Existence and approximate controllability for the Hilfer fractional neutral stochastic hemivariational inequality with Rosenblatt process

The approximate controllability of the Hilfer fractional neutral stochastic system with Rosenblatt process is discussed in this article along with the hemivariational inequalities. Initially, we demonstrate the existence of mild solution using fractional calculus, semigroup theory, stochastic analysis, Hilfer fractional derivatives, multivalued maps, and the fixed point theorem. Moreover, we prove the conditions under which the Hilfer fractional system is approximately controllable. Finally, we provide our key findings using an example.Abbreviation: FDEs: Fractional differential equations; R–L: Riemann–Liouville; HF: Hilfer fractional; HFD: Hilfer fractional derivative; SDEs: Stochastic differential equations; HVI: Hemivariational Inequality; HFNSHVI: Hilfer fractional neutral stochastic hemi-variational inequality