Qualitative results on nonlinear neutral-type fractional delayed-differential system: existence, uniqueness and controllability

The main concern of this work is to study the sufficient conditions for existence, uniqueness, and approximate controllability results for the nonlinear Caputo conformable fractional neutral-type delayed integro-differential system with nonlocal conditions in a Hilbert space. Since, the conformable derivative retains several fundamental properties of classical calculus including mean value theorem, Rolle's theorem, product, quotient, and linearity rules. This distinguishes it from traditional fractional derivatives such as Riemann-Liouville, Caputo, and Hilfer. Therefore, the conformable derivative is simpler and faster but ignores history while the Caputo conformable fractional derivative offers a balance capturing some memory with easier calculations. Firstly, the proposed system is reformulated into an equivalent fixed point problem implementing the Riemann-Liouville conformable fractional integral operator. The Schauder fixed point theorem is used to derive the existence of mild solution. The Banach contraction principle is then applied to show the uniqueness of mild solution. The main tools applied to derive the results are theory of fractional calculus, semigroup of bounded linear operators, and fixed point theorems. Further, the approximate controllability result for the proposed system is established under the consideration that the corresponding linear system is approximate controllable. An illustrative example is presented to demonstrate the applicability of theoretical results.