It is known that there is no non-constant periodic solutions on a closed bounded interval for differential equations with fractional order. Therefore, many researchers investigate the existence of asymptotically periodic solution for differential equations with fractional order. In this paper, we demonstrate the existence and uniqueness of the $S$-asymptotically $\omega$-periodic mild solution to non-instantaneous impulsive semilinear differential equations of order $1<\alpha <2$, and its linear part is an infinitesimal generator of a strongly continuous cosine family of bounded linear operators. In addition, we consider the case of differential inclusion. Examples are given to illustrate the applicability of our results.