n})^{-1}F(x_{n})\] \[x_{n+1} = z_{n}-(3I-\frac{5}{2}F^{\prime}(x_{n})^{-1}F^{\prime}(y_{n})+( \frac{1}{2}F^{\prime}(x_{n})^{-1}F^{\prime}(y_{n}))^{2})F^{\prime}(x_{n})^{-1 }F(z_{n}),\]  
respectively. If \(\alpha=\frac{2}{3}\) methods (2) and (3) are reduced to the methods designed in [12; 14], respectively. The motivation and the benefits of using these methods have been well explained in [13; 14]. These methods require the evaluation of two derivatives, one inverse, and two operator evaluations per iteration. The convergence analysis was given in the special case when \(X=Y=\mathbb{R}^{i}\). The local convergence of these methods is shown with the application of expensive Taylor formulas. Moreover, the existence of derivatives up to order seven is assumed. These derivatives do not appear in the methods. However, this approach reduces their applicability. **Motivation for writing this paper. ** Let us look at the following function to explain a viewpoint  
\[F(t)=\left\{ \begin{array}{ll}0,&\text{if }t=0,\\ 2t^{3}\ln(t)+t^{5}-t^{4},&\text{if }t\
eq 0,\end{array} \right. \tag{4}\]  
where \(X=Y=\mathbb{R}\) and the \(F\) is defined on \(\Omega=[-0.5,1.5]\). Then, the unboundedness of \(F^{\prime\prime\prime}\) makes the previous functions' convergence results ineffective for methods (2) and (3). Notice also that the results in [15; 16] can not be used to solve equations with operators that are not at least seven times differentiable. However, these methods may converge.