Approximate Controllability of Neutral Integro-Differential Inclusions Associated with Hemivariational Inequalities in Banach Spaces with Uniformly Convex Dual

Abstract

In this paper, we investigate the approximate controllability of neutral integro-differential inclusions driven by hemivariational inequalities with nonlocal inclusion conditions in a Banach space $\mathbb{V}$ whose dual ${\mathbb{V}}^{\ast }$ is uniformly convex, using resolvent operator theory. We first establish an equivalence between the original problem and an associated differential inclusion involving the Clarke subdifferential. Under suitable assumptions, we then prove the existence of mild solutions corresponding to admissible controls in ${\mathrm{L}}^2(I,\mathbb{U})$. A key contribution of this work lies in the analysis of the control construction in Banach spaces, where the duality mapping $\mathcal{J}=\partial (\frac{1}{2}\Vert \cdot {\Vert }_{\mathbb{V}}^2)$ is generally nonlinear, in contrast to the Hilbert space setting, while such nonlinearity has been viewed as a technical obstruction in earlier studies. Based on this framework, we prove approximate controllability of the original system by relating it to the controllability of the associated linearized inclusion and constructing appropriate controls involving the duality mapping. Finally, we present applications to illustrate the effectiveness of the proposed approach.