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Convergence results for primal and dual history-dependent quasivariational inequalities

Part of: Special kinds of problems Equations and inequalities involving nonlinear operators Existence theories Equilibrium (steady-state) problems Thin bodies, structures

Published online by Cambridge University Press:  27 December 2018

Mircea Sofonea  and
Ahlem Benraouda
Mircea Sofonea*
Affiliation:
Laboratoire de Mathématiques et Physique Université de Perpignan Via Domitia 52 Avenue Paul Alduy, 66860 Perpignan, France ([email protected])
Ahlem Benraouda
Affiliation:
Laboratoire de Mathématiques et Physique Université de Perpignan Via Domitia 52 Avenue Paul Alduy, 66860 Perpignan, France ([email protected])
*
*Corresponding author.
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Abstract

We consider a class of history-dependent quasivariational inequalities for which we prove the continuous dependence of the solution with respect to the set of constraints. Then, under additional assumptions, we associate with each inequality in the class a new inequality, the so-called dual variational inequality, for which we state and prove existence, uniqueness, equivalence and convergence results. The proofs are based on various estimates, monotonicity and fixed-point arguments for history-dependent operators. Our abstract results are useful in the study of various mathematical models of contact. To provide an example, we consider a boundary value problem which describes the equilibrium of a viscoelastic body in contact with an elastic-rigid foundation. We list the assumptions on the data and derive both the primal and the dual variational formulation of the problem. Then, we state and prove existence, uniqueness and convergence results. We also provide the link between the two formulations, together with their mechanical interpretation.

Keywords

history-dependent quasivariational inequalityprimal and dual variational formulationconvergence resultunilateral constraintfrictionless contactnormal complianceweak solution

MSC classification

Primary: 47J20: Variational and other types of inequalities involving nonlinear operators (general)
Secondary: 49J40: Variational methods including variational inequalities 49J45: Methods involving semicontinuity and convergence; relaxation 74G25: Global existence of solutions 74G30: Uniqueness of solutions 74K10: Rods (beams, columns, shafts, arches, rings, etc.) 74M15: Contact
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 471 - 494
Copyright
Copyright © Royal Society of Edinburgh 2018 

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