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We deal with a class of elliptic quasivariational inequalities with constraints in a reflexive Banach space. We use arguments of monotonicity, convexity and compactness in order to prove a convergence criterion for such inequalities.This criterion allows us to consider a new well-posedness concept in the study of the corresponding inequalities,which extends the classical Tykhonov and Levitin-Polyak well-posedness concepts used in the literature.Then, we introduce a new penalty method, for which we provide a convergence result. Finally, we consider a variational inequality which describes the equilibrium of a spring-rods system, under the action of external forces.We apply our abstract results in the study of this inequality and provide the corresponding mechanical interpretations. We also present numerical simulations which validate our convergence results. 2010 Mathematics Subject Classification: 47J20, 49J27, 49J40, 49K20, 74M15, 74M10.

We consider a mathematical model that describes the frictional contact of an electro-elastic body with a semi-insulator foundation. The process is static; the contact is bilateral and associated to Tresca’s friction law. We list the assumptions on the data and derive a variational formulation of the model, in the form of a system that couples two inclusions in which the unknowns are the strain field and the electric field. Then we prove the unique solvability of the system, as well as the continuous dependence of its solution with respect to the data. We use these results in the study of an associated optimal control problem, for which we prove an existence result. The proofs are based on arguments of monotonicity, compactness, convex analysis, and lower semicontinuity.

"In this paper we present a unified theory of convergence results in the study of abstract problems. To this end we introduce a new mathematical object, the so-called Tykhonov triple $\cT=(I,\Omega,\cC)$, constructed by using a set of parameters $I$, a multivalued function $\Omega$ and a set of sequences $\cC$. Given a problem $\cP$ and a Tykhonov triple $\cT$, we introduce the notion of well-posedness of problem $\cP$ with respect to $\cT$ and provide several preliminary results, in the framework of metric spaces. Then we show how these abstract results can be used to obtain various convergences in the study of a nonlinear equation in reflexive Banach spaces. "

We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem havealreadybeen obtained in the literature. In this current paper we complete these results by proving the well-posedness of the problem, in the sense of Tykhonov. To this end we introduce a family of approximating problems for which we state and prove various equivalence and convergence results. We illustrate these abstract results in the study of a frictionless contact model with elastic materials. The process is assumed to be static and the contact is with unilateral constraints. We derive a weak formulation of the model which is in the form of a mixed variational problem with unknowns being the displacement field and the Lagrange multiplier. Then, we prove various results on the corresponding mixed problem, including its well-posedness in the sense of Tykhonov, under various assumptions on the data. Finally, we provide mechanical interpretation of our results.

We consider a fixed point problem $\mathcal {S}u=u$ S u = u where ${\mathcal {S}:C(\mathbb{R}_{+};X)\to C(\mathbb{R}_{+};X)}$ S : C ( R + ; X ) → C ( R + ; X ) is an almost history-dependent operator. First, we recall the unique solvability of the problem. Then, we introduce the concept of Tykhonov triple, provide several relevant examples, and prove the corresponding well-posedness results for the considered fixed point problem. This allows us to deduce various consequences which illustrate the stability of the solution with respect to perturbations of the operator $\mathcal {S}$ S . Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in solid and contact mechanics. To give some examples, we consider two mathematical models which describe the equilibrium of a viscoelastic body in frictionless contact with a foundation. We state the mechanical problems, list the assumptions on the data, and derive their associated fixed point formulation. Then, we illustrate the use of our abstract results in order to deduce the continuous dependence of the solution with respect to the data and parameters. We also provide the corresponding mechanical interpretations.

We consider an abstract inclusion in a real Hilbert space, governed by an almost history-dependent operator and a time-dependent multimapping with prox-regular values. We establish the unique solvability of the inclusion under appropriate assumptions on the data. The proof is based on the arguments of monotonicity, fixed point, and prox-regularity. We then use our result in order to deduce some direct consequences, including an existence and uniqueness result for a class of sweeping processes associated with prox-regular sets. Finally, we provide an example in a finite dimensional case inspired by a rheological model in solid mechanics.

We consider a differential variational-hemivariational inequality with constraints, in the framework of reflexive Banach spaces. The existence of a unique mild solution of the inequality, together with its stability, was proved in [1]. Here, we complete these results with existence, uniqueness and convergence results for an associated penalty-type method. To this end, we construct a sequence of perturbed differential variational-hemivariational inequalities governed by perturbed sets of constraints and penalty coefficients. We prove the unique solvability of each perturbed inequality as well as the convergence of its solution to the solution of the original inequality. Then, we consider a mathematical model which describes the equilibrium of a viscoelastic rod in unilateral contact. The weak formulation of the model is in a form of a differential variational-hemivariational inequality in which the unknowns are the displacement field and the history of the deformation. We apply our abstract penalty method in the study of this inequality and provide the corresponding mechanical interpretations.

In this paper, an alternative approach is provided in the well-posedness analysis of elliptic variational–hemivariational inequalities in real Hilbert spaces. This includes the unique solvability and continuous dependence of the solution on the data. In most of the existing literature on elliptic variational–hemivariational inequalities, well-posedness results are obtained by using arguments of surjectivity for pseudomonotone multivalued operators, combined with additional compactness and pseudomonotonicity properties. In contrast, following (Han in Nonlinear Anal B Real World Appl 54:103114, 2020; Han in Numer Funct Anal Optim 42:371–395, 2021), the approach adopted in this paper is based on the fixed point structure of the problems, combined with minimization principles for elliptic variational–hemivariational inequalities. Consequently, only elementary results of functional analysis are needed in the approach, which makes the theory of elliptic variational–hemivariational inequalities more accessible to applied mathematicians and engineers. The theoretical results are illustrated on a representative example from contact mechanics.

We start with a mathematical model which describes the frictionless contact of an elastic body with an obstacle and prove that it leads to a stationary inclusion for the strain field. Then, inspired by this contact model, we consider a general stationary inclusion in a real Hilbert space, governed by three parameters. We prove the unique solvability of the inclusion as well as the continuous dependence of its solution with respect to the parameters. We use these results in the study of an associated optimal control problem for which we prove existence and convergence results. The proofs are based on arguments of monotonicity, compactness, convex analysis and lower semicontinuity. Then, we apply these abstract results to the mathematical model of contact and provide the corresponding mechanical interpretations.

We consider a history-dependent variational–hemivariational inequality with unilateral constraints in a reflexive Banach space. The unique solvability of the inequality follows from an existence and uniqueness result obtained in Sofonea and Migórski (2016, 2018). In this current paper we introduce and study a generalized penalty method associated to the inequality. To this end we consider a sequence of generalized penalty problems, governed by a parameter and an operator . We prove the unique solvability of the penalty problems as well as the convergence of corresponding solutions sequence to the solution of original problem. These results extend the previous results in Sofonea etal. (2018) and Xiao and Sofonea (2019). Finally, we illustrate them in the study of a history-dependent problem with unilateral boundary conditions which describes the quasistatic evolution of a rod–spring system under the action of given applied force.

We consider an initial and boundary value problem \begin{document}$ {{\mathcal{P}}} $\end{document} which describes the frictionless contact of a viscoplastic body with an obstacle made of a rigid body covered by a layer of elastic material. The process is quasistatic and the time of interest is \begin{document}$ \mathbb{R}_+ = [0,+\infty) $\end{document} . We list the assumptions on the data and derive a variational formulation \begin{document}${{\mathcal{P}}}_V $\end{document} of the problem, in a form of a system coupling an implicit differential equation with a time-dependent variational-hemivariational inequality, which has a unique solution. We introduce the concept of Tykhonov triple \begin{document}$ {{\mathcal{T}}} = (I,\Omega, {{\mathcal{C}}}) $\end{document} where \begin{document}$ I $\end{document} is set of parameters, \begin{document}$ \Omega $\end{document} represents a family of approximating sets and \begin{document}${{\mathcal{C} }} $\end{document} is a set of sequences, then we define the well-posedness of Problem \begin{document}${{\mathcal{P}}}_V $\end{document} with respect to \begin{document}$ {{\mathcal{T}}}$\end{document} . Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of \begin{document}$ {{\mathcal{P} }}_V $\end{document} with respect to a specific Tykhonov triple. We use this theorem in order to provide the continuous dependence of the solution with respect to the data. Finally, we state and prove additional convergence results which show that the weak solution to problem \begin{document}$ {{\mathcal{P}}} $\end{document} can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results.

We consider a frictionless contact problem, Problem [Formula: see text], for elastic materials. The process is assumed to be static and the contact is modelled with unilateral constraints. We list the assumptions on the data and derive a variational formulation of the problem, Problem [Formula: see text]. Then we consider a perturbation of Problem [Formula: see text], which could be frictional, governed by a small parameter [Formula: see text]. This perturbation leads in a natural way to a family of sets [Formula: see text]. We prove that Problem [Formula: see text] is well-posed in the sense of Tykhonov with respect to the family [Formula: see text]. The proof is based on arguments of monotonicity, pseudomonotonicity and various estimates. We extend these results to a time-dependent version of Problem [Formula: see text]. Finally, we provide examples and mechanical interpretation of our well-posedness results, which, in particular, allow us to establish the link between the weak solutions of different contact models.

We consider a mathematical model which describes the equilibrium of an elastic body in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth interface law which involves unilateral constraints and subdifferential conditions. The weak formulation of the model is in the form of an elliptic hemivariational inequality governed by a number of parameters. We prove the unique weak solvability of the problem, then we state and prove a continuous dependence result of the solution with respect to the data and parameters. Next, we formulate a boundary optimal control problem for which we prove the existence of optimal pairs. We also study the dependence of the optimal pairs with respect to the data and parameters and prove a convergence result. The proofs are based on arguments of monotonicity, lower semicontinuity and Clarke subdifferential calculus.

Consider an abstract Problem P in a metric space (X; d) assumed to have a unique solution u. The aim of this paper is to compare two convergence results u'n → u and u''n → u, both in X, and to construct a relevant example of convergence result un → u such that the two convergences above represent particular cases of this third convergence. To this end, we use the concept of Tykhonov triple. We illustrate the use of this new and nonstandard mathematical tool in the particular case of hemivariational inequalities in reflexive Banach space. This allows us to obtain and to compare various convergence results for such inequalities. We also specify these convergences in the study of a mathematical model, which describes the contact of an elastic body with a foundation and provide the corresponding mechanical interpretations.

We consider an abstract minimization problem in reflexive Banach spaces together with a specific family of approximating sets, constructed by perturbing the cost functional and the set of constraints. For this problem, we state and prove various well-posedness results in the sense of Tykhonov, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, compactness and Mosco convergence of sets. Our results are useful in the study of various mathematical models in contact mechanics. To provide examples, we introduce 2 models, which describe the equilibrium of an elastic body in contact with a rigid body covered by a rigid-plastic and an elastic material, respectively. The weak formulation of each model is in the form of a minimization problem for the displacement field. We use our abstract well-posedness results in the study of these minimization problems. In this way, we obtain existence, uniqueness and convergence results, and moreover, we provide their mechanical interpretations.

We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset\R^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $\cP$. We associate to Problem $\cP$ an optimal control problem, denoted by $\cQ$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\wK$, we provide results concerning the well-posedness of problems $\cP$ and $\cQ$. Our main results are Theorems 14 and 18, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem $\cP$, constructed with particular choices of $G$ and $\wK$. We prove that Theorems 14 and 18 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results., Comment: 28 pages. arXiv admin note: text overlap with arXiv:2008.12730

We consider an elliptic variational–hemivariational inequality with constraints in a reflexive Banach space, denoted $$\mathcal{P}$$ , to which we associate a sequence of inequalities $$\{\mathcal{P}_n\}$$ . For each $$n\in \mathbb {N}$$ , $$\mathcal{P}_n$$ is a variational–hemivariational inequality without constraints, governed by a penalty parameter $$\lambda _n$$ and an operator $$P_n$$ . Such inequalities are more general than the penalty inequalities usually considered in literature which are constructed by using a fixed penalty operator associated to the set of constraints of $$\mathcal{P}$$ . We provide the unique solvability of inequality $$\mathcal{P}_n$$ . Then, under appropriate conditions on operators $$P_n$$ , we state and prove the convergence of the solution of $$\mathcal{P}_n$$ to the solution of $$\mathcal{P}$$ . This convergence result extends the results previously obtained in the literature. Its generality allows us to apply it in various situations which we present as examples and particular cases. Finally, we consider a variational–hemivariational inequality with unilateral constraints which arises in Contact Mechanics. We illustrate the applicability of our abstract convergence result in the study of this inequality and provide the corresponding mechanical interpretations.

We consider a general mathematical model which describes the quasistatic contact of a deformable body with an obstacle, the so-called foundation. The material’s behaviour is modeled with a visco-elastic-type constitutive law and the contact is described with a general interface law associated to a version of Coulomb’s law of dry friction. We list the assumptions on the data and provide relevant examples of constitutive laws and boundary conditions. Then, we derive two different variational formulations of the model in which the unknowns are the displacement and the strain field, respectively. We prove the equivalence of these formulations. Finally, we use recent arguments of sweeping process in order to obtain the existence of a unique weak solution to the contact model.

We consider a rate-type constitutitve law given by an implicit nonlinear differential equation in the space of second order symmetric tensors on in which the unknowns are the stress and the linearized strain fields. We list the assumptions on the constitutive functions then we state and prove its well-posedness with respect to two different Tykhonov triples. We use these well-posedness properties in order to deduce two convergence results. Finally, we provide the mechanical interpretation of these results as well as some concluding remarks.

This work presents a unified approach to the analysis of contact problems with various interface laws that model the processes involved in contact between a deformable body and a rigid or reactive foundation. These laws are then used in the formulation of a general static frictional contact problem with unilateral constraints for elastic materials, which is governed by three parameters. A weak formulation of the problem is derived, which is in the form of an elliptic variational inequality, and the Tykhonov well-posedness of the problem is established, under appropriate assumptions on the data and parameters, with respect to a special Tykhonov triple. The proof is based on arguments on coercivity, compactness, and lower-semicontinuity. This abstract result leads to different convergence results, which establish the continuous dependence of the weak solution on the data and the parameters. Moreover, these results elucidate the links among the weak solutions of the different models. Finally, the corresponding mechanical interpretations of the conditions and the results are provided. The novelty in this work is the application of the Tykhonov well-posedness concept, which allows a unified and elegant framework for this class of static contact problems.

We consider a differential quasivariational inequality for which we state and prove the continuous dependence of the solution with respect to the data. This convergence result allows us to prove the existence of at least one optimal pair for an associated control problem. Finally, we illustrate our abstract results in the study of a free boundary problem which describes the equilibrium of a viscoelastic body in frictionless contact with a foundation made of a rigid body coveblack by a rigid-elastic layer., 23 pages, 0 figures

We consider a boundary value problem which describes the frictional antiplane shear of an elastic body. The process is static and friction is modeled with a slip-dependent version of Coulomb's law of dry friction. The weak formulation of the problem is in the form of a quasivariational inequality for the displacement field, denoted by $\cP$. We associated to problem $\cP$ a boundary optimal control problem, denoted by $\cQ$. For Problem $\cP$ we introduce the concept of well-posedness and for Problem $\cQ$ we introduce the concept of weakly and weakly generalized well-posedness, both associated to appropriate Tykhonov triples. Our main result are Theorems \ref{t1} and \ref{t2}. Theorem \ref{t1} provides the well-posedness of Problem $\cP$ and, as a consequence, the continuous dependence of the solution with respect to the data. Theorem \ref{t2} provides the weakly generalized well-posedness of Problem $\cQ$ and, under additional hypothesis, its weakly well posedness. The proofs of these theorems are based on arguments of compactness, lower semicontinuity, monotonicity and various estimates. Moreover, we provide the mechanical interpretation of our well-posedness results., 21 pages

In this paper, we introduce a new Tykhonov-type well-posedness concept for elliptic hemivariational inequalities, governed by an approximating function h. We characterize the well-posedness in terms of the metric properties of the family of approximating sets, under various assumptions on h. Then, we use the well-posedness properties in order to obtain convergence results of the solution with respect to the data. The proofs are based on arguments of monotonicity combined with the properties of the Clarke directional derivative. Our results provide mathematical tools in the study of a large number of static problems in Contact Mechanics. To provide an example, we consider a mathematical model which describes the equilibrium of a rod–spring system with unilateral constraints. We prove the unique weak solvability of the model, and then we illustrate our abstract convergence results in the study of this contact problem and provide the corresponding mechanical interpretations.

In (J. Optim. Theory Appl. 183:139–157, 2019) we introduced and studied the concept of well-posedness in the sense of Tykhonov for abstract problems formulated on metric spaces. Our aim of this current paper is to extend the results in (J. Optim. Theory Appl. 183:139–157, 2019) to a system which consists of two independent problems denoted by P and Q, coupled by a nonlinear equation. Following the terminology used in literature we refer to such a system as a split problem. We introduce the concept of well-posedness for the abstract split problem and provide its characterization in terms of metric properties for a family of approximating sets and in terms of the well-posedness for the problems P and Q, as well. Then we illustrate the applicability of our results in the study of three relevant particular cases: a split variational–hemivariational inequality, an elliptic variational inequality and a history-dependent variational inequality. We describe each split problem and clearly indicate the family of approximating sets. We provide necessary and sufficient conditions which guarantee the well-posedness of the split variational–hemivariational inequality. Moreover, under appropriate assumptions on the data, we prove the well-posedness of the split elliptic variational inequality as well as the well-posedness of the split history-dependent variational inequality. We illustrate our abstract results with various examples, part of them arising in contact mechanics.

This paper is devoted to the study of a class of sweeping processes with history-dependent operators. A well-posedness result is obtained, including the existence, uniqueness, and stability of the ...

We consider a hemivariational inequality of elliptic type (HVI, for short) in a reflexive Banach space, prove its solvability and the compactness of its set of solutions. To this end we employ a surjectivity theorem for multivalued mappings that we use for the sum of a maximal monotone operator and a bounded pseudomonotone operator. Next, we introduce the concepts of strongly and weakly well-posedness in the generalized sense for the HVI and provide two characterizations for the strongly well-posedness, under different assumptions on the data. These characterizations are formulated in terms of the metric properties of the approximating sets. We also provide sufficient conditions which guarantee the weakly and strongly well-posedness in the generalized sense of the HVI. Finally, we consider two perturbations of the HVI for which we obtain convergence results in the sense of Kuratowski.

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.

We study a new mathematical model which describes the equilibrium of a locking material in contact with a foundation. The contact is frictionless and is modeled with a nonsmooth multivalued interface law which involves unilateral constraints and subdifferential conditions. We describe the model and derive its weak formulation, which is in the form of an elliptic variational–hemivariational inequality for the displacement field. Then, we establish the existence of a unique weak solution to the problem. Next, we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a particular version of the model for which we prove the continuous dependence of the solution on the bounds which govern the locking and the normal displacement constraints, respectively. We apply this convergence result in the study of an optimization problem associated to the contact model.

This article constructs and analyzes a mathematical model that describes the quasistatic evolution of a 2D elastic bar that may come in frictional contact with a deformable foundation. The model and the underlying mechanical assumptions are described in detail and so are the assumptions on the problem data. The variational formulation of the problem is derived and, since friction is taken into account, it is in the form of an evolutionary variational inequality for the displacement field. Existence of solutions for the problem is established by using arguments of evolutionary variational inequalities.

Variational-hemivariational inequalities refer to the inequality problems where both convex and nonconvex functions are involved. In this paper, we consider the numerical solution of a family of stationary variational-hemivariational inequalities by the finite element method. For a variational-hemivariational inequality of a general form, we prove convergence of numerical solutions. For some particular variational-hemivariational inequalities, we provide error estimates of numerical solutions, which are of optimal order for the linear finite element method under appropriate solution regularity assumptions. Numerical results are reported on solving a variational-hemivariational inequality modeling the contact between an elastic body and a foundation with the linear finite element, illustrating the theoretically predicted optimal first order convergence and providing their mechanical interpretations.

We consider a new class of differential quasivariational inequalities, i.e. a nonlinear system that couples a differential equation with a time-dependent quasivariational inequality, both defined on abstract Banach spaces. We state and prove a general fixed principle that provides the existence and the uniqueness of the solution of the system. Then we consider a relevant particular setting for which our abstract result holds. We proceed with two examples that arise in Contact Mechanics. For each example, we describe the physical setting, the mathematical model and the assumption on the data. Then we state the variational formulation of each model, which is in the form of a differential quasivariational inequality. Finally, we apply our abstract results to provide the unique weak solvability of the corresponding contact problems.

The present paper deals with new results in the study of a class of elliptic hemivariational inequalities in reflexive Banach spaces. We start with an existence and uniqueness result. We complete it with a convergence result which describes the dependence of the solution with respect to the data. To this end, we use the notion of Mosco convergence for the set of constraints. Next, we formulate an optimal control problem for which we prove the existence of optimal pairs and state a convergence result. Finally, we exemplify the use of our results in the study of a two-dimensional boundary value problem which describes the frictionless contact of an elastic body with two reactive foundations.

The present paper represents a continuation of Migorski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force.

We consider a frictionless contact problem with unilateral constraints for a 2D bar. We describe the problem, then we derive its weak formulation, which is in the form of an elliptic variational inequality of the first kind. Next, we establish the existence of a unique weak solution to the problem and prove its continuous dependence with respect to the applied tractions and constraints. We proceed with the study of an associated control problem for which we prove the existence of an optimal pair. Finally, we consider a perturbed optimal control problem for which we prove a convergence result.

We consider an optimal control problem $\cQ$ governed by an elliptic quasivariational inequality with unilateral constraints. The existence of optimal pairs of the problem is a well known result, see \cite{SS}, for instance. We associate to $\cQ$ a new optimal control problem $\wQ$, 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 $\wQ$ to a solution of Problem $\cQ$. 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., Comment: 24 pages, manuscript accepted in the Journal: Numerical Functional Analysis and Optimization

We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraintsK, a nonlinear operatorA, and an elementf. We associate to this inequality a sequence {𝓟n} of variational-hemivariational inequalities such that, for eachn∈ ℕ, inequality 𝓟nis obtained by perturbing the dataKandAand, moreover, it contains an additional term governed by a small parameterεn. The unique solvability of 𝓟 and, for eachn∈ ℕ, the solvability of its perturbed version 𝓟n, are guaranteed by an existence and uniqueness result obtained in literature. Denote byuthe solution of Problem 𝓟 and, for eachn∈ ℕ, letunbe a solution of Problem 𝓟n. The main result of this paper states the strong convergence ofun→uinX, asn→ ∞. We show that the main result extends a number of results previously obtained in the study of Problem 𝓟. Finally, we illustrate the use of our abstract results in the study of a mathematical model which describes the contact of an elastic body with a rigid-deformable foundation and provide the corresponding mechanical interpretations.

We deal with the Tykhonov well-posedness of a time-dependent variational inequality defined on the unbounded interval of time ℝ+= [0, +∞ ), governed by a history-dependent operator. To this end we introduce the concept of Tykhonov triple, provide three relevant examples, then we state and prove the corresponding well-posedness results. This allows us to deduce various corollaries which illustrate the continuous dependence of the solution with respect to the data. Our results provide mathematical tools in the analysis of a large number of history-dependent problems which arise in Mechanics, Physics and Engineering Sciences. To give an example, we consider a mathematical model which describes the equilibrium of a viscoelastic body in frictionless contact with a rigid foundation.

We consider a semilinear differential variational inequality P in reflexive Banach spaces, governed by a set of constraints K. We associate to P a sequence of problems {Pn} where, for each n∈N, Pn is a differential variational inequality governed by a set of constraints Kn and a penalty parameter ρn. We use a result in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities. Appl Anal. 2019;1–16. doi:10.1080/00036811.2019.1652736] to prove the unique solvability of problems {P} and {Pn}. Then, we prove that, under appropriate assumptions, the sequence of solutions to Problem Pn converges to the solution of the original problem P. The proof is based on arguments of compactness, pseudomonotonicity and Mosco convergence. We also present two relevant particular case of our convergence result, including a recent result obtained in [Liu ZH, Zeng SD. Penalty method for a class of differential variational inequalities. Appl Anal. 2019;1–16. doi:10.1080/00036811.2019.1652736], in the case Kn=V. Finally, we provide an example of initial and boundary value problem for which our abstract results can be applied.

The present paper represents a continuation of [26] . There, a continuous dependence result for the solution of an elliptic variational–hemivariational inequality was obtained and then used to prove the existence of optimal pairs for two associated optimal control problems. In the current paper we complete this study with more general results. Indeed, we prove the continuous dependence of the solution with respect to a parameter which appears in all the data of the problem, including the set of constraints, the nonlinear operator and the two functionals which govern the variational–hemivariational inequality. This allows us to consider a general associated optimal control problem for which we prove the existence of optimal pairs, together with a new convergence result. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to elliptic variational–hemivariational inequalities. To provide an example, we illustrate our results in the study of an inequality which describes the equilibrium of an elastic body in frictional contact with a foundation made of a rigid body covered by a layer of soft material.

We consider a mixed variational problem in real Hilbert spaces, defined on the unbounded interval of time $[0,+\infty)$ and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2015). Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.

The present paper concerns a class of abstract mixed variational problems governed by a strongly monotone Lipschitz continuous operator. With the existence and uniqueness results in the literature for the problem under consideration, we prove a general convergence result, which shows the continuous dependence of the solution with respect to the data by using arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Then we consider an associated optimal control problem for which we prove the existence of optimal pairs. The mathematical tools developed in this paper are useful in the analysis and control of a large class of boundary value problems which, in a weak formulation, lead to mixed variational problems. To provide an example, we illustrate our results in the study of a mathematical model which describes the equilibrium of an elastic body in frictional contact with a foundation.

We consider a class of time-dependent inclusions in Hilbert spaces for which we state and prove an existence and uniqueness result. The proof is based on arguments of variational inequalities, convex analysis and fixed point theory. Then we use this result to prove the unique weak solvability of a new class of Moreau's sweeping processes with constraints in velocity. Our results are useful in the study of mathematical models which describe the quasistatic evolution of deformable bodies in contact with an obstacle. To provide some examples we consider three viscoelastic contact problems which lead to time-dependent inclusions and sweeping processes in which the unknowns are the displacement and the velocity fields, respectively. Then we apply our abstract results in order to prove the unique weak solvability of the corresponding contact problems.