scholarly journals On semilocal convergence analysis for two-step Newton method under generalized Lipschitz conditions in Banach spaces

2021 ◽  
Author(s):  
Yonghui Ling ◽  
Juan Liang ◽  
Weihua Lin
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Newton Method ◽  
Semilocal Convergence ◽  
Lipschitz Conditions

scholarly journals Semilocal convergence of a Secant-type method under weak Lipschitz conditions in Banach spaces

2018 ◽  
Vol 330 ◽  
pp. 732-741 ◽  
Author(s):  
Abhimanyu Kumar ◽  
D.K. Gupta ◽  
Eulalia Martínez ◽  
Sukhjit Singh
Keyword(s):  
Banach Spaces ◽  
Semilocal Convergence ◽  
Type Method ◽  
Lipschitz Conditions

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 Analysis for Inexact Newton Method under Weak Condition

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Xiubin Xu ◽  
Yuan Xiao ◽  
Tao Liu
Keyword(s):  
Newton Method ◽  
Convergence Theorem ◽  
Semilocal Convergence ◽  
Weak Condition ◽  
Inexact Newton Method ◽  
Convergence Criteria ◽  
First Derivative ◽  
Special Cases ◽  
Lipschitz Conditions ◽  
Semilocal Convergence Theorem

Under the hypothesis that the first derivative satisfies some kind of weak Lipschitz conditions, a new semilocal convergence theorem for inexact Newton method is presented. Unified convergence criteria ensuring the convergence of inexact Newton method are also established. Applications to some special cases such as the Kantorovich type conditions andγ-Conditions are provided and some well-known convergence theorems for Newton's method are obtained as corollaries.

Semilocal Convergence Analysis of S-iteration Process of Newton–Kantorovich Like in Banach Spaces

2016 ◽  
Vol 172 (1) ◽  
pp. 102-127 ◽  
Author(s):  
Daya Ram Sahu ◽  
Jen Chih Yao ◽  
Vipin Kumar Singh ◽  
Satyendra Kumar
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Iteration Process ◽  
Semilocal Convergence

SEMILOCAL CONVERGENCE OF A STIRLING-LIKE METHOD IN BANACH SPACES

2010 ◽  
Vol 07 (02) ◽  
pp. 215-228 ◽  
Author(s):  
S. K. PARHI ◽  
D. K. GUPTA
Keyword(s):  
Convergence Analysis ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Recurrence Relations ◽  
Lipschitz Continuity ◽  
Semilocal Convergence ◽  
Continuity Condition ◽  
Uniqueness Of Solutions ◽  
Third Order ◽  
Numerical Examples

The aim of this paper is to establish the semilocal convergence of a third order Stirling–like method employed for solving nonlinear equations in Banach spaces by using the first Fréchet derivative, which satisfies the Lipschitz continuity condition. This makes it possible to avoid the evaluation of higher order Fréchet derivatives which are computationally difficult at times or may not even exist. The recurrence relations are used for convergence analysis. A convergence theorem is given for deriving error bounds and the domains of existence and uniqueness of solutions. The R order of the method is also established to be equal to 3. Finally, two numerical examples are worked out, and the results obtained are compared with the existing results. It is observed that our convergence analysis is more effective.

scholarly journals On the Local Convergence of Two-Step Newton Type Method in Banach Spaces Under Generalized Lipschitz Conditions

Mathematics ◽  
2021 ◽  
Vol 9 (6) ◽  
pp. 669
Author(s):  
Akanksha Saxena ◽  
Ioannis K. Argyros ◽  
Jai P. Jaiswal ◽  
Christopher Argyros ◽  
Kamal R. Pardasani
Keyword(s):  
Convergence Rate ◽  
Convergence Analysis ◽  
Banach Spaces ◽  
Local Convergence ◽  
Nonlinear Operator ◽  
Integrable Function ◽  
Type Method ◽  
First Order ◽  
Average Condition ◽  
Lipschitz Conditions

The motive of this paper is to discuss the local convergence of a two-step Newton-type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e., L-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak L-average particularly it is assumed that L is positive integrable function but not necessarily non-decreasing. Our new idea gives a tighter convergence analysis without new conditions. The proposed technique is useful in expanding the applicability of iterative methods. Useful examples justify the theoretical conclusions.

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.

Improved semilocal convergence analysis in Banach space with applications to chemistry

2017 ◽  
Vol 56 (7) ◽  
pp. 1958-1975 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Elena Giménez ◽  
Á. A. Magreñán ◽  
Í. Sarría ◽  
Juan Antonio Sicilia
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Semilocal Convergence
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