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The objective of the paper is to investigate a dynamical system called a differential variational-hemivariational inequality (DVHVI) which couples an abstract variational-hemivariational inequality of elliptic type and a nonlinear evolution inclusion problem in a Banach space. Under appropriate assumptions, the nonemptiness and compactness of the solution set for DVHVI are established by using the Fan-Knaster-Kuratowski-Mazurkiewicz principle, the Minty approach, and the methods of nonsmooth analysis. Then, we explore properties of solution mapping for DVHVI which involve the relative compactness, continuity, and convergence in the Kuratowski sense. Employing these properties, we prove existence of a solution to the optimal control problem driven by a DVHVI. Next, well-posedness results for DVHVI are obtained, including the existence, uniqueness, and stability of the solution. Furthermore, we study sensitivity of a perturbed problem with multiparameters corresponding to DVHVI. Finally, a comprehensive parabolic-elliptic system with obstacle effect is considered as an illustrative application. [ABSTRACT FROM AUTHOR]

The primary objective of this paper is to study a large class of differential variational-hemivariational inequalities involving history-dependent operators and constraints in a Banach space. First, we establish a well-posedness result, which includes existence, uniqueness, and continuous dependence on the initial data. Second, related penalized and regularized problems without constraints are introduced whose solutions approach the solution to the original inequality. Finally, these results are applied to an obstacle parabolic-elliptic system consisting of a nonlinear reaction-diffusion equation and a time-dependent mixed boundary value problem with generalized gradient and Volterra integral terms. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a generalized nonlinear quasi-hemivariational inequality (QHI) involving a multivalued map in a Banach space. Under general assumptions, by using a fixed point theorem combined with the theory of nonsmooth analysis and the Minty technique, we prove that the set of solutions for the hemivariational inequality associated to the QHI problem is nonempty, bounded, closed, and convex. Then, we prove the existence of a solution to QHI. Furthermore, an optimal control problem governed by QVI is introduced, and a solvability result for the optimal control problem is established. Finally, an approximation of an elastic contact problem with the constitutive law involving a convex subdifferential inclusion is studied as an illustrative application, in which approximate contact boundary conditions are described by a multivalued version of the normal compliance contact condition with frictionless effect and a frictional contact law with the slip dependent coefficient of friction. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a class of sweeping processes with history-dependent operators. A well-posedness result is obtained, including the existence, uniqueness, and stability of the solution. Our approach is based on the variable time step-length discrete approximation method combined with a fixed point principle for history-dependent operators. Then, a quasi-static frictional contact problem for viscoelastic materials with unilateral constraints in velocity is considered. The abstract result is applied in the study of this problem in order to provide its unique weak solvability as well as the continuous dependence of the solution with respect to the initial data. [ABSTRACT FROM AUTHOR]