Approximate controllability of Atangana-Baleanu fractional neutral delay integrodifferential stochastic systems with nonlocal conditions^☆

The study of approximate controllability results ensures the essential conditions required for a solution. Keeping the importance of the study, we initiate the existence and approximate controllability an Atangana-Baleanu-Caputo (ABC)-fractional order neutral delay integrodifferential stochastic systems with nonlocal conditions. For this purpose, the suggested ABC-fractional order neutral integrodifferential stochastic system is transferred into an equivalent fixed point problem via an integral operator. The operator is then analyzed for boundedness, continuity and equicontinuity. Then Arzela-Ascolli theorem ensures the compact multivalued function of the operator and Martelli's fixed point theorem is utilized for the existence of the mild solution. Based on the fixed point theorem and Dunford-Pettis theorem joined with ζ-resolvent operators, we develop the approximate controllability results. Finally, an example is given to justify the theoretical results. The achieved results reveal that the proposed method is systematic and suitable for dealing with the stochastic fractional problems arising in physics, technology, and engineering in terms of the ABC fractional derivative.