Class Of Hemivariational Inequalities Research Articles

Hemivariational inequalities on graphs

In this paper, a new class of hemivariational inequalities is introduced. It concerns Laplace operator on locally finite graphs together with multivalued nonmonotone nonlinearities expressed in terms of Clarke’s subdifferential. First of all, we state and prove some results on the subdifferentiability of nonconvex functionals defined on graphs. Thereafter, an elliptic hemivariational inequality on locally finite graphs is considered and the existence and uniqueness of its weak solutions are proved by means of the well-known surjectivity result for pseudomonotone mappings. In the end of this paper, we tackle the problem of hemivariational inequalities of parabolic type on locally finite graphs and we prove the existence of its weak solutions.

Trifunction Bihemivariational Inequalities

In this paper, we consider a new class of hemivariational inequalities, which is called the trifunction bihemivariational inequality. We suggest and analyze some iterative methods for solving the trifunction bihemivariational inequality using the auxiliary principle technique. The convergence analysis of these iterative methods is also considered under some mild conditions. Several special cases are also considered. Results proved in this paper can be viewed as a refinement and improvement of the known results.

Multivalued nonmonotone dynamic boundary condition

In this paper, we introduce a new class of hemivariational inequalities, called dynamic boundary hemivariational inequalities, reflecting the fact that the governing operator is also active on the boundary. In our context, it concerns the Laplace operator with Wentzell (dynamic) boundary conditions perturbed by a multivalued nonmonotone operator expressed in terms of Clarke subdifferentials. We show that one can reformulate the problem so that standard techniques can be applied. We use the well-established theory of boundary hemivariational inequalities to prove that under growth and general sign conditions, the dynamic boundary hemivariational inequality admits a weak solution. Moreover, in the situation where the functionals are expressed in terms of locally bounded integrands, a “filling in the gaps” procedure at the discontinuity points is used to characterize the subdifferential on the product space. Finally, we prove that, under a growth condition and eventually smallness conditions, the Faedo–Galerkin approximation sequence converges to a desired solution.

Convergence of solutions to inverse problems for a class of variational-hemivariational inequalities

The paper investigates an inverse problem for a stationary variational-hemivariational inequality. The solution of the variational-hemivariational inequality is approximated by its penalized version. We prove existence of solutions to inverse problems for both the initial inequality problem and the penalized problem. We show that optimal solutions to the inverse problem for the penalized problem converge, up to a subsequence, when the penalty parameter tends to zero, to an optimal solution of the inverse problem for the initial variational-hemivariational inequality. The results are illustrated by a mathematical model of a nonsmooth contact problem from elasticity.

A Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces

We study a new class of elliptic variational-hemivariational inequalities in reflexive Banach spaces. 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. We deliver a result on existence and uniqueness of a solution to the inequality. Next, we show the continuous dependence of the solution on the data of the problem and we introduce a penalty method, for which we state and prove a convergence result. Finally, we consider a mathematical model which describes the equilibrium of an elastic body in unilateral contact with a foundation. The model leads to a variational-hemivariational inequality for the displacement field, that we analyse by using our abstract results.

A class of history-dependent variational-hemivariational inequalities

We consider a new class of variational-hemivariational inequalities which arise in the study of quasistatic models of contact. The novelty lies in the special structure of these inequalities, since each inequality of the class involve unilateral constraints, a history-dependent operator and two nondifferentiable functionals, of which at least one is convex. We prove an existence and uniqueness result of the solution. The proof is based on arguments on elliptic variational-hemivariational inequalities obtained in our previous work [23], combined with a fixed point result obtained in [30]. Then, we prove a convergence result which shows the continuous dependence of the solution with respect to the data. Finally, we present a quasistatic frictionless problem for viscoelastic materials in which the contact is modeled with normal compliance and finite penetration and the elasticity operator is associated to a history-dependent Von Mises convex. We prove that the variational formulation of the problem cast in the abstract setting of history-dependent quasivariational inequalities, with a convenient choice of spaces and operators. Then we apply our general results in order to prove the unique weak solvability of the contact problem and its continuous dependence on the data.

A Class of Variational-Hemivariational Inequalities with Applications to Frictional Contact Problems

A class of variational-hemivariational inequalities is studied in this paper. An inequality in the class involves two nonlinear operators and two nondifferentiable functionals, of which at least one is convex. An existence and uniqueness result is proved for a solution of the inequality. Continuous dependence of the solution on the data is shown. Convergence is established rigorously for finite element solutions of the inequality. An error estimate is derived which is of optimal order for the linear finite element method under appropriate solution regularity assumptions. Finally, the results are applied to a variational-hemivariational inequality arising in the study of some frictional contact problems.

On a Class of Variational-Hemivariational Inequalities Involving Upper Semicontinuous Set-Valued Mappings

This paper is devoted to the various coercivity conditions in order to guarantee existence of solutions and boundedness of the solution set for the variational-hemivariational inequalities involving upper semicontinuous operators. The results presented in this paper generalize and improve some known results.

Well-posedness by perturbations of variational-hemivariational inequalities with perturbations

In this paper, we consider an extension of the notion of well-posedness by perturbations, introduced by Zolezzi for a minimization problem, to a class of variational-hemivariational inequalities with perturbations in Banach spaces, which includes as a special case the class of mixed variational inequalities. Under very mild conditions, we establish some metric characterizations for the well-posed variational-hemivariational inequality, and show that the well-posedness by perturbations of a variational-hemivariational inequality is closely related to the well-posedness by perturbations of the corresponding inclusion problem. Furthermore, in the setting of finite-dimensional spaces we also derive some conditions under which the variational-hemivariational inequality is strongly generalized well-posed-like by perturbations.

Iterative schemes for trifunction hemivariational inequalities

In this paper, we consider and study a new class of hemivariational inequalities, which is called trifunction hemivariational inequality. We suggest and analyze a class of iterative methods for solving trifunction hemivariational inequalities using the auxiliary principle technique. We prove that the convergence of these new methods either requires partially relaxed strongly monotonicity or pseudomonotonicity, which is a weaker condition than monotonicity. Results obtained in this paper include several new and known results as special cases.

On a class of hemivariational inequalities at resonance

We consider a class of noncoercive hemivariational inequalities involving the p-Laplacian at resonance. We use the unilateral growth condition so the energy functional is nonsmooth, nonconvex and its effective domain does not coincide with the whole space W 1,p 0(Ω) . To avoid this difficulty we study the problem in finite-dimensional spaces using the mountain-pass theorem for locally Lipschitz functionals and then we pass to the limit to obtain the existence of solutions.

Multiple solutions for a class of eigenvalue problems involving a nonlinear monotone operator in hemivariational inequalities

In this paper we prove two multiplicity results for a class of hemi-variational inequalities involving a monotone operator. This type of eigenvalue problems was firstly studied by D. Motreanu and P.D. Panagiotopoulos in[8]. The study is made in the nonresonant case by using some techniques in critical point theory which was developed in [4].

Multiple Solutions for a Class of Hemivariational Inequalities Involving Periodic Energy Functionals

In this paper we prove firstly that if f:X → R is a locally Lipschitz function, bounded from below and invariant to a discrete group of dimension N is a suitable sense, acting on a Banach space X, then the problem: find u ∈ X such that o ∈ ∂f(u) (here ∂f(u) denotes Clarke's generalized gradient of fat x) admits at least N + 1 orbits of solutions. Then, for a class of discrete groups G of isometries of a Hilbert space X, we establish an existence result for infinitely many G-orbits of eigensolutions to the problem: find u ∈ X such that λΛu ∈ ∂f(u) for some λ ∈ R, where Λ:X → X * stands for the duality map. The last two sections are devoted to applications of the existence results to hemivariational inequalities possessing invariance properties.

Solvability theory for a class of hemivariational inequalities involving copositive plus matrices applications in robotics

The study of the equilibrium of an object-robotic hand system including nonmonotone adhesive effects and nonclassical friction effects leads to new inequality methods in robotics. The aim of this paper is to describe these inequality methods and provide a corresponding suitable mathematical theory.

On a class of Hemivariational inequalities involving hemicontinuous monotone operators. Application in unilateral mechanics

The hemivariational inequality approach of P.D. Panagiotopoulos [18]-[22] is used so as to establish the existence of solutions to a class of noncoercive and nonconvex constrained problems in a reflexive Banach space.

Homoclinic orbits for a class of hemivariational inequalities

Using a new compact imbedding theorem of C. De Coster and M.Willem [5], we prove the existence of homoclinic orbits for a class of hemivariational inequalities.

A numerical method for a class of hemivariational inequalities

Nonmonotone possibly multivalued stress-strain or reaction-displacement laws give rise to hemivariational inequalities. due to the lack of convexity and the nonsmoothness of the underlying (super) potentials the problems have generally nonunique solutions (stable or unstable). Here we propose a method for the decomposition of the hemivaritional inequality into a finite number of variational inequalities (monotone problems). The method is based on the decomposition of the contingent cone of the superpotential into convex constituents and gives rise to a very effective and reliable numerical algorithm for large scale hemivariational inequalities. Numerical results from the delamination of composites and discussion of the numerical aspects illustrate the theory.