Convergence Analysis of a Differential Variational Inequality

This paper deals with the study of a special class of systems which couple an implicit differential equation with an evolutionary variational inequality. Following the terminology used in the literature, we refer to such systems as differential variational inequalities (DVI). Inequalities of this form arise in a large number of mathematical models in Solid and Contact Mechanics. We start with a one-dimensional rheological example which leads to a DVI. Inspired by this example, we formulate the problem in the abstract framework of a real Hilbert space X, and then we state and prove an existence and uniqueness result. The proof is based on a result of evolutionary variational inequalities combined with a fixed-point argument for history-dependent operators. We proceed with a convergence criterion, that is, we identify necessary and sufficient conditions which guarantee the convergence of a sequence of elements of X to the solution. Then, we introduce a well-posedness concept for the DVI we study and we state and prove the corresponding well-posedness result. Finally, we apply these abstract results in the study of the one-dimensional rheological model and provide the corresponding mechanical interpretations.