Error bounds for a class of history-dependent variational inequalities controlled by $\mathcal{D}$-gap~functions

Abstract

In the present paper, we are concerned with investigating error bounds for history-dependent variational inequalities controlled by the difference gap (for brevity, D-gap) functions. First, we recall a class of elliptic variational inequalities involving the history-dependent operators (for brevity, HDVI). Then, we introduce a new concept of gap functions to the HDVI and propose the regularized gap function for the HDVI via the optimality condition for the concerning minimization problem. Consequently, the D-gap function for the HDVI depends on these regularized gap functions is established. Finally, error bounds for the HDVI controlled by the regularized gap function and the D-gap function are derived under suitable conditions.

Upper error bounds of DG-functions for history-dependent variational-hemivariational inequalities

“…Very recently, Cen-Nguyen-Zeng [3] considered upper error bounds for the generalized variationalhemivariational inequalities with history-dependent operators by using RG-functions. Chen-Tam [5] proposed upper error bounds for a class of history-dependent variational inequalities controlled by the DG-functions. Especially, Tam-Chen [39] derived upper error bounds for abstract elliptic variational-hemivariational inequalities based on generalized DG-functions with applications to contact mechanics.…”