Hemivariational Inequalities Research Articles

On variational–hemivariational inequalities in Banach spaces

This paper is devoted to a well-posedness analysis of elliptic variational–hemivariational inequalities in Banach spaces. The differential operator associated with the variational–hemivariational inequality is assumed to be strongly monotone of a general order, in contrast to that in the majority of existing references on this subject where the differential operator is assumed to be strongly monotone of order 2. Moreover, the solution existence is proved with an approach more accessible to applied mathematicians and engineers, instead of through an abstract surjectivity result for pseudomonotone operators in existing references. Equivalent minimization principles are established for certain variational–hemivariational inequalities, which are valuable for developing efficient numerical algorithms. The theoretical results are applied to the analysis of a mixed hemivariational inequality in the study of a generalized Newtonian fluid flow problem involving a nonsmooth slip boundary condition of friction type. Existence and uniqueness of both the velocity and pressure unknowns are shown for the mixed hemivariational inequalities.

Periodic solutions to history-dependent differential hemivariational inequalities with applications

In this article, we investigate a hybrid model combined by a parabolic differential equation and a parabolic hemivariational inequality (so-called differential hemivariational inequality of parabolic–parabolic type) in general infinite dimensional spaces which includes the history-dependent operator. The solvability of initial value problems as well as the periodic problems of the hemivariational inequality and the differential hemivariational inequality have been proved. In application, we study a contact problem with normal compliance driven by a history-dependent dynamical system.

Discontinuous Galerkin Methods for Hemivariational Inequalities in Contact Mechanics

Discontinuous Galerkin Methods for Hemivariational Inequalities in Contact Mechanics

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

Sensitivity analysis for optimal control problems governed by Hilfer fractional differential hemivariational inequalities

The Existence Problems of Solutions for a Class of Differential Variational–Hemivariational Inequality Problems

In this work, we used reflexive Banach spaces to study the differential variational—hemivariational inequality problems with constraints. We established a sequence of perturbed differential variational–hemivariational inequality problems with perturbed constraints and penalty coefficients. Then, for each perturbed inequality, we proved the unique solvability and convergence of the solutions to the problems. Following that, we proposed a mathematical model for a viscoelastic rod in unilateral contact equilibrium, where the unknowns were the displacement field and the history of the deformation. We used the abstract penalty method in the analysis of this inequality and provided the corresponding mechanical interpretations.

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.

Quasilinear differential inclusions driven by degenerated p-Laplacian with weight

"The main result of the paper provides the existence of a solution to a quasilinear inclusion problem with Dirichlet boundary condition which exhibits a term with full dependence on the solution and its gradient (convection term) and is driven by the degenerated p-Laplacian with weight. The multivalued term in the differential inclusion is in form of the generalized gradient of a locally Lipschitz function expressed through the primitive of a locally essentially bounded function, which makes the problem to be of a hemivariational inequality type. The novelty of our result is that we are able to simultaneously handle three major features: degenerated leading operator, convection term and discontinuous nonlinearity. Results of independent interest regard certain nonlinear operators associated to the differential inclusion."

Existence results for Dirichlet double phase differential inclusions

"In this paper we consider a class of double phase differential inclusions of the type $$\left\{ \begin{array}{ll} -{\rm div\;}\left(|\nabla u|^{p-2}\nabla u+\mu(x)|\nabla u|^{q-2}\nabla u\right) \in \partial_C^2 f(x,u) , & \mbox{ in }\Omega,\\ u=0, & \mbox{ on }\partial\Omega, \end{array} \right.$$ where $\Omega \subset \mathbb{R}^N$, with $N\ge 2$, is a bounded domain with Lipschitz boundary, $f(x,t)$ is measurable w.r.t. the first variable on $\Omega$ and locally Lipschitz w.r.t. the second variable and $\partial_C^2 f(x,\cdot)$ stands for the Clarke subdifferential of $t\mapsto f(x,t)$. The variational formulation of the problem gives rise to a so-called hemivariational inequality and the corresponding energy functional is not differentiable, but only locally Lipschitz. We use nonsmooth critical point theory to prove the existence of at least one weak solution, provided the $\partial_C^2 f(x,\cdot)$ satisfies an appropriate growth condition."

Existence of Sobolev-Type Hilfer Fractional Neutral Stochastic Evolution Hemivariational Inequalities and Optimal Controls

This article concentrates on a control system with a nonlocal condition that is driven by neutral stochastic evolution hemivariational inequalities (HVIs) of Sobolev-type Hilfer fractional (HF). In order to illustrate the necessary requirements for the existence of mild solutions to the required control system, we first use the characteristics of the modified Clarke sub-differential and a fixed point approach for multivalued functions. Then, we show that there are optimal state-control sets that are driven by Sobolev-type HF neutral stochastic evolution HVIs utilizing constrained Lagrange optimal systems. The optimal control (OC) results are created without taking the uniqueness of the control system solutions into account. Finally, the main finding is shown by an example.

Noncoercive hyperbolic variational–hemivariational inequalities with an application to contact problem

In this paper we study the solvability of a coupled system which consists of a hyperbolic variational–hemivariational inequality and an equation. The inequality is in the form of a second order evolution inclusion involving both convex and Clarke’s subdifferentials. Unlike most of inequality problems in literature, the one here is noncoercive due to the absence of a leading operator acting on the first order derivative of solutions with respect to time. As a result, the known surjectivity theorem for evolution inclusion of bounded and coercive L-pseudomonotone operator is ineffective to our problem. Based on the Rothe method and monotone operator theory, we establish an existence theorem for the coupled system. Moreover, an application to contact problem of beam is given.

A new general class of systems of elliptic quasi-variational–hemivariational inequalities

In this paper we study a general class of systems of strongly coupled quasi-variational–hemivariational inequalities with implicit constraints in reflexive Banach spaces. Each inequality contains a convex potential, a locally Lipschitz superpotential, and a implicit obstacle set. The solvability of the system is established based on a fixed point approach, monotonicity arguments, and nonsmooth analysis. The result generalizes several theorems on various particular cases on variational, variational–hemivariational, and quasi-variational inequalities. We illustrate the applicability of the abstract theory by a nonlinear static thermoelastic frictional contact problem with implicit obstacles for which we provide existence and regularity results.

A nonsmooth optimization approach for time-dependent hemivariational inequalities

This paper is devoted to the study of time-dependent hemivariational inequality. We prove the existence and uniqueness of its solution, provide a fully discrete scheme, and reformulate this scheme as a series of nonsmooth optimization problems. The introduced theory is later applied to a sample quasistatic contact problem that describes a viscoelastic body in frictional contact with a foundation. This contact is governed by a nonmonotone friction law with dependence on the normal component of displacement and the tangential component of velocity. Finally, computational simulations are performed to illustrate the obtained results.

A priori error estimate of virtual element method for a quasivariational–hemivariational inequality

In this paper, the virtual element method is used to solve a frictional contact problem which can be characterized by an elliptic quasivariational–hemivariational inequality. After reviewing the existence and uniqueness of the solution, we present the construction of virtual element spaces and corresponding numerical scheme. Furthermore, a Céa’s inequality for error estimation is established, and an optimal order error bound is derived under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence order.

TÍNH CHẤT NGHIỆM CHO MỘT LỚP CÁC BẤT ĐẲNG THỨC HEMI-BIẾN PHÂN KIỂU PARABOLIC

In this paper, we study a class of parabolic hemivariational inequalities in infinite dimensional spaces. We prove the solvability and the continuous dependence on initial conditions for this problem based on surjective lemma, monotone theory and estimates.

The virtual element method for general variational–hemivariational inequalities with applications to contact mechanics

This paper mainly analyzes the general elliptic variational–hemivariational inequalities with or without constraints by using the virtual element method. The approximations can be internal or external and a Céa’s type inequality is derived for a priori error estimates. Then, we apply the results to a variational–hemivariational inequality arising in frictional contact problems, and the optimal order error estimate is obtained for the linear virtual element solution under appropriate solution regularity assumptions. Finally, numerical simulation results are reported to show the performance of the proposed method, in particular, numerical convergence orders are in good agreement with the theoretical predictions.

Second order evolutionary problems driven by mixed quasi-variational–hemivariational inequalities

In this paper we introduce a differential quasi-variational inequality which consists of a second order partial differential equation involving history-dependent operators and a mixed quasi-variational–hemivariational inequality in Banach spaces. At first, by using the KKM theorem and monotonicity arguments, we show that the solution set of the mixed quasi-variational–hemivariational inequality is nonempty, bounded, closed and convex. Then, we establish the measurability and upper semicontinuity of the solution set with respect to the time variable and state variable. Finally, based on the theory of strongly continuous cosine operators and a fixed point theorem for condensing set-valued operators, we build the existence of mild solutions for the differential quasi-variational–hemivariational inequality.

A class of differential hemivariational inequalities constrained on nonconvex star-shaped sets

The purpose of this paper is to investigate a class of nonconvex-constrained differential hemivariational inequalities consisting of nonlinear evolution equations and evolutionary hemivariational inequalities. The admissible set of constraints is closed and star-shaped with respect to a certain ball in a reflexive Banach space. We construct an auxiliary inclusion problem and obtain the existence results by applying a surjectivity theorem for multivalued pseudomonotone operators and the properties of Clarke subgradient operator. Moreover, the existence of a solution to the original problem is established by hemivariational inequality approach and a penalization method in which a small parameter does not have to tend to zero. Finally, an application of the main results is provided.

A new class of fully history-dependent variational-hemivariational inequalities with application to contact mechanics

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.

Solvability and pullback attractor for a class of differential hemivariational inequalities with its applications

In this paper, we consider an abstract system which consists of a nonlinear differential inclusion and a parabolic hemivariational inequality (DPHVI) in Banach spaces. The objective of this paper is four fold. The first target is to deal with the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to parabolic hemivariational inequality. The second aim is to investigate the existence of mild solutions to DPHVI by means of a fixed point technique. The third one is to study the existence of a pullback attractor for the multivalued processes governed by DPHVI. Finally, the fourth goal is to demonstrate a concrete application of our main results arising from the dynamic thermoviscoelasticity problems.

Analysis and Approximation of Hemivariational Inequality for a Frictional Thermo-electro-visco-elastic Contact Problem with Damage

The aim of this paper is to investigate a contact problem involving thermo-electro-visco-elastic body with damage and a rigid foundation. The friction is modelled with a subgradient of a locally Lipschitz mapping, and the contact is described by the Signorini's unilateral contact condition. A parabolic differential inclusion for the damage function is used to include the damaging effect in the model. We establish the model's variational formulation using four systems of three hemivariational inequalities and a parabolic equation, we prove an existence and uniqueness result of this problem. The proof is based on a fixed point argument and a recent finding from hemivariational inequality theory. Finally, by employing the finite element approach, we investigate a fully discrete approximation of the model and we derive error estimates on the approximate solution.