A global representation of all solutions to a nonlinear equation and its applications
type: Item
instance of: scholarly article
title: A global representation of all solutions to a nonlinear equation and its applications
zbMATH Open document ID: 0782.47052
author: Shige Peng
author: Jiongmin Yong
published in: Chinese Annals of Mathematics. Series B
publication date: 1993-04-01T00:00:00Z
review text: Let \(U\) be a Hilbert space and \(Z\) be a Banach space. The equation \[ h(u)=0\qquad h: U\to Z\tag{1} \] with continuous map \(h\) is considered. For affine \(h\), i.e. \(h(u)= Gu-z\), \(\forall u\in U\), for some \(G\in{\mathcal L}(U,Z)\) and some \(z\in Z\), with invertible \(GG^*\in{\mathcal L}(Z)\) all possible solutions of (1) can be represented in the form \(u=G^*(GG^*)^{-1} z+ (I-G^*(GG^*)^{-1} G)v\) with \(v\in U\) as a parameter [see \textit{J. P. Aubin}, Applied functional analysis (1979; Zbl 0424.46001)]. With the aid of this representation the authors obtain a nonlinear generalized inverse theorem and give applications to control systems with mixture constraints.
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 148498
zbMATH Keywords: nonlinear generalized inverse theorem
zbMATH Keywords: control systems with mixture constraints
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 148498