Caputo fractional differential variational-hemivariational inequalities involving history-dependent operators: Global error bounds and convergence

In this paper, we introduce and investigate a new differential variational-hemivariational inequality (for brevity, FDVHI) involving history-dependent operators and Caputo fractional order derivative operator. We first propose the new concept of gap functions for the history -dependent variational control system of FDVHI. Furthermore, using the estimate technologies involving Caputo fractional derivatives and history-dependent operators, we establish a RG-function of the Fukushima type (in which RG is a brevity of "regularized gap") and provide some nice properties of the RG-functions. Then, global error bounds for the FDVHI controlled by the Fukushima RG-function are derived under some suitable assumptions. Finally, by applying penalty methods, we formulate FDVHI to a family penalized problems that are fractional differential variational-hemivariational inequalities without constraints, and establish the convergence result that the solution to the original problem FDVHI can be approached by the solutions of the corresponding penalized problems, as the penalty parameter converges to zero.