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In the present paper, we consider a mathematical model describing the dynamic Coulomb's frictional contact between a thermo-viscoelastic body and a thermally conductive rigid foundation. We employ the nonlinear constitutive viscoelastic law with long-term memory and thermal effects. We describe some contact and thermal conditions with the Clarke subdifferential boundary conditions. We derive the weak formulation of the problem as a system coupling two variational-hemivariational inequalities. We provide results on the existence and uniqueness of a weak solution to the model by using recent results from the theory of variational-hemivariational inequalities. Finally, the continuous dependence of the solution on the data is derived by applying an abstract result that we demonstrate. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate a model of incompressible fluid of Bingham type in a bounded domain. We obtain the variational formulation of the model of incompressible fluid which is a variational-hemivariational inequality. The existence and uniqueness of the solution are proven utilizing recent advancements in the theory of hemivariational inequalities. Additionally, employing the finite element method, we analyze a fully discrete approximation of the model and provide error estimates for the approximate solutions. Finally, we demonstrate a continuous dependence result and establish the existence of optimal pairs for the incompressible fluid of Bingham type. [ABSTRACT FROM AUTHOR]

In this paper we examine a class of elliptic quasi-variational inequalities, which involve a constraint set and a set-valued map. First, we establish the existence of a solution and the compactness of the solution set. The approach is based on results for an elliptic variational inequality and the Kakutani-Ky Fan fixed point theorem. Next, we prove an existence and compactness result for a quasi-variational-hemivariational inequality. The latter involves a locally Lipschitz continuous functional and a convex potential. Finally, we present an application to the stationary incompressible Navier-Stokes equation with mixed boundary conditions which model a generalized Newtonian fluid of Bingham type. [ABSTRACT FROM AUTHOR]

In this paper, we study the well-posedness of a class of evolutionary variational–hemivariational inequalities coupled with a nonlinear ordinary differential equation in Banach spaces. The proof is based on an iterative approximation scheme showing that the problem has a unique mild solution. In addition, we established the continuity of the flow map with respect to the initial data. Under the general framework, we consider two new applications for modeling of frictional contact for viscoelastic materials. In the first application, we consider Coulomb's friction with normal compliance, and in the second, normal damped response. The structure of the friction coefficient μ is new with motivation from geophysical applications in earth sciences with dependence on an external state variable α and the slip rate | u · τ |. [ABSTRACT FROM AUTHOR]

This paper is devoted to studying a dynamic adhesive contact model where the body is composed of a viscoelastic material with long memory. The adhesion process is modeled by a bonding field on part of the contact surface and on which the contact and the friction are modeled by nonmonotone Clarke subdifferential conditions with adhesion. Meanwhile, on another part of the contact surface, the contact is comprised of a normal compliance contact condition while the memory effects of the obstacle are considered, the friction is modeled by a version of Coulomb's law of dry friction and the friction bound is dependent on the normal stress. The variational form of the model generates a system of a doubly-history dependent variational–hemivariational inequality as well as a differential equation. Then, on the basis of the proof of abstract variational–hemivariational inequality, the existence and uniqueness results of the contact problem are established. [ABSTRACT FROM AUTHOR]

A variational-hemivariational inequality that describes stationary Stokes equations for the incompressible fluid with mixed nonlinear boundary conditions is studied. We discuss the equivalence of different weak formulations together with the results on unique solvability. The mixed finite element method of P1b/P1 pair is used to discretize the inequality problem and error bounds are derived. Numerical simulations based on Uzawa method are provided to confirm the predicted results of convergence order. [ABSTRACT FROM AUTHOR]

This paper is dedicated to the discussion of a new dynamical system involving a history-dependent variational-hemivariational inequality coupled with a non-linear evolution equation. The existence and uniqueness of the solution to this problem are established using the Rothe method and a surjectivity result for a pseudo-monotone perturbation of a maximal operator. Additionally, we derive the regularity solution for such a history-dependent variational-hemivariational inequality. Furthermore, the main results obtained in this study are applied to investigate the unique solvability of a dynamical viscoelastic frictional contact problem with long memory and wear. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the difference gap function (for brevity, DG-function) and upper error bounds for an abstract class of variational-hemivariational inequalities with history-dependent operators (for brevity, HDVHIs). First, we propose a new concept of gap functions to the HDVHIs and consider the regularized gap function (for brevity, RG-function) for the HDVHIs using the optimality condition for the concerning minimization problem. Then, the DG-function for the HDVHIs depending on these RG-functions is constructed. Finally, we establish upper error bounds for the HDVHIs controlled by the RG-function and the DG-function under suitable conditions. [ABSTRACT FROM AUTHOR]

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities arising in contact problems. A fully discrete scheme is introduced for the inequality problem, in which the history-dependent operator is approximated by the trapezoidal rule and the spatial variable is approximated by the lowest-order virtual element method. An optimal order error estimate is derived under appropriate solution regularity assumptions. Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this article, we study a large class of evolutionary variational–hemivariational inequalities of hyperbolic type without damping terms, in which the functional framework is considered in an evolution triple of spaces. The inequalities contain both a convex potential and a locally Lipschitz superpotential. The results on existence, uniqueness, and regularity of solution to the inequality problem are provided through the Rothe method.

The aim of this paper is to investigate a nonlinear optimal control problem governed by a complicated dynamic variational-hemivariational inequality (DVHVI, for short) with history-dependent operators in the framework of an evolution triple of spaces. First, we prove a generalized existence and uniqueness theorem to a class of dynamic variational-hemivariational inequalities, which extends the recent results established by Han-Migórski-Sofonea [12, Theorem 9] and Migórski-Bai [23, Theorem 5]. Then, an optimal control problem driven by a (DVHVI) is studied and an existence theorem is delivered. Finally, we explore the sensitivity properties of the optimal control problem under the consideration depending on the initial data ξ ∈ H and some further parameter λ ∈ E. [ABSTRACT FROM AUTHOR]

A class of quasi variational–hemivariational inequalities in reflexive Banach spaces is studied. The inequalities contain a convex potential, a locally Lipschitz superpotential and an implicit obstacle set of constraints. Results on the well-posedness including existence, uniqueness, dependence of solution on the data and the compactness of the solution set in the strong topology are established. The applicability of the results is illustrated by the steady-state generalized Stokes model of a generalized Newtonian incompressible fluid with a non-monotone slip boundary condition. This article is part of the theme issue 'Non-smooth variational problems and applications'. [ABSTRACT FROM AUTHOR]

In this paper we provide a new approach in the study of a variational-hemivariational inequality in Hilbert space, based on the theory of maximal monotone operators and the Banach fixed point theorem. First, we introduce the inequality problem we are interested in, list the assumptions on the data and show that it is governed by a multivalued maximal monotone operator. Then, we prove that solving the variational-hemivariational inequality is equivalent to finding a fixed point for the resolvent of this operator. Based on this equivalence result, we use the Banach contraction principle to prove the unique solvability of the problem. Moreover, we construct the corresponding Picard, Krasnoselski and Mann iterations and deduce their convergence to the unique solution of the variational-hemivariational inequality. [ABSTRACT FROM AUTHOR]

This paper is devoted to the numerical analysis of a general elliptic variational-hemivariational inequality. After a review of a solution existence and uniqueness result, we introduce a family of Galerkin methods to solve the problem. We prove the convergence of the numerical method under the minimal solution regularity condition available from the existence result and derive a Céa’s inequality for error estimation of the numerical solutions. Then, we apply the results for the numerical analysis of a variational-hemivariational inequality in the study of a static problem which models the contact of an elastic body with a reactive foundation. In particular, under appropriate solution regularity conditions, we derive an optimal order error estimate for the linear finite element solution. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational-hemivariational inequality P in a p-uniformly smooth Banach space. We prove that the inequality is governed by a multivalued maximal monotone operator, and, for each λ > 0, we use the resolvent of this operator to construct an auxiliary fixed point problem, denoted Pλ . Next, we perform a parallel study of problems P and Pλ based on their intrinsic equivalence. In this way, we prove existence, uniqueness, and well-posedness results with respect to specific Tykhonov triples. The existence of a unique common solution to problems P and Pλ is proved by using the Banach contraction principle in the study of Problem Pλ. In contrast, the well-posedness of the problems is obtained by using a monotonicity argument in the study of Problem P. Finally, the properties of Problem Pλ allow us to deduce a convergence criterion in the study of Problem P. [ABSTRACT FROM AUTHOR]

Abstract We study the Stokes problem for the incompressible fluid with mixed nonlinear boundary conditions of subdifferential type. The latter involve a unilateral boundary condition, the Navier slip condition, a nonmonotone version of the nonlinear Navier–Fujita slip condition, and the threshold slip and leak condition of frictional type. The weak form of the problem leads to a new class of variational–hemivariational inequalities on convex sets for the velocity field. Solution existence and the weak compactness of the solution set to the inequality problem are established based on the Schauder fixed point theorem.

A class of evolutionary variational–hemivariational inequalities with a convex constraint is studied in this paper. An inequality in this class involves a first-order derivative and a history-dependent operator. Existence and uniqueness of a solution to the inequality is established by the Rothe method, in which the first-order temporal derivative is approximated by backward Euler's formula, and the history-dependent operator is approximated by a modified left endpoint rule. The proof of the result relies on basic results in functional analysis only, and it does not require the notion of pseudomonotone operators and abstract surjectivity results for such operators, used in other papers on the Rothe method for other evolutionary variational–hemivariational inequalities. Moreover, a Lipschitz continuous dependence conclusion of the solution on the right-hand side is proved. Finally, a new frictional contact problem for viscoelastic material is discussed, which illustrates an application of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with numerical analysis on optimal control of a dynamic contact model. Our work builds on some existing results of this model. We derive a weak formulation, and present an approximate problem for the weak formulation by discretization in space. We prove that the approximate problem has a unique solution, which converges to the unique solution of the original problem. The existence of optimal solutions to the control problem is established based on a known continuous dependence result. We consider the numerical solutions of the optimal control problem and prove a convergence result of numerical solutions. [ABSTRACT FROM AUTHOR]

In this paper we investigate a new class of elliptic variational–hemivariational inequalities without the relaxed monotonicity condition of the generalized subgradient. The inequality describes the mathematical model of the steady state flow of incompressible fluid of Bingham type in a bounded domain. The boundary condition represents a generalization of the no leak condition, and a multivalued and nonmonotone version of a nonlinear Navier–Fujita frictional slip condition. The analysis provides results on existence of solution to a variational–hemivariational inequality, continuous dependence of the solution on the data, existence of solutions to optimal control problems, and the dependence of the solution on the yield limit. The proofs profit from results of nonsmooth analysis and the theory of multivalued pseudomontone operators. [ABSTRACT FROM AUTHOR]

This paper presents a multiplicity theorem for a kind of non-smooth functionals. The proof of this theorem relies on a suitable deformation lemma and the perturbation methods. We also apply this result to prove a multiplicity theorem for elliptic variational-hemivariational inequality problems.

We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraints K, a nonlinear operator A, and an element f. We associate to this inequality a sequence {𝓟n} of variational-hemivariational inequalities such that, for each n ∈ ℕ, inequality 𝓟n is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter εn. The unique solvability of 𝓟 and, for each n ∈ ℕ, the solvability of its perturbed version 𝓟n, are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem 𝓟 and, for each n ∈ ℕ, let un be a solution of Problem 𝓟n. The main result of this paper states the strong convergence of un → u in X, as n → ∞. 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.

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. [ABSTRACT FROM AUTHOR]

In this paper we study a new abstract evolutionary variational–hemivariational inequality which involves unilateral constraints and history–dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected functions together with a fixed-point principle for history–dependent operators. Then, we apply the abstract result to show the unique weak solvability to a dynamic viscoelastic frictional contact problem. The contact law involves a unilateral Signorini-type condition for the normal velocity combined with the nonmonotone normal damped response condition while the friction condition is a version of the Coulomb law of dry friction in which the friction bound depends on the accumulated slip. [ABSTRACT FROM AUTHOR]

The primary objective is to investigate a class of noncoercive variational–hemivariational inequalities on a Banach space. We start with several new existence results for the abstract inequalities in which our approach is based on arguments of recession analysis and the theory of pseudomonotone operators. A nonsmooth elastic contact problem is considered as an illustrative application. [ABSTRACT FROM AUTHOR]

We consider a class of elliptic variational-hemivariational inequalities in an abstract Banach space for which we introduce the concept of well-posedness in the sense of Tykhonov. We characterize the well-posedness in terms of metric properties of a family of associated sets. Our results, which provide necessary and sufficient conditions for the well-posedness of inequalities under consideration, are valid under mild assumptions on the data. Their proofs are based on arguments of monotonicity, lower semicontinuity and properties of the Clarke directional derivative. For well-posed inequalities we also prove a continuous dependence result of the solution with respect to the data. We illustrate our abstract results in the study of one-dimensional examples, then we focus on some relevant particular cases, including variational-hemivariational inequalities with strongly monotone operators. Finally, we consider a model variational-hemivariational inequality which arises in Contact Mechanics for which we discuss its well-posedness and provide the corresponding mechanical interpretations.

We study a system of differential variational-hemivariational inequality arising in the modelling of adhesive viscoelastic contact problems for locking materials. The system consists of a variational-hemivariational inequality for the displacement field and an ordinary differential equation for the adhesion field. The contact is described by the unilateral constraint and normal compliance contact condition in which adhesion is taken into account and the friction is modelled by the nonmonotone multivalued subdifferential condition with adhesion. The problem is governed by a linear viscoelastic operator, a nonconvex locally Lipschitz friction potential and the subdifferential of the indicator function of a convex set which describes the locking constraints. The existence and uniqueness of solution to the coupled system are proved. The proof is based on a time-discretization method, known as the Rothe method. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational–hemivariational inequality with constraints in a reflexive Banach space, denoted P , to which we associate a sequence of inequalities { P n } . For each n ∈ N , P n is a variational–hemivariational inequality without constraints, governed by a penalty parameter λ 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 P . We provide the unique solvability of inequality P n . Then, under appropriate conditions on operators P n , we state and prove the convergence of the solution of P n to the solution of 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. [ABSTRACT FROM AUTHOR]

In this paper, we provide an alternative approach to establish the solution existence and uniqueness for elliptic variational-hemivariational inequalities. The new approach is based on elementary results from functional analysis, and thus removes the need of the notion of pseudomonotonicity and the dependence on surjectivity results for pseudomonotone operators. This makes the theory of elliptic variational-hemivariational inequalities more accessible to applied mathematicians and engineers. In addition, equivalent minimization principles are further explored for particular elliptic variational-hemivariational inequalities. Representative examples from contact mechanics are discussed to illustrate application of the theoretical results. [ABSTRACT FROM AUTHOR]

We consider an elliptic variational-hemivariational inequality 𝓟 in a reflexive Banach space, governed by a set of constraints K, a nonlinear operator A, and an element f. We associate to this inequality a sequence {𝓟n} of variational-hemivariational inequalities such that, for each n ∈ ℕ, inequality 𝓟n is obtained by perturbing the data K and A and, moreover, it contains an additional term governed by a small parameter εn. The unique solvability of 𝓟 and, for each n ∈ ℕ, the solvability of its perturbed version 𝓟n, are guaranteed by an existence and uniqueness result obtained in literature. Denote by u the solution of Problem 𝓟 and, for each n ∈ ℕ, let un be a solution of Problem 𝓟n. The main result of this paper states the strong convergence of un → u in X, as n → ∞. 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. [ABSTRACT FROM AUTHOR]

We consider an initial and boundary value problem P 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 R+ = [0,+∞). We list the assumptions on the data and derive a variational formulation PV 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 T = (I,Ω,C) where I is set of parameters, Ω represents a family of approximating sets and C is a set of sequences, then we define the well-posedness of Problem PV with respect to T. Our main result is Theorem 3.4, which provides sufficient conditions guaranteeing the well-posedness of PV 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 P can be approached by the weak solutions of different contact problems. Moreover, we provide the mechanical interpretation of these convergence results. [ABSTRACT FROM AUTHOR]

In this paper we study a new class of abstract evolution first order hemivariational inequalities which involves constraints and history-dependent operators. First, we prove the existence and uniqueness of solution by using a mixed equilibrium formulation with suitable selected bifunctions combined with a fixed-point principle for history-dependent operators. Next, we deduce existence, uniqueness and regularity results for some special subclasses of problems which include a constrained history-dependent variational–hemivariational inequality, an evolution quasi-variational inequality with constraints, and an evolution second order hemivariational inequality with constraints. Then, we provide an application of the results to a dynamic unilateral viscoelastic frictional contact problem and show its unique weak solvability. [ABSTRACT FROM AUTHOR]

This paper deals with an evolution inclusion which is an equivalent form of a variational-hemivariational inequality arising in quasistatic contact problems for viscoelastic materials. Existence of a weak solution is proved in a framework of evolution triple of spaces via the Rothe method and the theory of monotone operators. Comments on applications of the abstract result to frictional contact problems are made. The work extends the known existence result of a quasistatic hemivariational inequality by S. Migórski and A. Ochal [SIAM J. Math. Anal., 41 (2009) 1415-1435]. One of the linear and bounded operators in the inclusion is generalized to be a nonlinear and unbounded subdifferential operator of a convex functional, and a smallness condition of the coefficients is removed. Moreover, the existence of a hemivariational inequality is extended to a variational-hemivariational inequality which has wider applications. [ABSTRACT FROM AUTHOR]

In this paper, we consider a mathematical model which describes the quasistatic frictionless contact between a viscoplastic body and a foundation. The contact is modeled with normal compliance and unilateral constraint. We present the variational-hemivariational formulation of the model and prove its unique solvability. Then we introduce a fully discrete scheme to solve the problem and derive an error estimate. Under appropriate regularity assumptions of the exact solution, we obtain the optimal order error estimate. Finally, numerical results are reported to show the performance of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper, numerical analysis is carried out for a class of history-dependent variational-hemivariational inequalities by arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study boundary optimal control of an evolutionary system governed by a history‐dependent variational‐hemivariational inequality. The inequality is a weak formulation of a dynamic frictional contact problem for a viscoelastic body with a multivalued normal damped response condition and a simplified version of the Coulomb law of dry friction. A continuous dependence result for the solution map is proved and the existence of optimal solutions to the control problem is established. [ABSTRACT FROM AUTHOR]

The paper investigates control problems for a class of nonlinear elliptic variational–hemivariational inequalities with constraint sets. Based on the well posedness of a variational–hemivariational inequality, we prove some results on continuous dependence and existence of optimal pairs to optimal control problems. We obtain some continuous dependence results in which the strong dependence and weak dependence are considered, respectively. A frictional contact problem is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper we investigate the gap functions and regularized gap functions for a class of variational–hemivariational inequalities of elliptic type. First, based on regularized gap functions introduced by Yamashita and Fukushima, we establish some regularized gap functions for the variational–hemivariational inequalities. Then, the global error bounds for such inequalities in terms of regularized gap functions are derived by using the properties of the Clarke generalized gradient. Finally, an application to a stationary nonsmooth semipermeability problem is given to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper is devoted to an analysis of singular perturbations of inequality problems. For a general variational-hemivariational inequality, it is shown rigorously that under appropriate conditions, as the singular perturbation parameter approaches zero, the solution of the singularly perturbed problem converges to the solution of the limiting problem. As corollaries of this general result, we have similar convergence results for singularly perturbed problems of "pure"" hemivariational inequalities and of variational inequalities. The results are illustrated in the study of an obstacle plate bending problem. [ABSTRACT FROM AUTHOR]

The paper investigates inverse problems for a class of nonlinear elliptic variational-hemivariational inequalities. We prove results on the well posedness of a variational-hemivariational inequality and on the existence of solution to inverse problems. We illustrate our findings by an inverse problem for a frictional unilateral contact problem in nonlinear elasticity. [ABSTRACT FROM AUTHOR]

A quasistatic nonsmooth frictional contact problem for a viscoelastic material is studied. The contact is modeled by a multivalued normal damped response condition with the Clarke generalized gradient of a locally Lipschitz superpotential and the friction is described by a version of the Coulomb law of dry friction with the friction bound depending on the regularized normal stress. The weak formulation of the contact problem is a history-dependent variational–hemivariational inequality coupled with an operator equation for the stress field. A result on the unique weak solvability to this coupled system is proved through a recent contribution on history-dependent subdifferential inclusions and a fixed point approach. • New model in contact mechanics. • Multivalued contact conditions depending on each other. • Existence and uniqueness of a local solution. • Application of existence and uniqueness result to real world problems. [ABSTRACT FROM AUTHOR]

In this paper, we provide a comprehensive stability analysis for stationary variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. With contact mechanics as application background, stability is analyzed for solutions with respect to combined or separate perturbations in constitutive relations, external forces, constraints, and non-smooth contact boundary conditions of the inequality problems. The stability result is first proved for a general variational-hemivariational inequality. Then, stability results are obtained for various variational inequalities and hemivariational inequalities as special cases. Finally, we illustrate applications of the theoretical results for the stability analysis of model problems in contact mechanics. [ABSTRACT FROM AUTHOR]

Abstract Hemivariational inequalities have been successfully employed for mathematical and numerical studies of application problems involving nonsmooth, nonmonotone and multivalued relations. In recent years, error estimates have been derived for numerical solutions of hemivariational inequalities under additional solution regularity assumptions. Since the solution regularity properties have not been rigorously proved for hemivariational inequalities, it is important to explore the convergence of numerical solutions of hemivariational inequalities without assuming additional solution regularity. In this paper, we present a general convergence result enhancing existing results in the literature. [ABSTRACT FROM AUTHOR]

The present paper represents a continuation of Migórski 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. [ABSTRACT FROM AUTHOR]