Existence results for evolution hemivariational inequalities of second order
type: Item
instance of: scholarly article
title: Existence results for evolution hemivariational inequalities of second order
zbMATH Open document ID: 1082.34052
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-05-12T00:00:00Z
review text: The author proves existence of solutions for the second-order nonlinear evolution inclusion \[ \begin{gathered} y''(t)+ A(t,y'(t))+ By(t)+\partial J(t,y(t))\ni f(t),\quad\text{a.e. on }(0,T),\\ y(0)= y_0,\quad y'(0)= y_1\end{gathered} \] which can be expressed as the hemivariational inequality \[ \langle y''(t)+ A(t,y'(t))+ By(t)- f(t),v\rangle+ J^0(t, y(t)lv)\geq 0 \] for every \(v\in V\), for almost every \(t\in(0,T)\) \[ y(0)= y_0,\quad y'(0)= y_1. \] Here, \(\partial J\) represents the Clarke subdifferential of \(J\), \(J(t,x; v)\) is the generalized directional derivative of \(J(t,\cdot)\) at a point \(x\) in the direction \(v\) and \(V\) is a reflexive Banach space. This problem can be thought of as an evolution inclusion with a multi-valued perturbation. Physical applications of hemivariational inequalities are mentioned, including references, and a survey of the theoretical results involving these problems is also given. The proof involves an application of the result in [\textit{N. S. Papageorgiou}, \textit{F. Papalini} and \textit{F. Renzacci}, Rend. Circ. Mat. Palermo, II. Ser. 48, No. 2, 341--364 (1999; Zbl 0931.34043)]. Among other hypotheses, \(A\) is assumed to be pseudomonotone in its second variable, \(B\) is a bounded, linear, monotone and symmetric operator, and \(J\) is locally Lipschitz in its second variable. A result is also stated in which the \(\partial J\) term depends on \(y'\) rather than \(y\). Two specific applications from mechanics are given to which the author's existence result can be applied.
reviewed by: Daniel C. Biles
Mathematics Subject Classification ID: 34G25
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 74H20
Mathematics Subject Classification ID: 47J20
zbMATH DE Number: 2167838
zbMATH Keywords: Hemivariational inequality
zbMATH Keywords: Hyperbolic
zbMATH Keywords: Compact embedding
zbMATH Keywords: Multifunction
zbMATH Keywords: Selection
zbMATH Keywords: Semicontinuity
zbMATH Keywords: Pseudomonotone operator
author: Anna Ochal
MaRDI profile type: Publication
full work available at URL: j.na.2004.10.021
OpenAlex ID: W2082204197
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DOI: 10.1016/J.NA.2004.10.021
description: scientific article; zbMATH DE number 2167838