Enlarging the convergence domain in local convergence studies for iterative methods in Banach spaces

Abstract

In this work we introduce a new form of setting the general assumptions for the local convergence studies of iterative methods in Banach spaces that allows us to improve the convergence domains. Specifically a local convergence result for a family of higher order iterative methods for solving nonlinear equations in Banach spaces is established under the assumption that the Fréchet derivative satisfies the Lipschitz continuity condition. For some values of the parameter, these iterative methods are of fifth order. The importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained.

We have considered a number of numerical examples including a nonlinear Hammerstein equation and computed the radii of the convergence balls. It is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods.

The global convergence properties of the family are explored by analyzing the dynamics of the corresponding operator on complex quadratic polynomials.

Introduction

One of the most important problems of Numerical Functional Analysis is to solve nonlinear equations in Banach spaces. This has become necessitated as the mathematical modeling [7], [8], [13], [17] of a large number of problems of science and engineering involving scalar equations, system of equations, differential equations, integral equations, etc., reduce to thousands of such equations. One such example is the dynamical systems which are mathematically modeled by difference or differential equations and their solutions usually represent the states of the systems. With the advancement in computer S/W and H/W, this problem has gained an added advantage. Generally, iterative methods along with their local and semi local convergence analysis are used for them. The local convergence analysis [2], [9] uses information around the solution whereas semi local convergence analysis [6], [14], [22] is based on information around an initial point. Another important problem which is to be considered for these iterative methods is the convergence domains/radii of convergence balls. In general, the convergence domain of an iterative method is small and one always tries to enlarge it by considering additional hypothesis. It is worth mentioning that most of the local convergence results are obtained under general conditions that, by using Taylor’s expansions allow us to find the convergence order but not the radii of the convergence balls (see, [20], [21], [23] and references cited therein).

The aim of this paper is to describe local convergence analysis for a unique solution x^* of nonlinear operator equation

$$\begin{array}{c}\hfill F\left(x\right)=0\end{array}$$

where, F is a Fréchet-differentiable operator defined on a subset D of a Banach space X with values in a Banach space Y. Starting from one or several initial approximations of x^*, a sequence {x_k} of approximations is constructed so that it converges to x^*. The sequence {x_k} can be obtained in different ways depending on the iterative method that is applied. The well known quadratically convergent Newton’s method is the most widely used iterative method to solve (1.1). Starting with x₀, it is given, for $k=0,1,2,\dots ,$ by

$$x_{k+1}=x_k-F^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(x_k\right)$$

Iterative methods of higher order convergence require evaluation of higher order derivatives which are very expansively in general. For example, the third order Chebyshev–Halley type methods [18] require evaluation of second Fréchet derivative which either does not exist or is computationally difficult to evaluate. But higher order methods have their importance as in some applications involving stiff system of equations that need faster convergence. Also, there are integral equations where the second Fréchet derivative is diagonal by blocks and inexpensive [3]. The local convergence analysis of a family of third order iterative methods for nonlinear equations in Banach spaces is established in [5] for the method described in [16]. Argyros et al. [10] considered multi-point-parametric Chebyshev–Halley-type methods of high convergence order involving Fréchet derivative and discussed their local convergence analysis in Banach spaces. The local convergence analysis of a modified Halley-like method of high convergence order is described in [11]. Starting from an initial point x₀, it is defined for $k=0,1,2,\dots$ by

$$\begin{array}{cc}\hfill y_k& =x_k-F^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(x_k\right),\hfill \\ \hfill u_k& =y_k+(1-a)F^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(x_k\right),\hfill \\ \hfill z_k& =y_k-\gamma A_{a,k}{F}^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(x_k\right),\hfill \\ \hfill x_{k+1}& =z_k-\alpha B_{a,k}{F}^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(z_k\right),\hfill \end{array}$$

where, $\alpha ,\gamma ,a\in (-\infty ,\infty )-\left\{0\right\},$$H_{a,k}=\frac{1}{a}{F}^{{}^{\prime }}{\left(x_k\right)}^{-1}(F^{{}^{\prime }}\left(u_k\right)-F^{{}^{\prime }}\left(x_k\right)),$$A_{a,k}=I-\frac{1}{2}{H}_{a,k}(I-\frac{1}{2}{H}_{a,k})$ and $B_{a,k}=I-H_{1,k}+H_{a,k}^2$. Recently, a local convergence analysis along with the dynamics of Chebyshev–Halley-type methods free from second derivatives is described in [12]. Starting with an initial approximation x₀, it is given for $k=0,1,2\dots$ by

$$\begin{array}{cc}\hfill y_k& =x_k-F^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(x_k\right),\hfill \\ \hfill z_k& =x_k-(1+{(F\left(x_k\right)-2\alpha F\left(y_k\right))}^{-1}F\left(y_k\right))F^{{}^{\prime }}{\left(x_k\right)}^{-1}F\left(x_k\right),\hfill \\ \hfill x_{k+1}& =z_k-{(F^{{}^{\prime }}\left(x_k\right)+{\overline{F}}^{\prime \prime }\left(x_k\right)(z_k-x_k))}^{-1}F\left(z_k\right),k\ge 0,\hfill \end{array}$$

where, ${\overline{F}}^{{}^{\prime \prime }}\left(x_k\right)=2F\left(y_k\right)F^{{}^{\prime }}{\left(x_k\right)}^{2}F{\left(x_k\right)}^{-2}$ and α is a parameter.

In this paper, a local convergence analysis of a family of iterative methods for solving nonlinear equations in Banach spaces is established under the assumption that the first Fréchet derivative satisfies the Lipschitz continuity condition. For the values of the parameter $a=\pm 1,$ these iterative methods are of fifth order. The importance of our work is that it avoids the usual practice of boundedness conditions of higher order derivatives which is a drawback for solving some practical problems. The existence and uniqueness theorem that establishes the convergence balls of these methods is obtained.

We have considered a number of numerical examples including a nonlinear Hammerstein equation and computed the radii of the convergence balls. It is found that the radius of convergence ball obtained by our approach is much larger when compared with some other existing methods.

Finally, the complex dynamics of the family is studied for some parameter values, by analyzing the attraction basins of the iterative scheme for complex quadratic polynomials.