The goal of this paper is to consider fractional differential hemivariational inequalities (FDHVIs, for short) in the framework of Banach spaces. Our first aim is to investigate the existence of mild solutions to FDHVIs by means of a fixed point technique avoiding the hypothesis of compactness on the semigroup. The second step of the paper is to study the existence of decay mild solutions to FDHVIs via giving asymptotic behavior of Mittag-Leffler function.

In this paper, we are concerned with a generalized evolution dynamical system, called fractional differential variational-hemivariational inequality (FDVHVI, for short), which is composed of a nonlinear fractional evolution inclusion and a time-dependent mixed variational-hemivariational inequality in the framework of Banach spaces. The objective of this paper is four fold. The first one is to investigate the nonemptiness as well as the compactness of the mild solutions set to the FDVHVI. The second aim is to study the optimal control problems described by the FDVHVI. The third goal is to establish the well-posedness results of the FDVHVI, including the existence, uniqueness, and stability. Furthermore, the sensitivity analysis of a perturbed problem associated to the FDVHVI with respect to the initial state and the two parameters is also obtained. Finally, a comprehensive fractional model is given to illustrate the validity of our main results.