New exploration on the nonlinear fractional neutral-type integro-differential delayed system with nonlocal conditions: Existence, uniqueness and controllability

This study primarily focuses on establishing the sufficient conditions for the existence and uniqueness of the mild solution along with the approximate and trajectory controllability results for a new class of the nonlinear $ \Psi $-Caputo fractional neutral-type integro-differential system with finite delay and nonlocal conditions in a Hilbert space. A key advantage of the $ \Psi $-Caputo fractional derivative is that it allows to choose a suitable kernel function $ \Psi $. First, we derive the existence of the mild solution for the proposed control system by using a fixed point approach. For this purpose, the proposed control system is transformed into an equivalent fixed point problem using the $ \Psi $-Riemann-Liouville fractional integral operator. Then, the existence of the mild solution is established by Schauder's fixed point theorem. Then, the uniqueness of the mild solution is studied with the help of the Banach contraction principle. Moreover, the approximate controllability result of the proposed control system is established under the consideration that the corresponding linear system is approximate controllable. Further, the trajectory controllability result is studied by using the Grönwall's inequality. The set of sufficient conditions is derived by using the concepts of fractional calculus, Laplace transform, fixed point techniques, and semigroup theory of bounded linear operators. Finally, an illustrative example is presented to validate and demonstrate the applicability of the theoretical results.