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In this paper, we consider the behaviour of solutions to a class of fully history-dependent variational-hemivariational inequalities with respect to the perturbation of the data. First, the existence and uniqueness of the solution to a class of fully history-dependent variational-hemivariational inequalities is obtained by using a fixed point theorem. Second, we obtain continuous dependence result of solutions with respect to all the data of variational-hemivariational inequalities. Meanwhile, the convergence results of the solutions for the special case of abstract variational inequalities are also given. Finally, to illustrate our main results, we consider a class of viscoelastic contact problem with a long memory. By using our abstract result, we get the continuous dependence of the solutions to frictional contact problem with respect to all the data. [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]

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]

In this paper, quasi-statical boundary contact problems of couple-stress viscoelasticity for inhomogeneous anisotropic bodies with regard to friction are investigated. We prove the uniqueness theorem of weak solutions using the corresponding Green's formulas and positive definiteness of the potential energy. To analyze the existence of solutions, we equivalently reduce the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter, and study its solvability by the Galerkin approximate method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure, the existence theorem for the original contact problem with friction is proved. [ABSTRACT FROM AUTHOR]

We consider the Volterra integro-differential equation with a time-dependent prox-regular constraint that changes in an absolutely continuous way in time (a Volterra absolutely continuous time-dependent sweeping process). The aim of our paper is twofold. The first one is to show the solvability of the initial value problem by setting up an appropriate catching-up algorithm (full discretization). This part is a continuation of our paper (Bouach et al. in arXiv: 2102.11987. 2021) where we used a semi-discretization method. Obviously, strong solutions and convergence of full discretization scheme are desirable properties, especially for numerical simulations. Applications to non-regular electrical circuits are provided. The second aim is to establish the existence of optimal solution to an optimal control problem involving the Volterra integro-differential sweeping process. [ABSTRACT FROM AUTHOR]

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]

The goal of this paper is to introduce a new class of parametric elliptical variational-hemivariational inequality problems together with parametric inverse problems. We show the existence of solution for parametric inverse problem and well-posedness of parametric elliptical variational-hemivariationl inequality problems. An implementation in nonlinear elasticity, we address the parametric inverse problem for a parametric frictional unilateral contact problem. [ABSTRACT FROM AUTHOR]

In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a (u , v − u) + b (v − u , λ) + j (v) − j (u) + J (u , v) − J (u , u) ≥ (f , v − u) X , b (u , μ − λ) − ψ (μ) + ψ (λ) ≤ 0 , (v ∈ K ⊆ X , μ ∈ Λ ⊂ Y ), where (X , (⋅ , ⋅) X) and (Y , (⋅ , ⋅) Y) are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ (⋅) = ϵ ψ ¯ (⋅) (ϵ > 0) . Then, we deliver a convergence result for ϵ → 0 , the case ψ ≡ 0 appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones. [ABSTRACT FROM AUTHOR]

This paper starts off by the celebrated Knaster–Kuratowski–Mazurkiewicz principle in the formulation by Ky Fan. We provide a novel variant of this principle and build an existence theory for extended real-valued equilibrium problems with general, then monotone and pseudomonotone bifunctions. We develop our existence theory first in general topological vector spaces, then in reflexive Banach spaces, where we investigate the issue of coerciveness for existence on unbounded sets. Thereafter we use the Clarke generalized differential calculus for locally Lipschitz functions and derive existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. As application, we treat a unilateral contact problem in solid mechanics with nonmonotone friction. [ABSTRACT FROM AUTHOR]

The present paper analyzes and provides the existence of solutions to integro-differential sweeping processes involving perturbations with multimappings. The closed moving sets describing the processes are non-convex and only assumed to be prox-regular sets. [ABSTRACT FROM AUTHOR]

In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a (u , v − u) + b (v − u , λ) + j (v) − j (u) + J (u , v) − J (u , u) ≥ (f , v − u) X , b (u , μ − λ) − ψ (μ) + ψ (λ) ≤ 0 , (v ∈ K ⊆ X , μ ∈ Λ ⊂ Y ), where (X , (⋅ , ⋅) X) and (Y , (⋅ , ⋅) Y) are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ (⋅) = ϵ ψ ¯ (⋅) (ϵ > 0) . Then, we deliver a convergence result for ϵ → 0 , the case ψ ≡ 0 appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones. [ABSTRACT FROM AUTHOR]

We consider a mixed variational problem governed by a nonlinear operator and a set of constraints. Existence, uniqueness and convergence results for this problem have already been 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. [ABSTRACT FROM AUTHOR]

In the present paper, we study inverse problems for a class of nonlinear hemivariational inequalities. We prove the existence and uniqueness of a solution to inverse problems. Finally, we introduce an inverse problem for an electro-elastic frictional contact problem to illustrate our results. [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]

In this paper, we consider a penalty method for a class of differential hemivariational inequalities in reflexive Banach spaces, governed by a set of constraints. First, we recall the existence and uniqueness result of the differential hemivariational inequality. Further, we introduce a penalized problem without constraints and we prove that, as a penalty parameter tends to zero, the solution of the original inequality can be approached to the solution of the penalized problem. Finally, we illustrate our results by an example of a contact problem for which our abstract results can be applied. [ABSTRACT FROM AUTHOR]

In this paper we introduce an abstract nonsmooth optimization problem and prove existence and uniqueness of its solution. We present a numerical scheme to approximate this solution. The theory is later applied to a sample static contact problem describing an elastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on normal and tangential components of displacement. Finally, computational simulations are performed to illustrate obtained results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study a quasistatic contact problem between an electro-elastic viscoplastic body with damage and an electrically conductive foundation. The contact is modelled with an electrical condition, normal compliance and the associated version of Coulomb's law of dry friction in which slip dependent friction is included. We derive a variational formulation for the model and, under a smallness assumption, we prove the existence and uniqueness of a weak solution. [ABSTRACT FROM AUTHOR]

The aim of the paper is to introduce and investigate a dynamical system which consists of a variational–hemivariational inequality of hyperbolic type combined with a non-linear evolution equation. Such a dynamical system arises in studies of complicated contact problems in mechanics. Existence, uniqueness and regularity of a global solution to the system are established. The approach is based on a new semi-discrete approximation with an application of a surjectivity result for a pseudomonotone perturbation of a maximal monotone operator. A new dynamic viscoelastic frictional contact model with adhesion is studied as an application, in which the contact boundary condition is described by a generalised normal damped response condition with unilateral constraint and a multivalued frictional contact law. [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]

In this paper, we study a differential hemivariational inequality (DHVI, for short) in the framework of reflexive Banach spaces. Our aim is three fold. The first one is to investigate the existence and the uniqueness of mild solution, by applying a general fixed-point principle. The second one is to study its exponential stability, by employing the formula for the variation of parameters and inequality techniques. Finally, the third aim is to illustrate an application of our abstract results in the study of an initial and boundary value problem which describes the contact of an elastic rod with an obstacle. [ABSTRACT FROM AUTHOR]

Hemivariational inequalities arise in nonsmooth mechanics of solid involving nonmonotone and multi-valued mechanical relations. Typically, after the finite-element discretization, they lead to constrained nonsmooth nonconvex optimization problems with objective functions being the sum of quadratic functions and nonsmooth terms. In this paper, smoothing approximations are employed to solve the constrained nonsmooth nonconvex optimization problems. After properties of the smoothing functions are analysed, a smoothing quadratic regularization algorithm is presented and studied. The proposed algorithm can be implemented efficiently since the closed form solution is available at each iteration. Convergence of the algorithm is shown, and the worst-case complexity is investigated for reaching an ε-Clarke stationary point. A numerical example is reported to show the performance of the proposed method. [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]

We consider a mixed variational problem in real Hilbert spaces, defined on the unbounded interval of time [ 0 , + ∞) 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. [ABSTRACT FROM AUTHOR]

In this paper, we consider a dynamic viscoelastic contact problem with friction and wear, and describe it as a system of nonlinear partial differential equations. We formulate the previous problem as a hyperbolic quasi-variational inequality by employing the variational method. We adopt the Rothe method to show the existence and uniqueness of solution for the hyperbolic quasi-variational inequality under mild conditions. We also give a fully discrete scheme for solving the hyperbolic quasi-variational inequality and obtain error estimates for the fully discrete scheme. [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]

This paper deals with the existence and uniqueness of solutions for a class of state-dependent sweeping processes with constrained velocity in Hilbert spaces without using any compactness assumption, which is known to be an open problem. To overcome the difficulty, we introduce a new notion called hypomonotonicity-like of the normal cone to the moving set, which is satisfied by many important cases. Combining this latter notion with an adapted Moreau's catching-up algorithm and a Cauchy technique, we obtain the strong convergence of approximate solutions to the unique solution, which is a fundamental property. Using standard tools from convex analysis, we show the equivalence between this implicit state-dependent sweeping processes and quasistatic evolution quasi-variational inequalities. As an application, we study the state-dependent quasistatic frictional contact problem involving viscoelastic materials with short memory in contact mechanics. [ABSTRACT FROM AUTHOR]

This paper represents a continuation of our previous work, where a mathematical model which describes the equilibrium of an elastic body in frictional contact with a moving foundation was considered. An existence and uniqueness result was proved, together with a convergence result. The proofs were carried out by using arguments of elliptic variational inequalities. In this current paper, we complete our model by taking into account the wear of the foundation. This makes the problem evolutionary and leads to a new and nonstandard mathematical model, which couples a time-dependent variational inequality with an integral equation. We provide the unique weak solvability of the model by using a fixed point argument, among others. Then, we penalize the unilateral contact condition and prove that the penalized problem has a unique solution which converges to the solution of the original problem, as the penalization parameter converges to zero. [ABSTRACT FROM AUTHOR]

In this paper, we consider a new class of fully history-dependent quasivariational inequalities which arise in the study of quasistatic models of contact and involve two history-dependent operators. By using a fixed-point theorem and arguments of monotonicity and convexity, we prove an existence and uniqueness result of the solution, which includes as special cases some results already obtained in some papers. Then, the obtained result is applied to two problems of quasistatic frictional contact for viscoelastic materials and the unique weak solvability of each contact problem is obtained. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the process of contact with adhesion between a piezoelectric body and an obstacle, the so-called foundation. The material's behavior is assumed to be electro-viscoelastic; the process is quasistatic, the contact is modeled by the Signorini condition. The adhesion process is modeled by a bonding field on the contact surface. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on a general result on evolution equations with maximal monotone operators and fixed-point arguments. [ABSTRACT FROM AUTHOR]

The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics. [ABSTRACT FROM AUTHOR]

In this paper, we first gather existence results for linear and for pseudo-monotone variational inequalities in reflexive Banach spaces. We discuss the necessity of the involved coerciveness conditions and their relationship. Then, we combine Mosco convergence of convex closed sets with an approximation of pseudo-monotone bifunctions and provide a convergent approximation procedure for pseudo-monotone variational inequalities in reflexive Banach spaces. Since hemivariational inequalities in linear elasticity are pseudo-monotone, our approximation method applies to nonmonotone contact problems. We sketch how regularization of the involved nonsmooth functionals together with finite element approximation lead to an efficient numerical solution method for these nonconvex nondifferentiable optimization problems. To illustrate our theory, we give a numerical example of a 2D linear elastic block under a given nonmonotone contact law. [ABSTRACT FROM AUTHOR]

The paper investigates the contact between a nonlinear dynamic Gao beam and a rigid or reactive foundation. The contact is modeled with the normal compliance condition for the deformable foundation and with the Signorini condition for the rigid foundation. The existence and uniqueness of the weak solution for the problem with normal compliance are obtained. The solution of the Signorini condition for the rigid foundation is obtained by passing to the limit when the normal compliance approaches infinity. [ABSTRACT FROM AUTHOR]

In the present paper we investigate a three-dimensional boundary-contact problem of dynamics for a homogeneous hemitropic elastic medium with regard to friction. We prove the uniqueness theorem using the corresponding Green formulas and positive definiteness of the potential energy. To analyze the existence of solutions we reduce equivalently the problem under consideration to a spatial variational inequality. We consider a special parameter-dependent regularization of this variational inequality which is equivalent to the relevant regularized variational equation depending on a real parameter and study its solvability by the Faedo-Galerkin method. Some a priori estimates for solutions of the regularized variational equation are established and with the help of an appropriate limiting procedure the existence theorem for the original contact problem with friction is proved. [ABSTRACT FROM AUTHOR]

A model for the adhesive, quasi-static and frictionless contact between an electro-elastic body and a rigid foundation is studied in this paper. The contact is modelled with Signorini's conditions with adhesion. We provide variational formulation for the problem and prove the existence of a unique weak solution to the model. The proofs are based on arguments of time-dependent variational inequalities, differential equations and Banach fixed point. Then, a fully discrete scheme is introduced based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. Error estimates are derived on the approximative solutions from which the linear convergence of the algorithm is deduced under suitable regularity conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of evolution subdifferential inclusions involving history-dependent operators. We improve our previous theorems on existence and uniqueness and produce a continuous dependence result with respect to weak topologies under a weaker smallness condition. Two applications are provided to a frictional viscoelastic contact problem with long memory, and to a nonsmooth semipermeability problem. [ABSTRACT FROM AUTHOR]

In this paper, we study a quasi‐static contact problem between a thermo‐electro‐viscoelastic body and an electrically and thermally conductive foundation. The contact is modeled using a normal compliance contact condition with Coulomb's friction law, a regularized electrical contact condition and a heat flux contact condition taking into account the frictional heating effects. After introducing the model, the functional framework and the assumptions about the data, we derive the variational formulation of the model and prove its solvability. The proof is based on results of variational inequalities and fixed point argument. Finally, we investigate a fully discrete approximation using, respectively, Euler scheme and finite element method for the spatial variable and the time derivatives. Some error estimates are then derived, leading to convergence results under suitable additional regularity conditions. [ABSTRACT FROM AUTHOR]