A discussion on the approximate controllability of Hilfer fractional neutral stochastic integro-differential systems.

• This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. • The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. • Initially, we discuss the approximate controllability of the fractional evolution system. • Then, we extend our results to the concept of nonlocal conditions. • Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion. This manuscript is mainly focusing on the approximate controllability of Hilfer fractional neutral stochastic integro-differential equations. The principal results of this article are proved based on the theoretical concepts related to the fractional calculus and Schauder's fixed-point theorem. Initially, we discuss the approximate controllability of the fractional evolution system. Then, we extend our results to the concept of nonlocal conditions. Finally, we provide theoretical and practical applications to assist in the effectiveness of the discussion. [ABSTRACT FROM AUTHOR]