scholarly journals On semilocal convergence of three-step Kurchatov method under weak condition

2021 ◽  
Author(s):  
Himanshu Kumar
Keyword(s):  
Recurrence Relations ◽  
Divided Difference ◽  
Semilocal Convergence ◽  
Weak Condition ◽  
Theoretical Development ◽  
Difference Operators ◽  
Nonlinear System Of Equations ◽  
First Order ◽  
Type Integral ◽  
Weaker Conditions

AbstractThe purpose of this paper to establish the semilocal convergence analysis of three-step Kurchatov method under weaker conditions in Banach spaces. We construct the recurrence relations under the assumption that involved first-order divided difference operators satisfy the $$\omega $$ ω condition. Theorems are given for the existence-uniqueness balls enclosing the unique solution. The application of the iterative method is shown by solving nonlinear system of equations and nonlinear Hammerstein-type integral equations. It illustrates the theoretical development of this study.

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

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-11
Author(s):  
Abhimanyu Kumar ◽  
Dharmendra K. Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Local Convergence ◽  
Recurrence Relations ◽  
Semilocal Convergence ◽  
Convergence Domain ◽  
Differentiability Condition ◽  
Type Integral ◽  
Weaker Conditions ◽  
Theoretical Results

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.

scholarly journals Convergence of an Iteration of Fifth-Order Using Weaker Conditions on First Order Fréchet Derivative in Banach Spaces

2018 ◽  
Vol 15 (06) ◽  
pp. 1850048
Author(s):  
Sukhjit Singh ◽  
Dharmendra Kumar Gupta ◽  
Randhir Singh ◽  
Mehakpreet Singh ◽  
Eulalia Martinez
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Semilocal Convergence ◽  
Fréchet Derivative ◽  
Frechet Derivative ◽  
First Order ◽  
Weaker Conditions ◽  
Fifth Order ◽  
Derivatives Of ◽  
Text Condition

The convergence analysis both local under weaker Argyros-type conditions and semilocal under [Formula: see text]-condition is established using first order Fréchet derivative for an iteration of fifth order in Banach spaces. This avoids derivatives of higher orders which are either difficult to compute or do not exist at times. The Lipchitz and the Hölder conditions are particular cases of the [Formula: see text]-condition. Examples can be constructed for which the Lipchitz and Hölder conditions fail but the [Formula: see text]-condition holds. Recurrence relations are used for the semilocal convergence analysis. Existence and uniqueness theorems and the error bounds for the solution are provided. Different examples are solved and convergence balls for each of them are obtained. These examples include Hammerstein-type integrals to demonstrate the applicability of our approach.

scholarly journals Semilocal Convergence of the Extension of Chun’s Method

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 161
Author(s):  
Alicia Cordero ◽  
Javier G. Maimó ◽  
Eulalia Martínez ◽  
Juan R. Torregrosa ◽  
María P. Vassileva
Keyword(s):  
Iterative Method ◽  
Existence And Uniqueness ◽  
Recurrence Relations ◽  
Nonlinear Integral Equation ◽  
Difference Operator ◽  
Divided Difference ◽  
Semilocal Convergence ◽  
Starting Point ◽  
Domain Of Existence ◽  
Theoretical Results

In this work, we use the technique of recurrence relations to prove the semilocal convergence in Banach spaces of the multidimensional extension of Chun’s iterative method. This is an iterative method of fourth order, that can be transferred to the multivariable case by using the divided difference operator. We obtain the domain of existence and uniqueness by taking a suitable starting point and imposing a Lipschitz condition to the first Fréchet derivative in the whole domain. Moreover, we apply the theoretical results obtained to a nonlinear integral equation of Hammerstein type, showing the applicability of our results.

scholarly journals Memory in a New Variant of King’s Family for Solving Nonlinear Systems

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1251
Author(s):  
Munish Kansal ◽  
Alicia Cordero ◽  
Sonia Bhalla ◽  
Juan R. Torregrosa
Keyword(s):  
Nonlinear Systems ◽  
Convergence Analysis ◽  
Numerical Experiments ◽  
Order Of Convergence ◽  
Recent Literature ◽  
Divided Difference ◽  
Difference Operators ◽  
New Variant ◽  
Order Four ◽  
With Memory

In the recent literature, very few high-order Jacobian-free methods with memory for solving nonlinear systems appear. In this paper, we introduce a new variant of King’s family with order four to solve nonlinear systems along with its convergence analysis. The proposed family requires two divided difference operators and to compute only one inverse of a matrix per iteration. Furthermore, we have extended the proposed scheme up to the sixth-order of convergence with two additional functional evaluations. In addition, these schemes are further extended to methods with memory. We illustrate their applicability by performing numerical experiments on a wide variety of practical problems, even big-sized. It is observed that these methods produce approximations of greater accuracy and are more efficient in practice, compared with the existing methods.

scholarly journals Two-Step Solver for Nonlinear Equations

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 128 ◽  
Author(s):  
Ioannis Argyros ◽  
Stepan Shakhno ◽  
Halyna Yarmola
Keyword(s):  
Nonlinear Equations ◽  
Order Of Convergence ◽  
Semilocal Convergence ◽  
Numerical Example ◽  
Weaker Conditions ◽  
Theoretical Results ◽  
Nondifferentiable Operator

In this paper we present a two-step solver for nonlinear equations with a nondifferentiable operator. This method is based on two methods of order of convergence 1 + 2 . We study the local and a semilocal convergence using weaker conditions in order to extend the applicability of the solver. Finally, we present the numerical example that confirms the theoretical results.

Semilocal Convergence of a Seventh-Order Method in Banach Spaces Under Hölder Continuity Condition

2020 ◽  
Vol 87 (1-2) ◽  
pp. 56
Author(s):  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Banach Spaces ◽  
Error Bound ◽  
Order Of Convergence ◽  
Nonlinear Operator ◽  
Lipschitz Continuity ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Theoretical Development ◽  
Numerical Examples ◽  
Seventh Order

The motive of this article is to analyze the semilocal convergence of a well existing iterative method in the Banach spaces to get the solution of nonlinear equations. The condition, we assume that the nonlinear operator fulfills the Hölder continuity condition which is softer than the Lipschitz continuity and works on the problems in which either second order Frèchet derivative of the nonlinear operator is challenging to calculate or does not hold the Lipschitz condition. In the convergence theorem, the existence of the solution x<sup>*</sup> and its uniqueness along with prior error bound are established. Also, the <em>R</em>-order of convergence for this method is proved to be at least 4+3q. Two numerical examples are discussed to justify the included theoretical development followed by an error bound expression.

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