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Convergence domains under Zabrejko-Zinčenko conditions using recurrent functions

100%
Argyros I., Hilout S.
Applicationes Mathematicae
|
2011
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tom 38
|
nr 2
193-209
EN
We provide a semilocal convergence analysis for Newton-type methods using our idea of recurrent functions in a Banach space setting. We use Zabrejko-Zinčenko conditions. In particular, we show that the convergence domains given before can be extended under the same computational cost. Numerical examples are also provided to show that we can solve equations in cases not covered before.
2
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On the convergence of Newton's method under ω*-conditioned second derivative

100%
Argyros I., Hilout S.
Applicationes Mathematicae
|
2011
|
tom 38
|
nr 3
341-355
EN
We provide a new semilocal result for the quadratic convergence of Newton's method under ω*-conditioned second Fréchet derivative on a Banach space. This way we can handle equations where the usual Lipschitz-type conditions are not verifiable. An application involving nonlinear integral equations and two boundary value problems is provided. It turns out that a similar result using ω-conditioned hypotheses can provide usable error estimates indicating only linear convergence for Newton's method.
3
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Expanding the applicability of two-point Newton-like methods under generalized conditions

100%
Argyros I., Hilout S.
Applicationes Mathematicae
|
2013
|
tom 40
|
nr 1
63-90
EN
We use a two-point Newton-like method to approximate a locally unique solution of a nonlinear equation containing a non-differentiable term in a Banach space setting. Using more precise majorizing sequences than in earlier studies, we present a tighter semi-local and local convergence analysis and weaker convergence criteria. This way we expand the applicability of these methods. Numerical examples are provided where the old convergence criteria do not hold but the new convergence criteria are satisfied.
4
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Inexact Newton methods and recurrent functions

100%
Argyros I., Hilout S.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 1
113-126
EN
We provide a semilocal convergence analysis for approximating a solution of an equation in a Banach space setting using an inexact Newton method. By using recurrent functions, we provide under the same or weaker hypotheses: finer error bounds on the distances involved, and an at least as precise information on the location of the solution as in earlier papers. Moreover, if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained. Furthermore, numerical examples are provided using polynomial, integral and differential equations.
5
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On the convergence of two-step Newton-type methods of high efficiency index

100%
Argyros I., Hilout S.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 4
465-499
EN
We introduce a new idea of recurrent functions to provide a new semilocal convergence analysis for two-step Newton-type methods of high efficiency index. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar type, and a differential equation containing a Green's kernel are also provided.
6
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Newton's methods for variational inclusions under conditioned Fréchet derivative

100%
Argyros I., Hilout S.
Applicationes Mathematicae
|
2007
|
tom 34
|
nr 3
349-357
EN
Estimates of the radius of convergence of Newton's methods for variational inclusions in Banach spaces are investigated under a weak Lipschitz condition on the first Fréchet derivative. We establish the linear convergence of Newton's and of a variant of Newton methods using the concepts of pseudo-Lipschitz set-valued map and ω-conditioned Fréchet derivative or the center-Lipschitz condition introduced by the first author.
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