A convergence result for history-dependent quasivariational inequalities

Abstract

In this paper, we consider a general class of history-dependent quasivariational inequalities with constraints. Our aim is to study the behavior of the solution with respect to the set of constraints and, in this matter, we prove a continuous dependence result. The proof is based on various estimates and monotonicity arguments. Then, we consider two mathematical models which describe the equilibrium of a viscoplastic and viscoelastic body, respectively, in contact with a deformable foundation. The variational formulation of each model is in a form of a history-dependent quasivariational inequality for the displacement field, governed by a set of constraints. We prove the unique weak solvability of each model, then we use our abstract result to prove the continuous dependence of the solution with respect to the set of constraints.

Keywords:

• History-dependent operator
• quasivariational inequality
• convergence result
• unilateral constraint
• frictionless contact
• normal compliance
• weak solution

AMS Subject Classifications:

• 47J20
• 49J40
• 49J45
• 74G25
• 74G30
• 74M15

Notes

No potential conflict of interest was reported by the authors.