Convergence and Optimization Results for a History-Dependent Variational Problem

We consider a mixed variational problem in real Hilbert spaces, defined on the unbounded interval of time [0,+∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$[0,+\infty)$\end{document} and governed by a history-dependent operator. We state the unique solvability of the problem, which follows from a general existence and uniqueness result obtained in Sofonea and Matei (J. Glob. Optim. 61:591–614, 2015). Then, we state and prove a general convergence result. The proof is based on arguments of monotonicity, compactness, lower semicontinuity and Mosco convergence. Finally, we consider a general optimization problem for which we prove the existence of minimizers. The mathematical tools developed in this paper are useful in the analysis of a large class of nonlinear boundary value problems which, in a weak formulation, lead to history-dependent mixed variational problems. To provide an example, we illustrate our abstract results in the study of a frictional contact problem for viscoelastic materials with long memory.