Existence and approximate controllability of Hilfer fractional impulsive evolution equations

Our main concern is the existence of a new \(PC_{2-v}\)-mild solution for Hilfer fractional impulsive evolution equations of order \(\alpha \in (1,2)\) and \(\beta \in [0,1]\) as well as the approximate controllability problem in Banach spaces. Firstly, under the condition that the operator A is the infinitesimal generator of a cosine family, we give a new representation of \(PC_{2-v}\)-mild solution for the objective equations by the Laplace transform and probability density function. Secondly, we rely on the Banach contraction mapping principle to discuss a new existence and uniqueness result of \(PC_{2-v}\)-mild solution when the sine family is compact. Thirdly, a sufficient condition for the approximate controllability result of impulsive evolution equations is formulated and proved under the assumptions that the nonlinear item is uniformly bounded and the corresponding fractional linear system is approximately controllable. Finally, two examples are given to illustrate the validity of the obtained results in the application.