Approximate controllability for a class of fractional semilinear system with instantaneous and non-instantaneous impulses

This paper is mainly concerned with the existence of mild solutions and approximate controllability for a class of fractional semilinear systems with instantaneous and non-instantaneous impulses. By applying the Kuratowski measure of noncompactness and $ \rho $-set contractive fixed-point theorem, the results for the considered system were obtained. In the end, an example was studied to support the main results.