New results on newton-kantorovich approximations with applications to nonlinear integral equations

Appell, Jürgen; Pascale, Espedito De; Lysenko, Julia V.; Zabrejko, Petr P.

doi:10.1080/01630569708816744

Cited by 50 publications

(47 citation statements)

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“…Condition (5) is a scaled Riemannian analogue of the property used by Zabrejko and Nguen in [25] (see also [2,24]) to prove a local convergence result for Newton's method in Banach spaces, based on a radial parametrization of the original "majorant method" developed by Kantorovich [12]. Here, scaling means that the inverse of X (p 0 ) is incorporated in the distance between covariant derivatives of X, an idea that has been already used in the Banach space context (see [4,23]).…”

Section: Rr N°5381mentioning

confidence: 99%

“…This technique was originally developed by Zabrejko and Nguen in [25] (see also [2,24]) in order to obtain refinements of the Kantorovich theorem in Banach spaces. In fact, this approach is based on the construction of a real-valued function, namely the majorant function, using an adequate local Lipschitz-type radial estimate for the first derivative of X.…”

Section: Introductionmentioning

confidence: 99%

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A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

2006

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Abstract:We consider the problem of finding a singularity of a vector field X on a complete Riemannian manifold. In this regard we prove a unified result for local convergence of Newton's method. Inspired by previous work of Zabrejko and Nguen on Kantorovich's majorant method, our approach relies on the introduction of an abstract one-dimensional Newton's method obtained using an adequate Lipschitz-type radial function of the covariant derivative of X. The main theorem gives in particular a synthetic view of several famous results, namely the Kantorovich, Smale and Nesterov-Nemirovskii theorems. Concerning real-analytic vector fields an application of the central result leads to improvements of some recent developments in this area.

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Section: Rr N°5381mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

2006

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“…, we obtain the main theorem in [1]. Note however that uniqueness results are not given in either [1] or [12].…”

Section: Semilocal Convergence Analysismentioning

confidence: 96%

“…In this study we are concerned with the problem of approximating a locally unique solution x * of the equation (1) F (x) = 0, where F is a Fréchet-differentiable operator defined on a closed convex subset D of a Banach space X with values in a Banach space Y .…”

Section: Introductionmentioning

confidence: 99%

New unifying convergence criteria for Newton-like methods

Argyros

2002

Appl. Math. (Warsaw)

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Abstract. We present a local and a semilocal analysis for Newton-like methods in a Banach space. Our hypotheses on the operators involved are very general. It turns out that by choosing special cases for the "majorizing" functions we obtain all previous results in the literature, but not vice versa. Since our results give a deeper insight into the structure of the functions involved, we can obtain semilocal convergence under weaker conditions and in the case of local convergence a larger convergence radius.

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“…A survey of sufficient conditions for the local as well as the semilocal convergence of Newton-type methods as well as an error analysis for such methods can be found in [1]- [5], [9] and the references there.…”

Section: Introductionmentioning

confidence: 99%

On the "Terra Incognita" for the Newton-Kantrovich Method With Applications

Argyros¹,

Cho²,

George³

2014

Journal of the Korean Mathematical Society

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Abstract. In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]- [13], in some interesting cases, provided that the Fréchet-derivative of the operator involved is p-Hölder continuous (p ∈ (0, 1]). Numerical examples involving two boundary value problems are also provided.

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New results on newton-kantorovich approximations with applications to nonlinear integral equations

Cited by 50 publications

References 1 publication

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

A Unifying Local Convergence Result for Newton's Method in Riemannian Manifolds

New unifying convergence criteria for Newton-like methods

On the "Terra Incognita" for the Newton-Kantrovich Method With Applications

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