Research Article Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation D. K. Igobi and U. Abasiekwere Department of Mathematics and Statistics, University of Uyo, P.M.B. 1017, Nigeria Correspondence should be addressed to D. K. Igobi; dodiigobi@gmail.com Received 5 January 2019; Accepted 5 March 2019; Published 20 March 2019 Academic Editor: Xiaodi Li Copyright © 2019 D. K. Igobi and U. Abasiekwere. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term. Te fundamental matrix theorem is employed to derive the integral equivalent of the equation which is Lebesgue integrable. Te integral equivalent equation with impulses satisfying the Carath´ eodory and Lipschitz conditions is embedded in the space of generalized ordinary diferential equations (GODEs), and the correspondence between the generalized ordinary diferential equation and the nonhomogeneous retarded equation coupled with impulsive term is established by the construction of a local fow by means of a topological dynamic satisfying certain technical conditions. Te uniqueness of the equation solution is proved. Te results obtained follow the primitive Riemann concept of integration from a simple understanding. 1. Introduction Te dynamic of an evolving system is most ofen subjected to abrupt changes such as shocks, harvesting, and natural disasters. When the efects of these abrupt changes are trivial the classical diferential equation is most suitable for the modeling of the system. But for short-term perturbation that acts in the form of impulses, the impulsive delay diferential equation becomes handy. An impulsive retarded diferential equation is a delay equation coupled with a diference equation known as the impulsive term. Among the earliest research work on impulsive diferential equation was the arti- cle by Milman and Myshkis [1]. Tereafer, growing research interest in the qualitative analysis of the properties of the impulsive retarded equation increases, as seen in the works of Igobi and Ndiyo [2], Isaac and Lipcsey [3], Benchohra and Ntouyas [4], Federson and Schwabik [5], Federson and Taboas [6], Argawal and Saker [7], and Ballinger [8]. Te introduction of the generalized ordinary diferential equation in the Banach space function by Kurzweil [9] has become a valuable mathematical tool for the investigation of the qualitative properties of continuous and discrete systems from common sense. Te topological dynamic of the Kurzweil equation considers the limit point of the translate   → (, + ) under the assumptions that the limiting equation satisfying the Lipschitz and Carath´ eodory conditions is not an ordinary diferential equation, and the space of the ordinary equation is not complete. But if the ordinary diferential equation is embedded in the Kurzweil equations we obtained a complete and compact space, such that the techniques of the topological translate can be applied. A more relaxed Kurzweil condition was presented in the article by Artstein [10]. He considered the metric topology characterized by the convergence   →  0  ∫  0   (, )  → ∫  0   (, ) , (1) with the following properties: (i) ⊂ is a compact set; then there exists a locally Lebesgue integrable function   () such that      (, )     ≤  () , ∈, (2) Hindawi International Journal of Differential Equations Volume 2019, Article ID 2523615, 9 pages https://doi.org/10.1155/2019/2523615