𝔖 Bobbio Scriptorium
✦   LIBER   ✦

Newton methods on Banach spaces with a convergence structure and applications

✍ Scribed by I.K. Argyros

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
582 KB
Volume
40
Category
Article
ISSN
0898-1221

No coin nor oath required. For personal study only.

✦ Synopsis

In this study, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the metric properties of the examined problem can be analyzed more precisely. Moreover, this approach allows us to derive from the same theorem, on the one hand, semilocal results of Kantorovich-type, and on the other hand, global results based on monotonicity considerations. By imposing very general Lipschitz-like conditions on the operators involved, on the one hand, we cover a wider range of problems, and on the other hand, by choosing our operators appropriately we can find sharper error bounds on the distances involved than before. Finally, our methods are used to solve integral equations that cannot be solved with existing methods. (~) 2000 Elsevier Science Ltd. All rights reserved.

πŸ“œ SIMILAR VOLUMES

Improving the rate of convergence of New
Improving the rate of convergence of Newton methods on Banach spaces with a convergence structure and applications
✍ I.K. Argyros πŸ“‚ Article πŸ“… 1997 πŸ› Elsevier Science 🌐 English βš– 589 KB

this note, we use inexact Newton-like methods to find solutions of nonlinear operator equations on Banach spaces with a convergence structure. Our technique involves the introduction of a generalized norm as an operator from a linear space into a partially ordered Banach space. In this way, the met

On the semilocal convergence of inexact
On the semilocal convergence of inexact Newton methods in Banach spaces
✍ Ioannis K. Argyros πŸ“‚ Article πŸ“… 2009 πŸ› Elsevier Science 🌐 English βš– 489 KB

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134-144; I.K. Argyros, A

On the linear convergence of quasi-newto
On the linear convergence of quasi-newton methods in finite-and infinite-dimensional Hilbert spaces
✍ A. Lannes πŸ“‚ Article πŸ“… 1980 πŸ› Elsevier Science 🌐 English βš– 415 KB

This paper summarizes the main results concerning the analysis of the local convergence of quasi-newton methods in finite and infinite-dimensional Hilbert spaces. Although the physicist working on the computer is essentially concerned with the finite-dimensional case (i.e. the discrete case), it is

A Steffensen's type method in Banach spa
A Steffensen's type method in Banach spaces with applications on boundary-value problems
✍ Virginia AlarcΓ³n; Sergio Amat; Sonia Busquier; David J. LΓ³pez πŸ“‚ Article πŸ“… 2008 πŸ› Elsevier Science 🌐 English βš– 147 KB

In this paper, a modified Steffensen's type iterative scheme for the numerical solution of a system of nonlinear equations is studied. Two convergence theorems are presented. The numerical solution of boundary-value problems by the multiple shooting method using the proposed iterative scheme is anal

Invariant Submeans and Semigroups of Non
Invariant Submeans and Semigroups of Nonexpansive Mappings on Banach Spaces with Normal Structure
✍ Anthony To-Ming Lau; Wataru Takahashi πŸ“‚ Article πŸ“… 1996 πŸ› Elsevier Science 🌐 English βš– 345 KB

In this paper we study the relation between invariant submean and normal structure in a Banach space. This is used to give an improvement and different proof of a fixed point theorem of Lim (also of Belluce and Kirk for commutative semigroups) for left reversible semigroup of nonexpansive mappings o