An identification problem for a second-order abstract neutral functional differential equation with impulses

Abstract This manuscript investigates the identification problem for a second-order abstract neutral functional differential equation incorporating both instantaneous and non-instantaneous impulses in a Banach space. We establish distinct results for two fundamental cases based on the problem’s singularity. For the non-singular case, we prove the existence and uniqueness of a mild solution using perturbation theory, the properties of strongly continuous cosine families, and duality in functional analysis. For the singular case, we propose an iterative variable-replacement framework that successively modifies the original problem until a non-singular form is obtained, enabling the derivation of a unique mild solution via a cyclic vector. Our theoretical findings are validated with a concrete example.