The solution by iteration of a composed K-positive definite operator equation in a Banach space
type: Item
instance of: scholarly article
title: The solution by iteration of a composed K-positive definite operator equation in a Banach space
zbMATH Open document ID: 1228.47040
DOI: 10.1155/2010/376852
author: S. J. Aneke
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2010-09-28T00:00:00Z
full work available at URL: https://eudml.org/doc/230392
review text: The article deals with linear equations of the type \(Lu = f\) in a Banach space \(E\), where \(L = A + B\), \(A\) is a \(K\)-positive definite (or an asymptotically \(K\)-positive definite) operator, and \(B\) is a linear operator. In the article, an operator \(A\) is called \(K\) positive definite if \(\langle Au,jKu \rangle \geq \alpha\|Ku\|^2\), \(u \in D(A)\), where \(K\) is an operator defined on a dense subspace \(E_0\) of \(E\) and \(K\) has a bounded inverse on \(R(T)\); an operator \(A\) is called asymptotically \(K\) positive definite if \(\langle K^{n-1}Au,j(K^n) \rangle \geq ck_n\|K^nu\|^2\), \(u \in D(A)\), where \(c > 0\), \(k_n \geq 1\), \(k_n \to 1\) (in these definitions, \(j(\cdot) \in J(\cdot)\), \(J\) is the normalized duality mapping for \(E\)). The main results are concerned with conditions under which the iterations \(x_{n+1} = x_n + r_n\), \(r_n = K^{-1}f - K^{-1}Ar_n\), converge strongly to the unique solution of \(Ax = f\) (the original equation \(Lu = f\) is not considered in the correspondent Theorems 3.3 and 3.5). Reviewer's remarks. All these results seem to be erroneous; in order to verify this fact, it is sufficient to cite some sentences from the article: Since \(A\) is continuously invertible, the operator \(T = A^{-1}\) is completely continuous. Hence \(T\) is locally lipschitzian and accretive. It follows that (3.3) (i.e., \(u + Tu = g\)) has a unique solution. Further, the author considers an example when \(B = A\) and thus \(B^{-1}A = I\)! The definition of an asymptotically \(K\) positive definite operator is also senseless.
Mathematics Subject Classification ID: 47B99
Mathematics Subject Classification ID: 47B44
zbMATH DE Number: 5792474
zbMATH Keywords: operator equations
zbMATH Keywords: \(K\)-positive definite operators
zbMATH Keywords: inverse function theorem
zbMATH Keywords: iterative methods of approximate solving
Wikidata QID: Q58651758
reviewed by: Peter P. Zabreĭko
MaRDI profile type: Publication
OpenAlex ID: W2077725024
cites work: Direct and Iterative Methods for the Solution of Linear Operator Equations in Hilbert Space
cites work: Construction of fixed points of nonlinear mappings in Hilbert space
cites work: Q3759439
cites work: Q5832587
cites work: Existence, uniqueness and approximation of a solution for a k-positive definite operator equation
cites work: A LOCAL APPROXIMATION METHOD FOR THE SOLUTION OF K-POSITIVE DEFINITE OPERATOR EQUATIONS
cites work: Nonlinear semigroups and evolution equations
cites work: Q4358791
DBLP publication ID: journals/ijmmsc/Aneke10
description: scientific article; zbMATH DE number 5792474