Well-Posedness and Convergence Results for History-Dependent Inclusions

We consider an inclusion in a real Hilbert space governed by a time-dependent set of constraints and a history-dependent operator. We introduce the concept of T- well-posedness for this inclusion, associated to a given Tykhonov triple T. Next, we provide a T-well-posedness result that we use in order to deduce the continuous dependence of the solution with respect to the data. Then, we state and prove a convergence criterion to the solution of the inclusion that we use to prove a convergence result for an associate penalty problem. Moreover, we show that this criterion allows us to construct a Tykhonov triple T-which give rise to an optimal well-posedness concept for the corresponding inclusion. Finally, we use these abstract results in the study of a nonlinear viscoelastic constitutive law with long memory term and unilateral constraints.