A convergence result for history-dependent quasivariational inequalities

Benraouda, Ahlem; Sofonea, Mircea

doi:10.1080/00036811.2016.1236920

Cited by 8 publications

(6 citation statements)

References 24 publications

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“…Finally, we conjecture that under additional hypotheses the convergence result of Theorem 16(ii) can be strengthen to the uniform convergence of u ρ → u in C(R + ; X), as ρ → 0, which will be studied in the future. We note that a convergence result for Problem 15 with j ≡ 0, A, f and ϕ independent of ρ, and K ρ of the form (10) was provided in [1] under assumption that A is Lipschitz continuous and ϕ depends on a history-dependent operator.…”

Section: Problem 15 Find a Function Umentioning

confidence: 99%

“…The notation j 0 (u; v) stands for the generalized directional derivative of j at point u ∈ X in the direction v ∈ X. The goal of the paper is to study the convergence of solution of the variational-hemivariational inequality (1) when the data A, f , ϕ, j and K are subjected to perturbations. The dependence of solutions to elliptic variational-hemivariational inequalities on the data has been studied only recently.…”

Section: Introductionmentioning

confidence: 99%

“…For a class of elliptic historydependent variational-hemivariational inequalities studied in [21], the convergence result was obtained in a case when ϕ depends on a history-dependent operator, and A does not depend on perturbations. A result on the convergence with respect to the set of constraints K were studied for elliptic quasivariational inequalities in [1]. In all aforementioned papers the convergence results were applied to various mathematical models of deformable bodies in contact mechanics.…”

Section: Introductionmentioning

confidence: 99%

“…Finally, we underline that our convergence results of Sect. 3 are also applicable to various problems in contact mechanics like a nonlinear elastic contact problem with normal compliance condition with unilateral constraint, and a contact problem with the Coulomb friction law in which the friction bound is supposed to depend on the normal displacement, studied in, e.g., [1,6,13,19].…”

Section: Introductionmentioning

confidence: 99%

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On convergence of solutions to variational–hemivariational inequalities

Zeng

Liu

Migórski

2018

Z. Angew. Math. Phys.

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Section: Problem 15 Find a Function Umentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On convergence of solutions to variational–hemivariational inequalities

Zeng

Liu

Migórski

2018

Z. Angew. Math. Phys.

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“…Then, the existence of a unique solution of problem P ρ as well as its convergence to the solution of problem P as ρ → 0 could be obtained by using a result in [3], where a more general class of history-dependent variational inequalities has been considered. Nevertheless, in the current paper, we decided to present a different version of this convergence result, under different assumptions.…”

Section: A First Convergence Resultsmentioning

confidence: 99%

Convergence results for primal and dual history-dependent quasivariational inequalities

Sofonea¹,

Benraouda²

2018

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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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.

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Convergence results for elliptic quasivariational inequalities

Sofonea¹,

Benraouda²

2016

Z. Angew. Math. Phys.

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We consider an optimal control problem Q governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see [28], for instance. We associate to Q a new optimal control problem Q, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of Problem Q to a solution of Problem Q. The proofs are based on convergence results for elliptic quasivariational inequalities, obtained by using arguments of compactness, lower semicontinuity, monotonicity, penalty and various estimates. Finally, we illustrate the use of the abstract convergence results in the study of optimal control associated with two boundary value problems. The first one describes the equilibrium of an elastic body in frictional contact with an obstacle, the so-called foundation. The process is static and the contact is modeled with normal compliance and unilateral constraint, associated to a version of Coulomb's law of dry friction. The second one describes a stationary heat transfer problem with unilateral constraints. For the two problems we prove existence, uniqueness and convergence results together with the corresponding physical interpretation.We associate to Problem P the set of admissible pairs defined byand we consider a cost functional L : X × Y → R. Here and below, X × Y represents the product of the Hilbert spaces X and Y , equipped with the canonical inner product. Then, the optimal control problem we study in this paper is the following. Problem Q. Find (u * , f * ) ∈ V ad such that L(u * , f * ) = min (u,f )∈V ad L(u, f ). (1.3)Next, consider a set K ⊂ X, an operator A : X → X and an element f ∈ Y . With these data we construct the following perturbation of Problem P.

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A convergence result for history-dependent quasivariational inequalities

Cited by 8 publications

References 24 publications

On convergence of solutions to variational–hemivariational inequalities

On convergence of solutions to variational–hemivariational inequalities

Convergence results for primal and dual history-dependent quasivariational inequalities

Convergence results for elliptic quasivariational inequalities

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