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

Sharma, Debasis; Parhi, Sanjaya Kumar

doi:10.1007/s41478-019-00216-x

Cited by 12 publications

(5 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…0 L(u)udu is non-decreasing with respect to t. Next, on arbitrarily choosing x 0 ∈ V(x * , r) and using the non-decreasing property of 1 t 2 t 0 L(u)udu and the inequality (14), it follows that…”

Section: Local Convergence Of Newton Type Methods (3)mentioning

confidence: 99%

“…Numerous researchers studied the local convergence analysis for Newton-type, Jarratt-type, Weerakoon-type, etc. in Banach space setting in the articles [8][9][10][11][12][13][14] and reference therein. In most of articles, the local convergence have been discussed using the hypotheses of Lipschitz, Hölder or w-continuity conditions but sometimes, we will come across that the nonlinear problems do not fulfilled any of these three conditions which limits the applicability of nonlinear equations, but satisfy the generalized Lipschitz condition.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

et al. 2021

View full text Add to dashboard Cite

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.

show abstract

Section: Local Convergence Of Newton Type Methods (3)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

View full text Add to dashboard Cite

show abstract

“…The semilocal convergence study uses information around a starting point to provide criteria for ensuring the convergence of iterative methods, whereas the local one uses information around a solution to estimate the radii of the convergence balls. Numerous authors investigated the local convergence analysis for Newton-type, Jarratt-type, Weerakoon-type, and other types in Banach space in works [ [1], [3], [6], [8], [14]].…”

Section: Introductionmentioning

confidence: 99%

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

Saxena¹,

Jaiswal²,

Pardasani³

2022

Preprint

View full text Add to dashboard Cite

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.

show abstract

“…The local one determines the balls relying on the information surrounding * . A number of authors have investigated the convergence criteria in the local sense for Jarratt-like, Newton-like, Weerakoon-like, and various others in a Banach setup [3,[19][20][21][22].…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2022

View full text Add to dashboard Cite

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.

show abstract

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

Cited by 12 publications

References 17 publications

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

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

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

Contact Info

Product

Resources

About