The main motivation of our conversation is the existence and approximate controllability for fractional stochastic Volterra–Fredholm integro-differential systems having order 1 < r < 2. The primary outcomes are obtained by applying concepts and ideas from fractional calculus, multivalued maps, the theory of cosine family, Martelli and Dhage, and Leray–Schauder fixed point techniques. We begin by emphasizing the existence, and then demonstrate the approximate controllability of the considered system. Additionally, we determine the approximate controllability outcomes for the system with infinite delay. At last, an application is established for drawing the theoretical conclusions of primary outcomes.

中文翻译：

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1 < r < 2 阶分数随机 Volterra-Fredholm 积分-微分系统的可控性讨论

我们谈话的主要动机是阶数为 1 < 的分数随机 Volterra-Fredholm 积分-微分系统的存在性和近似可控性r< 2. 主要结果是通过应用分数微积分、多值映射、余弦族理论、Martelli 和 Dhage 以及 Leray-Schauder 不动点技术的概念和思想获得的。我们首先强调存在性，然后证明所考虑系统的近似可控性。此外，我们确定了具有无限延迟的系统的近似可控性结果。最后，建立了用于得出主要结果的理论结论的应用程序。