<p>This article focuses on the study of well-posedness for the class of abstract integro-differential equations by applying the neutral to the system. Initially, it explores the existence and uniqueness of both the mild solution and strict solution. To achieve this, we outline a set of conditions necessary for establishing the existence of mild solution. Later the study utilizes semigroup theory, Schauder fixed-point method, and techniques from functional analysis to establish the existence, uniqueness, and regularity of solutions for the functions involved. Moreover, the results demonstrate the extension of local solutions to global solutions. The main findings show that a unique strict solution exists within the space which satisfy the differential equation directly while maintaining both Hölder’s and Lipschitz continuity. The assumptions related to condition and Fréchet differentiability ensure the boundedness and regularity of the associated operators. Finally, the paper includes an application to the partial neutral integro-differential equations with state-dependent delay, showcasing the practical implications of the theoretical findings.</p>