Approximate controllability of impulsive fractional neutral hemivariational inequality with infinite delay via an approximating technique

In this paper, we investigate a class of impulsive fractional neutral hemivariational inequality with infinite delay in Banach spaces. First, we give a proper definition of the resolvent operator in Banach space, and define the concept of a mild solution by using fractional calculus, the semigroup of operators theory, and properties of generalized Clarke subdifferential. Second, an optimal control problem is constructed for the corresponding linear control system, and the optimal control expression is derived based on the resolvent operator consisting of duality mapping. Then, under the assumption that the impulsive function is only continuous, we prove the existence of mild solutions and approximate controllability results by using an approximating technique, the fractional power operator, and the fixed point theorem of condensing multivalued maps. Finally, an example is given to verify the validity of the main results.