Strongly nonlinear multivalued systems involving singular \(\Phi \)-Laplacian operators
type: Item
instance of: scholarly article
title: Strongly nonlinear multivalued systems involving singular \(\Phi \)-Laplacian operators
zbMATH Open document ID: 1205.34016
author: Francesca Papalini
published in: Communications on Pure and Applied Analysis
publication date: 2010-09-22T00:00:00Z
review text: The paper studies the following problem \[ (k(t)\Phi (x'(t)))'\in Ax(t)+F(t,x(t),x'(t))\text{ a.e. on }[0,b],\quad x(0)=x(b),\tag{1} \] where \(k:[0,b]\to {\mathbb R}\) is a positive continuous function, \(\Phi :B\subset {\mathbb R}^n\to {\mathbb R}^n\) is a strictly monotone homeomorphism defined on a bounded domain \(B\), \(A:{\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a maximal monotone operator and \(F:[0,b]\times {\mathbb R}^n\times {\mathbb R}^n\to {\mathcal P}({\mathbb R}^n)\) is a Carathéodory set-valued map with nonempty and compact values. Under certain technical hypotheses, the author proves the existence of solutions of problem (1) both in the cases when the set-valued map \(F\) has convex values. Afterwards, these results are used to obtain the existence of solutions for the more general problem \[ (a(t,x(t))\Phi (x'(t)))'\in Ax(t)+F(t,x(t),x'(t))\text{ a.e. on }[0,b],\quad x(0)=x(b), \] where \(a:[0,b]\times {\mathbb R}^n\to {\mathbb R}\) is a positive continuous function.
reviewed by: Aurelian Cernea
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 47N20
zbMATH DE Number: 5790298
zbMATH Keywords: differential inclusions
zbMATH Keywords: singular \(\Phi \)-Laplacian
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: maximal monotone map
zbMATH Keywords: usc and lsc multifunction
zbMATH Keywords: Yoshida approximation
zbMATH Keywords: variational inequalities
MaRDI profile type: Publication
full work available at URL: cpaa.2010.9.1025
OpenAlex ID: W2010048950
DOI: 10.3934/CPAA.2010.9.1025
description: scientific article; zbMATH DE number 5790298