Existence, uniqueness and controllability results of nonlinear neutral implicit $\mathcal{A}\mathcal{B}\mathcal{C}$ fractional integro-differential equations with delay and impulses

In this article, the necessary and sufficient conditions for the existence and uniqueness of the mild solutions for nonlinear neutral implicit integro-differential equations of non-integer order $0<\alpha <1$ in the sense of $\mathcal{A}\mathcal{B}\mathcal{C}$ derivative with impulses, delay, and integro initial conditions were established. The existence results were derived using the semi-group theory, measures of non-compactness, and the fixed-point theory in the sense of Arzel $\stackrel{`}{a}$–Ascoli theorem and Schauder's fixed-point theorem. We analyzed the controllability results of the proposed problem by incorporating the ideas of semi-group theory and fixed-point techniques. The Banach contraction principle was used to derive the uniqueness and controllability of the proposed problem. We provide an example to support the theoretical results.