Qualitative analysis of the nonlinear Ψ-Hilfer fractional neutral-type delayed integro-differential stochastic system: existence, uniqueness and controllability

This research work primarily aims to establish the sufficient conditions for the existence and uniqueness of the mild solution together with approximate controllability results for a class of the nonlinear Ψ-Hilfer fractional neutral-type integro-differential stochastic system with infinite delay in a separable Hilbert space. As a unified extension of the Caputo and Riemann–Liouville fractional derivatives and their Ψ-versions, the Ψ-Hilfer fractional derivative offers a more flexible tool for modelling. Its adaptable kernel allows more accurate descriptions of complex dynamical systems and effectively captures both local and nonlocal memory effects. First, the proposed control problem is transferred into an equivalent fixed point problem by implementing the Ψ-Riemann–Liouville fractional integral operator. The existence of the mild solution is established using the Krasnoselskii's fixed point theorem. Subsequently, the uniqueness of the mild solution is studied with the help of the Banach contraction principle. Further, the approximate controllability of the proposed control system is established under the consideration that the corresponding linear system is approximate controllable. The results are obtained using the concepts of fractional calculus, the general theory of stochastic analysis, semigroup theory of bounded linear operators and fixed point techniques. At the end, a concrete example is provided to validate the abstract results.