New Exploration on Approximate Controllability of Nonlinear Ψ $$ \Psi $$ -Hilfer Fractional Neutral-Type Delayed Integro-Differential Stochastic Inclusions

Abstract

The main objective of this paper is to investigate the sufficient conditions for the existence of mild solution and approximate controllability results for a class of nonlinear fractional neutral-type integro-differential stochastic inclusions with infinite delay in a separable Hilbert space. In the proposed control system, the $\Psi$-Hilfer fractional derivative has been considered, which has a quality to choose a suitable kernel function $\Psi$. First, by making use of the $\Psi$-Riemann–Liouville fractional integral operator, the proposed stochastic control problem is converted into an equivalent fixed point problem, and then, the Bohnenblust–Karlin fixed point theorem is applied to derive the existence of the mild solution. The approximate controllability result for the proposed control system has been established under the consideration that the corresponding linear system is approximate controllable. The theory of multivalued maps, fractional calculus, the concept of stochastic analysis and the fixed point technique have been used to bring off the main results. At the end, we provide a concrete example in order to validate the abstract findings.