On the local convergence of Newton-like methods with fourth and fifth order
of convergence under hypotheses only on the first Fréchet derivative

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.

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Local convergence of Super Halley’s method under weaker conditions on Fréchet derivative in Banach spaces

The Journal of Analysis ◽

10.1007/s41478-017-0034-9 ◽

2017 ◽

Vol 28 (1) ◽

pp. 35-44 ◽

Cited By ~ 1

Author(s):

Abhimanyu Kumar ◽

D. K. Gupta

Keyword(s):

Banach Spaces ◽

Local Convergence ◽

Fréchet Derivative ◽

Frechet Derivative ◽

Halley’S Method ◽

Halley's Method ◽

Weaker Conditions

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Local convergence of inexact Newton methods under affine invariant conditions and hypotheses on the second Fréchet derivative

Applicationes Mathematicae ◽

10.4064/am-26-4-457-465 ◽

1999 ◽

Vol 26 (4) ◽

pp. 457-465

Author(s):

Ioannis Argyros

Keyword(s):

Local Convergence ◽

Fréchet Derivative ◽

Affine Invariant ◽

Newton Methods ◽

Inexact Newton Methods ◽

Frechet Derivative

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Local convergence of deformed Euler–Halley-type methods in Banach space under weak conditions

Asian-European Journal of Mathematics ◽

10.1142/s1793557117500863 ◽

2017 ◽

Vol 10 (02) ◽

pp. 1750086

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Nonlinear Equation ◽

Convergence Analysis ◽

Error Estimates ◽

Local Convergence ◽

Fréchet Derivative ◽

Convergence Ball ◽

Numerical Examples ◽

Frechet Derivative ◽

Local Convergence Analysis

We present a unified local convergence analysis for deformed Euler–Halley-type methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Euler, Halley and other high order methods. The convergence ball and error estimates are given for these methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second Fréchet derivative. Numerical examples are also provided in this study.

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Local convergence of deformed Jarratt-type methods in Banach space without inverses

Asian-European Journal of Mathematics ◽

10.1142/s1793557116500157 ◽

2016 ◽

Vol 09 (01) ◽

pp. 1650015

Author(s):

Ioannis K. Argyros ◽

Santhosh George

Keyword(s):

Banach Space ◽

Convergence Analysis ◽

Local Convergence ◽

Computational Cost ◽

Fréchet Derivative ◽

Convergence Conditions ◽

Type Method ◽

Frechet Derivative ◽

The Third ◽

Local Convergence Analysis

We present a local convergence analysis for the Jarratt-type method of high convergence order in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Earlier studies use hypotheses up to the third Fréchet-derivative. Hence, the applicability of these methods is expanded under weaker hypotheses and less computational cost for the constants involved in the convergence analysis. Numerical examples are also provided in this study.

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On the semi-local convergence of Halley’s method under a center-Lipschitz condition on the second Fréchet derivative

Applied Mathematics and Computation ◽

10.1016/j.amc.2012.04.078 ◽

2012 ◽

Vol 218 (23) ◽

pp. 11488-11495 ◽

Cited By ~ 5

Author(s):

Hongmin Ren ◽

Ioannis K. Argyros

Keyword(s):

Lipschitz Condition ◽

Local Convergence ◽

Fréchet Derivative ◽

Frechet Derivative ◽

Halley’S Method ◽

Halley's Method

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A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

The Journal of the Australian Mathematical Society Series B Applied Mathematics ◽

10.1017/s0334270000010432 ◽

1995 ◽

Vol 36 (3) ◽

pp. 261-273 ◽

Cited By ~ 3

Author(s):

Mohammad A. Kazemi

Keyword(s):

Optimal Control ◽

Partial Differential Equations ◽

Differential Equations ◽

Optimal Control Problems ◽

Fréchet Derivative ◽

Necessary Condition ◽

Control Problems ◽

Partial Differential ◽

Frechet Derivative ◽

First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

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