New discussion on the approximate controllability of Sobolev-type Hilfer fractional stochastic mixed Volterra-Fredholm integrodifferential inclusions of order 1 < μ < 2

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. The primary focus of this article is to explore the existence and controllability of a new class of Sobolev-type Hilfer fractional stochastic Volterra-Fredholm integrodifferential systems in Hilbert spaces. Our main findings are grounded in ideas from fractional calculus, stochastic analysis, cosine families, and multivalued maps. To begin, we use Bohnenblust-Karlin’s fixed point theorem to examine the existence of mild solutions for the given problem. Furthermore, we introduce a novel set of sufficient conditions for the considered nonlinear Hilfer fractional stochastic differential systems, and assuming that the corresponding linear system is approximately controllable. Finally, an example is provided to illustrate the proposed theory.