Existence, uniqueness and controllability results of nonlinear neutral implicit $ \mathcal{ABC} $ fractional integro-differential equations with delay and impulses - Sivaranjani Ramasamy; Thangavelu Senthilprabu; Kulandhaivel Karthikeyan; Palanisamy Geetha; Saowaluck Chasreechai; Thanin Sitthiwirattham (AIMS Mathematics)
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 < α < 1 $ in the sense of $ \mathcal{ABC} $ 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$ \grave{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.