Partial differential hemivariational inequalities

This paper is devoted to the study of a new and complicated dynamical system, called a fractional differential hemivariational inequality, which consists of a quasilinear evolution equation involving the fractional Caputo derivative operator and a coupled generalized parabolic hemivariational inequality. Under certain general assumptions, existence and regularity of a mild solution to the dynamical system are established by employing a surjectivity result for weakly-weakly upper semicontinuous multivalued mappings, and a feedback iterative technique together with a temporally semi-discrete approach through the backward Euler difference scheme with quasi-uniform time-steps. To illustrate the applicability of the abstract results, we consider a nonstationary and incompressible Navier-Stokes system supplemented by a fractional reaction-diffusion equation, which is studied as a fractional hemivariational inequality. Bibliography: 57 titles.

<p style='text-indent:20px;'>This paper deals with a class of history-dependent frictional contact problem with the surface traction affected by the impulsive differential equation. The weak formulation of the contact problem is a history-dependent hemivariational inequality with the impulsive differential equation. By virtue of the surjectivity of multivalued pseudomonotone operator theorem and the Rothe method, existence and uniqueness results on the abstract impulsive differential hemivariational inequalities is established. In addition, we consider the stability of the solution to impulsive differential hemivariational inequalities in relation to perturbation data. Finally, the existence and uniqueness of weak solution to the contact problem is proved by means of abstract results.</p>

The purpose of this paper is to investigate an abstract system, a combination of a nonlinear evolutionary differential inclusion and a first-order evolutionary variational-hemivariational inequality (EDVHVI, for short) in the framework of Banach spaces. The central idea of our present paper consists of four parts. Initially, we are concerned with the results on the existence of solutions and the properties which involve the boundedness and continuous dependence results of the solution set to an evolutionary variational-hemivariational inequality (EVHVI). Subsequently, the second goal further studies the existence of mild solutions to EDVHVI by employing the fixed point theorem. Next, the existence of pullback attractors for the multivalued processes driven by EDVHVI is investigated in a general setting. At the end, we give a specific application about a comprehensive dynamic system generated by partial differential inclusions with obstacle effect, which is shown to substantiate our theory towards the conclusions.

The main goal of this paper is to analyze and study a new class of dual abstract systems that consists of differential hemivariational inequalities systematized by an evolutionary hemivariational inequality accumulated with a fractional differential equation of order 1<q<2 in a Banach space. The existence of a solution to the mentioned problem is proved by employing Rothe’s method and a surjectivity result. An example is provided to validate the results.

In this paper, we show the existence of at least three nontrivial solutions for a nonlinear elliptic equation driven by the p-Laplacian with a nonsmooth potential (hemivariational inequality) and Robin boundary condition. Two of these solutions are of constant sign (one is positive, the other negative). We mainly use a variational approach together with a sub-sup solution method.

In this paper, we study the existence of solutions for noncoercive stationary Navier–Stokes equations of heat-conducting fluids with nonmonotone boundary conditions modeled by hemivariational inequalities. Our method is new and differs from most of the existing techniques developed in the literature. It is based on a recent approach developed on the existence of solutions for mixed equilibrium problems described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brezis. We introduce a Browder–Tikhonov regularization method for mixed equilibrium problems by means of the duality mapping with gauge function $$\mu (t)$$. By using this regularization procedure and techniques from the recession analysis, we study the existence of solutions for the problem considered in this paper.

This paper is devoted to the existence of solution and Tikhonov regularization theory for a class of hemivariational inequalities. We give the existence of solutions for the class of hemivariational inequalities when the mappings satisfy the so-called hemivariational inequality property and a rather weak coercivity condition. The existence result allows us to deduce the Tikhonov regularization result. Our results generalize some results by He (Abstr. Appl. Anal. 2012, 172061, 2012) and others.

Hyperbolic hemivariational inequalities controlled by evolution equations with application to adhesive contact model

Analysis of an adhesive contact problem for viscoelastic materials with long memory

Exploiting recent results obtained in the theory of equilibrium problems, we focus on the study of the existence of antiperiodic solutions for a first order hemivariational inequality problem. The problem involved is associated to time dependent pseudomonotone and quasimonotone operators. Our results are new and lead to introduce a new and efficient approach for the study of hemivariational inequality problems.

Dynamic Solution of the Hamilton-Jacobi inequality in the ℒ2-disturbance Attenuation Problem

A general differential quasi variational–hemivariational inequality: Well-posedness and application

We consider a nonlinear hemivariational inequality with the p-Laplacian at resonance. Using an extension of the nonsmooth mountain pass theorem of Chang, which makes use of the Cerami compactness condition, we prove the existence of a nontrivial solution. Our existence results here extends a recent theorem on resonant hemivariational inequalities, by the authors in 1999.

Existence of periodic solutions for a differential inclusion systems involving the p(t)-Laplacian

The aim of this paper is to study an elliptic bilateral obstacle system (EBOS, for short) involving a nonlinear and nonhomogeneous partial differential operator and a multivalued term which is described by Clarke’s generalized gradient. First, we obtain the weak formulation of (EBOS) which is a variational-hemivariational inequality, and prove the unique solvability of the bilateral obstacle problem. Second, we consider a nonlinear optimal control problem governed by (EBOS) in which the control variable is the bilateral obstacle, and establish the existence of an optimal solution to the obstacle control problem under mild conditions. Then, employing the regularization technique and penalty approach, we introduce a family of approximating problems corresponding to the optimal control problem under consideration. Finally, a convergence result that any sequence of solutions for the approximating problems converges to an optimal solution of the original optimal control problem is delivered.