Solvability and controllability of Sobolev-type nonlocal Hilfer fractional stochastic evolution inclusions of order 1<μ<2: Solvability and controllability of Sobolev-type nonlocal Hilfer fractional stochastic

This article establishes the existence and controllability results for Sobolev-type higher-order Hilfer fractional stochastic differential inclusions with non-local conditions in Hilbert spaces. The primary contributions are grounded in fractional calculus, two multi-valued functions, stochastic analysis, non-local conditions, and fixed-point methods. By applying the fixed-point theorem for multi-valued operators, we derive existence results for Sobolev-type non-local Hilfer fractional stochastic differential systems of order 1 < μ < 2 . Notably, we identify a new set of sufficient conditions for the approximate controllability of nonlinear Hilfer fractional stochastic differential inclusions, assuming that the associated linear system is approximately controllable. Finally, an example is provided to illustrate the obtained results.