Generalized solutions of operator equations and extreme elements
type: Item
instance of: scholarly article
title: Generalized solutions of operator equations and extreme elements
zbMATH Open document ID: 1240.47002
DOI: 10.1007/978-1-4614-0619-8
author: Juan-Miguel Gracia
published in: Springer Optimization and Its Applications
publication date: 2011-08-29T00:00:00Z
review text: In the first part of this monograph, various notions of generalized solutions of linear equations \(\mathcal{L}u = f \) are introduced. Here, \( \mathcal{L}: D(\mathcal{L}) \to F \) is a linear injective operator between Banach spaces \( E \) and \( F \), with dense domain \( D(\mathcal{L}) \subset E \) and dense range in \( F \). Each of the considered generalized solutions belongs to a completion of \( D(\mathcal{L}) \) with respect to some Hausdorff topology. For example, consider the completion \( \overline{E} \) of \( D(\mathcal{L}) \) with respect to the norm \( \| u \|_{\overline{E}} = \| \mathcal{L} u \|_F \), where \( \| \cdot \|_F \) denotes the norm in the Banach space \( F \). An element \( u \in \overline{E} \) is then called a strong generalized solution if \( \overline{\mathcal{L}} u = f \) is satisfied, where \( \overline{\mathcal{L}}: \overline{E} \to F \) denotes the continuous continuation of \( \mathcal{L} \) with respect to the norms \( \| \cdot \|_{\overline{E}} \) and \( \| \cdot \|_F \). The authors also introduce other kinds of generalized solutions, e.g., weak generalized solutions, strong near-solutions and weak near-solutions. Existence and unique results are presented, respectively, and relations between the various notions are given. Subsequently, applications of the presented theory to special problems are considered, e.g., Volterra integral equations of the first kind, infinite systems of linear equations, partial differential equations of parabolic type, and wave equations. The next chapter is devoted to the approximate computation of classical and generalized solutions of linear operator equations. Special emphasis is put on the Neumann series type iteration method \( \widetilde{x}_k = \mu(I+U+U^2+\dots+ U^k) b, \;k = 0,1,2,\dots, \) where \( U = I -\mu A \) and \( \mu > 0 \), and \( A: H \to H \) is a linear bounded operator in a Hilbert space \( H \). The operator version of this iterative scheme, that is, Schultz's method, is als…
reviewed by: Robert Plato
Mathematics Subject Classification ID: 47-02
Mathematics Subject Classification ID: 49-02
Mathematics Subject Classification ID: 47A50
Mathematics Subject Classification ID: 47N20
Mathematics Subject Classification ID: 65J10
Mathematics Subject Classification ID: 47J05
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 49J27
Mathematics Subject Classification ID: 54E15
zbMATH DE Number: 5944101
zbMATH Keywords: linear unbounded operator
zbMATH Keywords: generalized solution
zbMATH Keywords: completion
zbMATH Keywords: Banach space
zbMATH Keywords: metric space
zbMATH Keywords: topological vector space
zbMATH Keywords: uniform structure
zbMATH Keywords: Volterra integral equation of the first kind
zbMATH Keywords: infinite system of linear equations
zbMATH Keywords: partial differential equation of parabolic type
zbMATH Keywords: wave equation
zbMATH Keywords: optimization
zbMATH Keywords: nonlinear equation
zbMATH Keywords: Neumann series
zbMATH Keywords: Schultz's method
zbMATH Keywords: Hoteling method
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-1-4614-0619-8
OpenAlex ID: W1260549156
description: scientific article; zbMATH DE number 5944101