scholarly journals On the Local Convergence of Regula-falsi-type Method for Generalized Equations

2017 ◽  
Vol 2 (3) ◽  
Author(s):  
Farhana Alam ◽  
◽  
M. H. Rashid ◽  
M. A. Alom ◽  
◽  
...  
Keyword(s):  
Local Convergence ◽  
Generalized Equations ◽  
Type Method

scholarly journals Convergence Properties of Extended Newton-type Iteration Method for Generalized Equations

2018 ◽  
Vol 10 (4) ◽  
pp. 1
Author(s):  
M. Khaton ◽  
M. Rashid ◽  
M. Hossain
Keyword(s):  
Banach Spaces ◽  
Local Convergence ◽  
Iteration Method ◽  
Generalized Equation ◽  
Generalized Equations ◽  
Convergence Properties ◽  
Closed Graph ◽  
Point Bar ◽  
Type Method ◽  
Set Valued Mapping

In this paper, we introduce and study the extended Newton-type method for solving generalized equation $0\in f(x)+g(x)+\mathcal F(x)$, where $f:\Omega\subseteq\mathcal X\to \mathcal Y$ is Fr\'{e}chet differentiable in a neighborhood $\Omega$ of a point $\bar{x}$ in $\mathcal X$, $g:\Omega\subseteq \mathcal X\to \mathcal Y$ is linear and differentiable at a point $\bar{x}$, and $\mathcal F$ is a set-valued mapping with closed graph acting in Banach spaces $\mathcal X$ and $\mathcal Y$. Semilocal and local convergence of the extended Newton-type method are analyzed.

scholarly journals Convergence of the Newton-Type Method for Generalized Equations

2016 ◽  
Vol 35 ◽  
pp. 27-40 ◽  
Author(s):  
MH Rashid ◽  
A Sarder
Keyword(s):  
Banach Spaces ◽  
Differentiable Function ◽  
Local Convergence ◽  
Generalized Equation ◽  
Generalized Equations ◽  
Closed Graph ◽  
Type Method ◽  
Set Valued Mapping ◽  
Fréchet Differentiable

Let X and Y be real or complex Banach spaces. Suppose that f: X->Y is a Frechet differentiable function and F: X => 2Yis a set-valued mapping with closed graph. In the present paper, we study the Newton-type method for solving generalized equation 0 ? f(x) + F(x). We prove the existence of the sequence generated by the Newton-type method and establish local convergence of the sequence generated by this method for generalized equation.GANIT J. Bangladesh Math. Soc.Vol. 35 (2015) 27-40

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.

scholarly journals On an Aitken-Steffensen-Newton type method

2018 ◽  
Vol 34 (1) ◽  
pp. 85-92
Author(s):  
ION PAVALOIU ◽  
Keyword(s):  
Local Convergence ◽  
Convergence Result ◽  
Convergence Order ◽  
Monotone Convergence ◽  
Type Method ◽  
Numerical Examples ◽  
Newton Iterations ◽  
Convergence Orders

We consider an Aitken-Steffensen type method in which the nodes are controlled by Newton and two-step Newton iterations. We prove a local convergence result showing the q-convergence order 7 of the iterations. Under certain supplementary conditions, we obtain monotone convergence of the iterations, providing an alternative to the usual ball attraction theorems. Numerical examples show that this method may, in some cases, have larger (possibly sided) convergence domains than other methods with similar convergence orders.

scholarly journals On the Semi-Local Convergence of a Traub-Type Method for Solving Equations

Foundations ◽  
2022 ◽  
Vol 2 (1) ◽  
pp. 114-127
Author(s):  
Samundra Regmi ◽  
Christopher I. Argyros ◽  
Ioannis K. Argyros ◽  
Santhosh George
Keyword(s):  
Banach Space ◽  
Local Convergence ◽  
Type Method ◽  
Lipschitz Constants ◽  
Solving Equations ◽  
Original Domain ◽  
Theoretical Results

The celebrated Traub’s method involving Banach space-defined operators is extended. The main feature in this study involves the determination of a subset of the original domain that also contains the Traub iterates. In the smaller domain, the Lipschitz constants are smaller too. Hence, a finer analysis is developed without the usage of additional conditions. This methodology applies to other methods. The examples justify the theoretical results.

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