Complexity

Volume 2018, Article ID 7352780, 11 pages

https://doi.org/10.1155/2018/7352780

## Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

^{1}Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India^{2}Instituto Universitario de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain^{3}Department of Mathematical Science, Punjab Technical University Jalandhar, Jalandhar, India

Correspondence should be addressed to Eulalia Martínez; se.vpu.tam@itramue

Received 6 February 2018; Accepted 29 March 2018; Published 7 May 2018

Academic Editor: Danilo Comminiello

Copyright © 2018 Abhimanyu Kumar et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The semilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided. Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.

#### 1. Introduction

Let be the unique solution ofwhere is a continuous nonlinear operator defined on a nonempty convex subset of a Banach with values in a Banach space . This problem frequently occurs in numerical analysis. Many scientific and real-life problems can be formulated mathematically in terms of integral equations, boundary value problems, equilibrium theory, optimization, and differential equations whose solutions require solving (1). The solutions of discrete dynamical systems also require solving them in order to represent the equilibrium states of these systems. With the existence of high-speed computational devices which solve them faster and with more accuracy, the problem for solving nonlinear equations has further gained and added advantages. Generally, iterative methods along with their convergence analysis are used to find the solutions of these equations. Many researchers [1–6] have extensively studied these problems and proposed many direct and iterative methods for their solutions along with their semilocal [1–3, 7], local [8, 9], and global convergence analysis [10]. In semilocal convergence analysis, we impose conditions on starting points while local convergence analysis requires the condition on the solution. Global study of convergence generally depends on the type of operators involved. The well-known quadratically convergent Newton’s method [11, 12] used for (1) is given by where and . Here, denotes the set of bounded linear operators from into . In [13], a family representing third-order iterative methods for (1) is given for by where is the starting iterate and . This family contains the Chebyshev method , the Halley method , and the Super-Halley method , respectively. These methods and many others use the differentiability of . Not much work is done by using nondifferentiability of which can be expressed in the formwhere are continuous and nonlinear operators. is differentiable while is continuous only and not differentiable. In [14], a quadratic order iterative method for (4) is given bywhere and are two starting points and satisfying for and .

Consider the following two-step difference differential method [15] for solving (4) given bywhere are two starting points and . Its local convergence analysis with super quadratic order is described. The differentiability condition on which restricts the applicability of (6) is also used. Moreover, no numerical examples were worked out. Recently, some special cases of (6) are also studied. One special case of (6) is given in [1, 16] for . Another one is given in [17] for . The importance of (6) lies in the fact that it uses the memory of in each iterate. If we consider the evaluation of as one function evaluation, then the total number of function evaluations of (6) for solving (4) coincides with that given by (5). However, the rate of convergence of (6) is much faster. This is shown by the following example.

*Example 1. *Consider where .

Take and .

A comparison of the absolute error approximation obtained by (6) and (5) with tolerance is given in Table 1.