New results on the existence and approximate controllability of neutral-type Ψ-Caputo fractional delayed stochastic differential inclusions

This research aims to establish the sufficient conditions for the existence of the mild solution and approximate controllability for a class of $\Psi$-Caputo fractional neutral-type integro-differential stochastic inclusions with infinite delay in a separable Hilbert space. In the proposed stochastic control system, the $\Psi$-Caputo fractional derivative is considered, which has the flexibility to choose a suitable kernel function $\Psi$. Firstly, we derive the existence of the mild solution for the $\Psi$-Caputo fractional neutral-type delayed integro-differential stochastic system by using the Karlin fixed point approach. For this purpose, the $\Psi$-Caputo fractional neutral-type delayed integro-differential stochastic inclusions is transferred into an equivalent fixed point problem by implementing the $\Psi$-Riemann–Liouville integral operator, and then the Karlin fixed point theorem is applied. Further, the approximate controllability results of the proposed stochastic control system are established under the consideration that the corresponding linear system is approximate controllable. The set of sufficient conditions is established by using the concepts of fractional calculus, the general theory of stochastic analysis, fixed point technique, semigroup theory of bounded linear operators, and the theory of multivalued maps. At the end of the paper, a concrete example is provided to validate the abstract results.