Search

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]