On the local convergence of Weerakoon–Fernando method with $$\omega $$ continuity condition in Banach spaces

SeMA Journal ◽  
2020 ◽  
Vol 77 (3) ◽  
pp. 291-304 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Debasis Sharma ◽  
Sanjaya Kumar Parhi
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Continuity Condition

scholarly journals Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers

2021 ◽  
Vol 5 (2) ◽  
pp. 27
Author(s):  
Debasis Sharma ◽  
Ioannis K. Argyros ◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Fourth Order ◽  
Local Convergence ◽  
Taylor Expansion ◽  
Iterative Algorithms ◽  
Continuity Condition ◽  
Dynamical Analysis ◽  
Local Analysis ◽  
Uniqueness Of The Solution

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

scholarly journals Extending the Applicability of a Two-Step Chord-Type Method for Non-Differentiable Operators

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 804
Author(s):  
Ioannis K. Argyros ◽  
Neha Gupta ◽  
J. P. Jaiswal
Keyword(s):  
Approximate Solution ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Recurrence Relation ◽  
Convergence Theorem ◽  
Local Convergence ◽  
Existence And Uniqueness ◽  
Numerical Illustration ◽  
Type Method ◽  
Local Convergence Analysis

The semi-local convergence analysis of a well defined and efficient two-step Chord-type method in Banach spaces is presented in this study. The recurrence relation technique is used under some weak assumptions. The pertinency of the assumed method is extended for nonlinear non-differentiable operators. The convergence theorem is also established to show the existence and uniqueness of the approximate solution. A numerical illustration is quoted to certify the theoretical part which shows that earlier studies fail if the function is non-differentiable.

Convergence of Traub's iteration under ω continuity condition in Banach spaces

2021 ◽  
pp. 61-79
Author(s):  
Debasis Sharma ◽  
◽  
Sanjaya Kumar Parhi ◽  
Shanta Kumari Sunanda ◽  
◽  
...  
Keyword(s):  
Banach Spaces ◽  
Continuity Condition

scholarly journals Convergence Behavior for Newton-Steffensen’s Method under -Condition of Second Derivative

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Yonghui Ling ◽  
Xiubin Xu ◽  
Shaohua Yu
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Nonlinear Operator ◽  
Convergence Criterion ◽  
Second Derivative ◽  
Operator Equations ◽  
Convergence Ball ◽  
Nonlinear Operator Equations ◽  
Steffensen’S Method ◽  
Convergence Problems

The present paper is concerned with the semilocal as well as the local convergence problems of Newton-Steffensen’s method to solve nonlinear operator equations in Banach spaces. Under the assumption that the second derivative of the operator satisfies -condition, the convergence criterion and convergence ball for Newton-Steffensen’s method are established.

On the local convergence of modified Weerakoon’s method in Banach spaces

2019 ◽  
Vol 28 (3) ◽  
pp. 867-877 ◽  
Author(s):  
Debasis Sharma ◽  
Sanjaya Kumar Parhi
Keyword(s):  
Banach Spaces ◽  
Local Convergence

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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