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In this paper we deal with a class of nonlinear quasi-variational inequalities involving a set-valued map and a constraint set. First, we prove that the set of weak solutions of the inequality is nonempty, weakly compact and upper semicontinuous with respect to perturbations in the data. Then, the results are applied to a quasi variational-hemivariational inequality of elliptic kind. Finally, as an illustrative applications we examine a mathematical model of a nonsmooth static frictional unilateral contact problem for ideally locking materials in nonlinear elasticity. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to study the existence of solutions for optimal control problems governed by mixed quasi-equilibrium problems under monotonicity type conditions. More precisely, the state control system takes the general form of a mixed quasi-equilibrium problem described by the sum of a maximal monotone bifunction and a pseudomonotone bifunction in the sense of Brézis. As applications, we study the existence of solutions for optimal control problems governed by quasi-variational inequalities. In particularly, we consider the optimal control of an evolutionary quasi-variational inequality described by a p-Laplacian type operator. The results obtained in this paper are new and can be applied to the study of optimal control of a variety of systems whose formulations can be presented as a mixed equilibrium problem. [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]

This paper deals with the study of weak sharp solutions for nonsmooth variational inequalities and finite convergence property of the proximal point method. We present several characterizations for weak sharpness of the solutions set of nonsmooth variational inequalities without using the gap functions. We show that under weak sharpness of the solutions set, the sequence generated by proximal point methods terminates after a finite number of iterations. We also give an upper bound for the number of iterations for which the sequence generated by the exact proximal point methods terminates. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a new class of fractional delay differential mixed variational inequalities (FDDMVI, for short) formulated by a fractional delay evolution inclusion and a mixed variational inequality in infinite Banach spaces. By applying a new measure of noncompactness and fixed point theorem for a condensing set-valued map, we obtain the global solvability of a FDDMVI on the half-line. Furthermore, we apply the obtained results to establish a weakly asymptotic stability result of the zero mild solution to a FDDMVI. Finally, an example is given to demonstrate the main results. [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]

In this paper, we introduce and study an optimal control problem governed by mixed equilibrium problems described by the sum of a maximal monotone bifunction and a bifunction which is pseudomonotone in the sense of Brézis/quasimonotone. Our motivation comes from the fact that many control problems, whose state system is a variational inequality problem or a nonlinear evolution equation or a hemivariational inequality problem, can be formulated as a control problem governed by a mixed equilibrium problem. There are different techniques to study optimal control problems governed by nonlinear evolution equations, variational inequalities or hemivariational inequalities in the literature. However, our technique is completely different from existing ones. It is based on the Mosco convergence and recent results in the theory of equilibrium problems. As an application, we study optimal control problems governed by elliptic variational inequalities with additional state constraints. [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 existence of solution for systems of Fokker–Planck equations coupled through a common nonlinear congestion. Two different kinds of congestion are considered: a porous media congestion or soft congestion and the hard congestion given by the constraint ρ 1 + ρ 2 ⩽ 1 . We show that these systems can be seen as gradient flows in a Wasserstein product space and then we obtain a constructive method to prove the existence of solutions. Therefore it is natural to apply it for numerical purposes and some numerical simulations are included. [ABSTRACT FROM AUTHOR]