In this paper, we consider an abstract system which consists of a hemivariational inequality of parabolic type combined with a nonlinear differential inclusion (DPHVI, for short) in the framework of Banach spaces. The objective of this paper is fourfold. The first one is to deal with the existence of solutions and the properties of the solution set for parabolic hemivariational inequalities (PHVIs, for short). The second aim is to investigate the existence of mild solutions for DPHVI by means of a fixed point technique. The third target is to study the existence of a global attractor for the $m$-semiflow governed by DPHVI. Finally, the fourth goal is to illustrate an application of our abstract results.

In this paper, we consider fractional differential hemivariational inequalities (FDHVIs, for short) in the framework of Banach spaces. The aim of this paper is three folds. The first one is to investigate the existence of mild solutions for FDHVIs and by means of a fixed point technique we are able to avoid the hypothesis of compactness on the semigroup. The second aim is to study the existence of decay mild solutions for FDHVI via giving asymptotic behavior of Mittag-Leffler function. Finally, a mathematical model is provided to illustrate our abstract results.