Semilocal Convergence of Fourth Order Iterative Method in Banach Spaces by Using Recurrence Relations

2019 ◽  
Vol 7 (1) ◽  
pp. 11-16
Author(s):  
Deepak Kumar ◽  
Janak Raj Sharma
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Fourth Order ◽  
Recurrence Relations ◽  
Semilocal Convergence

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 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.

Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces

2010 ◽  
Vol 56 (4) ◽  
pp. 497-516 ◽  
Author(s):  
Xiuhua Wang ◽  
Chuanqing Gu ◽  
Jisheng Kou
Keyword(s):  
Banach Spaces ◽  
Fourth Order ◽  
Semilocal Convergence

scholarly journals A NOTE ON THE SEMILOCAL CONVERGENCE OF CHEBYSHEV’S METHOD

2012 ◽  
Vol 88 (1) ◽  
pp. 98-105 ◽  
Author(s):  
MANUEL A. DILONÉ ◽  
MARTÍN GARCÍA-OLIVO ◽  
JOSÉ M. GUTIÉRREZ
Keyword(s):  
Iterative Method ◽  
Banach Spaces ◽  
Nonlinear Equations ◽  
Semilocal Convergence ◽  
Iterative Processes ◽  
Chebyshev’S Method

AbstractIn this paper we develop a Kantorovich-like theory for Chebyshev’s method, a well-known iterative method for solving nonlinear equations in Banach spaces. We improve the results obtained previously by considering Chebyshev’s method as an element of a family of iterative processes.

scholarly journals Semilocal Convergence for a Fifth-Order Newton's Method Using Recurrence Relations in Banach Spaces

2011 ◽  
Vol 2011 ◽  
pp. 1-15 ◽  
Author(s):  
Liang Chen ◽  
Chuanqing Gu ◽  
Yanfang Ma
Keyword(s):  
Banach Spaces ◽  
Newton’S Method ◽  
Newton's Method ◽  
Recurrence Relations ◽  
A Priori ◽  
Semilocal Convergence ◽  
Numerical Application ◽  
A Priori Error Bounds ◽  
Modified Newton’S Method ◽  
Fifth Order

We study a modified Newton's method with fifth-order convergence for nonlinear equations in Banach spaces. We make an attempt to establish the semilocal convergence of this method by using recurrence relations. The recurrence relations for the method are derived, and then an existence-uniqueness theorem is given to establish the R-order of the method to be five and a priori error bounds. Finally, a numerical application is presented to demonstrate our approach.

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