The main aim of this research is to study the sufficient conditions for the existence of an integral-form mild solution and approximate controllability results for a new class of the nonlinear Ψ-Caputo fractional Sobolev-type delayed stochastic multivalued system with extended impulsive dynamics in a separable Hilbert space. The Ψ-Caputo fractional derivative offers a versatile and unifying framework that extends the classical fractional differential operators through an appropriate choice of the kernel function Ψ. This adaptability enables more accurate modelling of memory and hereditary characteristics inherent in complex dynamical systems. Firstly, the proposed control system is transferred into an equivalent fixed point problem using the Ψ-Riemann-Liouville fractional integral operator. Then, the Karlin fixed point theorem is implemented to establish the existence of mild solution. Furthermore, the approximate controllability result of the proposed control system is investigated under the consideration that the corresponding linear system is approximate controllable. The main results are derived using fractional calculus, theory of multivalued map, the concepts of stochastic analysis, and fixed point approach. At the end of the paper, a concrete example is provided to illustrate the theoretical findings.
Sharma et al. (Wed,) studied this question.