Approximate Controllability of Fractional Hemivariational Inequalities in Banach Spaces

TL;DR

This paper establishes the approximate controllability of fractional hemivariational inequalities in Banach spaces, linking it to fractional differential inclusions and overcoming convexity challenges with a novel approach.

Contribution

It introduces a new method to prove approximate controllability of fractional hemivariational inequalities in Banach spaces, addressing convexity issues in multivalued maps.

Findings

Proved approximate controllability of fractional hemivariational inequalities.

Connected controllability to fractional differential inclusion problems.

Resolved convexity issues in the control map.

Abstract

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.