Approximate Controllability of Fractional Hemivariational Inequalities in Banach Spaces

In this paper, we derive the approximate controllability of fractional evolution hemivariational inequalities in reflexive Banach spaces involving Caputo fractional derivatives. We first show that the original problem is connected with a fractional differential inclusion problem involving the Clarke subdifferential operator, and then we prove the approximate controllability of the original problem via the approximate controllability of the connected fractional differential inclusion problem. In the meantime, we face the issue of convexity in the multivalued fixed point map due to the nonlinear nature of the duality map in reflexive Banach spaces involved in the expression of the control. We resolve this convexity issue and deduce main result. We also justify the abstract finding of this paper through an example.