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.

In this work, we develop certain coupled fixed-point results in Banach space by utilizing the Krasnosel'skii's expansive-type fixed-point results. Additionally, we establish existence and uniqueness conditions for coupled fixed points. Applying these results, we investigate the fractional-order HIV/AIDS model, demonstrating the existence of solutions. Our results contribute to the understanding of complex systems and non linear dynamics.