Strongly nonlinear, multivalued, periodic problems with maximal monotone terms
type: Item
instance of: scholarly article
title: Strongly nonlinear, multivalued, periodic problems with maximal monotone terms
zbMATH Open document ID: 1224.34024
author: Evgenia H. Papageorgiou
author: Nikolaos S. Papageorgiou
published in: Differential and Integral Equations
publication date: 2007-03-29T00:00:00Z
review text: The authors investigate solutions of the second-order differential inclusion for the function \(t \mapsto x(t)\) \[ \left (\alpha (x)\, \varphi (x')\right )' \in A(x) + F(t,x), \quad t \in [0,b], \qquad x(0) = x(b),\;x'(0) = x'(b), \] where \(\alpha \: \mathbb R^N \to \mathbb R\), \(\varphi \: \mathbb R^N \to \mathbb R^N\), \(\varphi (x) = | x| ^{p-2}x\), \(2 \leq p < \infty \), \(A\: D \subseteq \mathbb R^N \to 2^{\mathbb R^N}\) is a maximal monotone operator, and \(F\: [0,b] \times \mathbb R^N \to 2^{\mathbb R^N} \setminus \{0\}\) is a set-valued function (satisfying the Hartman condition). They extend results due to J. Mawhin. Making use of the theory of monotone operators, they find solutions \(x \in C^1 ([0,b], \mathbb R^N)\) of the corresponding equation with \(A\) replaced by its Yosida approximation \(A_{\lambda }\) and pass to the limit \(\lambda \to 0+\). In the particular case \(N=1 \) and \(p=2\), the nonlinearity \(F\) may depend more generally also on \(x'\). Another generalization is indicated if Dirichlet boundary conditions are considered instead of periodic ones and the operator \((\alpha (x)\, \varphi (x'))'\) is replaced by the more general operator \((\alpha (x,x'))'\).
reviewed by: L. Herrmann
Mathematics Subject Classification ID: 34A60
Mathematics Subject Classification ID: 34B15
zbMATH DE Number: 5138413
zbMATH Keywords: maximal monotone operators
zbMATH Keywords: periodic solutions
zbMATH Keywords: multivalued operators
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5138413