Approximate controllability of a multi-valued fractional impulsive stochastic partial integro-differential equation with infinite delay

In this paper, the approximate controllability of a new class of multi-valued fractional impulsive stochastic partial integro-differential equations with infinite delay in Hilbert spaces is studied. Firstly, by using stochastic analysis, analytic α-resolvent operator, fractional powers of closed operators and a fixed point theorem for multi-valued maps, we prove an existence result of mild solutions for the control systems under the mixed Lipschitz and Carathéodory conditions. Secondly, we discuss a new set of sufficient conditions for the approximate controllability of the systems. The results are obtained under the assumption that the associated linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory.