DSM of Newton type for solving operator equations \(F(u) = f\) with minimal smoothness assumptions on \(F\)
type: Item
instance of: scholarly article
title: DSM of Newton type for solving operator equations \(F(u) = f\) with minimal smoothness assumptions on \(F\)
zbMATH Open document ID: 1205.65188
DOI: 10.1504/IJCSM.2010.033926
published in: International Journal of Computing Science and Mathematics
publication date: 2010-08-17T00:00:00Z
review text: Authors' abstract: This paper is a review of the authors' results on the dynamical systems method (DSM) for solving operator equation (*) \(F(u) = f\). It is assumed that (*) is solvable. The novel feature of the results is the minimal assumption on the smoothness of \(F\). It is assumed that \(F\) is continuously Fréchet differentiable, but no smoothness assumptions on \(F^{\prime}(u)\) are imposed. The DSM for solving equation \((*)\) is developed. Under weak assumptions global existence of the solution \(u(t)\) is proved, the existence of \(u(\infty )\) is established, and the relation \(F(u(\infty )) = f\) is obtained. The DSM is developed for a stable solution of equation \((*)\) when noisy data \(f\delta \) are given, \(| f - f| \leq \delta \).
reviewed by: Boro Döring
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 47H05
zbMATH DE Number: 5771943
zbMATH Keywords: DSM
zbMATH Keywords: dynamical systems method
zbMATH Keywords: nonlinear operator equations
zbMATH Keywords: monotone operators
zbMATH Keywords: discrepancy principle
zbMATH Keywords: smoothness assumptions
author: Nguyen Si Hoang
author: A. G. Ramm
MaRDI profile type: Publication
full work available at URL: ijcsm.2010.033926
OpenAlex ID: W2123580826
description: scientific article; zbMATH DE number 5771943