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A new class of hyperbolic variational–hemivariational inequalities driven by non-linear evolution equations

Part of: Existence theories Equations and inequalities involving nonlinear operators Special kinds of problems Hyperbolic equations and systems Equilibrium (steady-state) problems

Published online by Cambridge University Press:  16 March 2020

STANISŁAW MIGÓRSKI ,
WEIMIN HAN  and
SHENGDA ZENG Open the ORCID record for SHENGDA ZENG [Opens in a new window]
STANISŁAW MIGÓRSKI
Affiliation:
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu610225, Sichuan Province, P.R. China, email: stanislaw.migorski@uj.edu.pl Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin537000, P.R. China Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30348Krakow, Poland, emails: shengdazeng@gmail.com; shdzeng@hotmail.com; zengshengda@163.com
WEIMIN HAN
Affiliation:
Department of Mathematics, University of Iowa, Iowa City, IA52242-1410, USA, email: weimin-han@uiowa.edu
SHENGDA ZENG
Affiliation:
Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing, Yulin Normal University, Yulin537000, P.R. China Jagiellonian University in Krakow, ul. Lojasiewicza 6, 30348Krakow, Poland, emails: shengdazeng@gmail.com; shdzeng@hotmail.com; zengshengda@163.com
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Abstract

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The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law.

Keywords

Hyperbolic variational–hemivariational inequalityClarke subgradientexistence and uniquenessdynamic contact problemnormal damped responseadhesion

MSC classification

Primary: 49J40: Variational methods including variational inequalities 47J20: Variational and other types of inequalities involving nonlinear operators (general) 47J22: Variational and other types of inclusions
Secondary: 74M10: Friction 74M15: Contact 74G25: Global existence of solutions 35L51: Second-order hyperbolic systems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Issue 1 , February 2021 , pp. 59 - 88
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Creative Commons
Creative Common License - CCCreative Common License - BYCreative Common License - NCCreative Common License - SA
This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike licence (http://creativecommons.org/licenses/by-nc-sa/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the same Creative Commons licence is included and the original work is properly cited. The written permission of Cambridge University Press must be obtained for commercial re-use.
Copyright
© The Author(s) 2020. Published by Cambridge University Press

Footnotes

†

This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement no. 823731 CONMECH. It is supported by the National Science Center of Poland under Preludium project no. 2017/25/N/ST1/00611. The first author is also supported by the Natural Science Foundation of Guangxi grant no. 2018GXNSFAA281353 and the Ministry of Science and Higher Education of Republic of Poland under grants no. 4004/GGPJII/H2020/2018/0 and 440328/PnH2/2019. The work of the second author was partially supported by NSF under grant DMS-1521684.

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