Approximate Controllability and Existence Results of the Sobolev-Type Fractional Stochastic Differential Equation Driven by a Fractional Brownian Motion

In this article, we investigate the existence and approximate controllability of a class of Sobolev-type fractional stochastic differential equations of order 1<δ<2 with infinite delay. The analysis is carried out in an abstract Hilbert space framework, incorporating fractional dynamics together with stochastic perturbations. By employing techniques from fractional calculus, semigroup theory, and fixed point theory, particularly the Banach contraction principle along with compactness arguments, we establish the existence of mild solutions for the proposed system. Subsequently, sufficient conditions for approximate controllability are derived by combining operator-theoretic methods with stochastic analysis. The novelty of this work lies in extending controllability results to Sobolev-type fractional stochastic systems of order 1<δ<2, where both the higher-order fractional structure and stochastic effects are treated simultaneously within a unified framework. This generalizes and complements several existing results in the literature that mainly address deterministic systems or fractional differential equations of order 0<δ≤1. Finally, an illustrative example is presented to demonstrate the applicability and effectiveness of the theoretical findings.