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

Saxena, Akanksha; Argyros, Ioannis K.; Jaiswal, J. P.; Argyros, Christopher I.; Pardasani, Kamal Raj

doi:10.3390/math9060669

Cited by 4 publications

(4 citation statements)

References 16 publications

(35 reference statements)

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“…Recently, the local convergence of a two-step Newton type method of convergence rate three under generalized Lipschitz conditions has been studied by Saxena et. al [12] whose definitions will be used in this article.…”

Section: Introductionmentioning

confidence: 99%

Convergence Criteria of a Three Step Scheme under generalized Lipschitz Condition in Banach Spaces

Saxena¹,

Jaiswal²,

Pardasani³

2022

Preprint

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The goal of this study is to investigate the local convergence of a three-step Newton-Traub technique for solving nonlinear equations in Banach spaces with a convergence rate of five. The first order derivative of a nonlinear operator is assumed to satisfy the generalized Lipschitz condition, i.e. the κ-average condition. Furthermore, a few results on the convergence of the same method in Banach spaces are developed under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak κ-average, and that κ is a positive integrable function but not necessarily nondecreasing. Our new notion provides a tighter convergence analysis without the need for new circumstances. As a result, we broaden the applicability of iterative approaches. Theoretical results are supported further by illuminating examples. The existence and uniqueness of the solution x * are examined in the convergence theorem. In the end, we achieve weaker sufficient convergence criteria and more specific information on the position of the solution than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for κ(u). A numerical test is carried out to corroborate the hypothesis established in this work.

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Section: Introductionmentioning

confidence: 99%

Convergence Criteria of a Three Step Scheme under generalized Lipschitz Condition in Banach Spaces

Saxena¹,

Jaiswal²,

Pardasani³

2022

Preprint

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“…Kantorovich [2] was the first to investigate this in a Banach space setting. Then, it was re-evaluated in a plethora of papers [1,[3][4][5][6][7][8][9][10][11][12].…”

Section: Introductionmentioning

confidence: 99%

“…Shakhno [25] explored the convergence criteria of the Secant-type with a two-step approach [2], when the generalized Lipschitz conditions were satisfied by the first-order division differences. Recently, a Newton-type two-step approach to the ball convergence, with a convergence rate of three under generalized Lipschitz conditions was demonstrated by Saxena et al [10], whose definitions will be used in this article.…”

Section: Introductionmentioning

confidence: 99%

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Convergence Criteria of a Three-Step Scheme under the Generalized Lipschitz Condition in Banach Spaces

et al. 2022

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In the given study, we investigate the three-step NTS’s ball convergence for solving nonlinear operator equations with a convergence order of five in a Banach setting. A nonlinear operator’s first-order derivative is assumed to meet the generalized Lipschitz condition, also known as the κ-average condition. Furthermore, several theorems on the convergence of the same method in Banach spaces are developed with the conditions that the derivative of the operators must satisfy the radius or center-Lipschitz condition with a weak κ-average and that κ is a positive integrable but not necessarily non-decreasing function. This novel approach allows for a more precise convergence analysis even without the requirement for new circumstances. As a result, we broaden the applicability of iterative approaches. The theoretical results are supported further by illuminating examples. The convergence theorem investigates the location of the solution ϵ* and the existence of it. In the end, we achieve weaker sufficient convergence criteria and more specific knowledge on the position of the ϵ* than previous efforts requiring the same computational effort. We obtain the convergence theorems as well as some novel results by applying the results to some specific functions for κ(u). Numerical tests are carried out to corroborate the hypotheses established in this work.

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Extended convergence ball for an efficient eighth order method using only the first derivative

Argyros

Sharma

Argyros

et al. 2022

SeMA

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

Cited by 4 publications

References 16 publications

Convergence Criteria of a Three Step Scheme under generalized Lipschitz Condition in Banach Spaces

Convergence Criteria of a Three Step Scheme under generalized Lipschitz Condition in Banach Spaces

Convergence Criteria of a Three-Step Scheme under the Generalized Lipschitz Condition in Banach Spaces

Extended convergence ball for an efficient eighth order method using only the first derivative

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