Results on Uniqueness of Solution of Nonhomogeneous Impulsive Retarded Equation Using the Generalized Ordinary Differential Equation

In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term.The fundamental matrix theorem is employed to derive the integral equivalent of the equation which is Lebesgue integrable. The integral equivalent equation with impulses satisfying the Carathéodory and Lipschitz conditions is embedded in the space of generalized ordinary differential equations (GODEs), and the correspondence between the generalized ordinary differential equation and the nonhomogeneous retarded equation coupled with impulsive term is established by the construction of a local flow by means of a topological dynamic satisfying certain technical conditions. The uniqueness of the equation solution is proved. The results obtained follow the primitive Riemann concept of integration from a simple understanding.


Introduction
The dynamic of an evolving system is most often subjected to abrupt changes such as shocks, harvesting, and natural disasters.When the effects of these abrupt changes are trivial the classical differential 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 differential equation becomes handy.An impulsive retarded differential equation is a delay equation coupled with a difference equation known as the impulsive term.Among the earliest research work on impulsive differential equation was the article by Milman and Myshkis [1].Thereafter, 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].
The introduction of the generalized ordinary differential 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.The topological dynamic of the Kurzweil equation considers the limit point of the translate   → (,  + ) under the assumptions that the limiting equation satisfying the Lipschitz and Carathéodory conditions is not an ordinary differential equation, and the space of the ordinary equation is not complete.But if the ordinary differential 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: for all  and   fixed The metric convergence of   fulfilling (i) and (ii) for some   and   guarantees the continuity of (, ) →   (, ) and the precompactness of the function space, but not completeness.That is, by equation ( 1), the Cauchy sequence   implies ∫  0 (, ) converges for all (, ).However, does not have the integral representation In summary, the Kurzweil equation addresses functions whose limit exists but are nowhere differentiable, but, by using the primitive definition of Riemann integral, a correspondence is established.Consider an ordinary equation with integral equivalent Suppose the integral is a Riemann integral, we can define a −fine partition If we defined The differential equation resulting from the primitive Kurzweil integral (11) is what is known as the generalized ordinary differential equation.
The correspondence between the generalized ordinary differential equation and other types of differential system is well established in the following articles: Federson and Taboas [6], Federson and Schwabik [5], Imaz and Vorel [11], Oliva and Vorel [12], and Schwabik [13].This was made possible by embedding the ordinary differential equation in the space of the generalized ordinary differential equation and constructing a local flow by means of a topological dynamic satisfying certain technical conditions.
In this work, we consider an initial value problem of a nonhomogeneous retarded functional equation of the form coupled with impulses Δ (  ) =   ( (  )) ,  = 1, 2, . . ., (14) where  0 × ,  We will employ the fundamental matrix theorem to derive the integral equivalent of Equation ( 13) and define Lebesgue integrable functions (() We embed the integral equivalent equation with impulses satisfying conditions (A) and (B) in the space of generalized ordinary differential equations (GODEs), and using similar argument as presented by Federson and Taboas [6] and Federson and Schwabik [5] to show the relationship between the solutions of the generalized ordinary differential equation and the equivalent impulsive retarded differential equation, and establish the uniqueness of the equation solution.
International Journal of Differential Equations 3

Generalized Ordinary Differential Equation
Let  be a Banach space and () a Banach space of bounded linear operators on , with ‖.‖  and ‖.‖ () defining the topological norms in  and (), respectively.A partition is any finite set  = { 0 ,  1 , . . .,  +1 } such that  =  0 <  1 < ⋅ ⋅ ⋅ <  +1 = .Given any finite step function () : [, ] → (), for () being a constant on we denote the set of all regulated functions () : [, ] → (), which is a Banach space when endowed with the usual supremum norm The Kurzweil integral is related to the Riemann integral when the space  is the set of real numbers such that  : [, ] →  and the Riemann sum is defined as for all  = [, ].The properties of the Kurzweil integral such as the linearity, additivity, and convergence with respect to the nearby interval have been extensively discussed in Artstein [10], Schwabik [13], and Federson and Schwabik [5].
We state here some of the fundamental results of the Kurzweil integral on a subinterval as proved in Kurzweil [9] and Artstein [10] which are the basic concepts to be employed in this work.21) abounds in Schwabik [13], Schwabik, Tvrdy, and Vejvoda [14], and Artstein [10].

Preliminary Results
In this section, we present results that are fundamental to the establishment of the main results in Section 4.

Conclusion
An initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term was considered.The integral equivalent of the equation which is Lebesgue integrable was obtained using the fundamental matrix theorem.The integral equivalent equation with impulses satisfying the Carathéodory and Lipschitz conditions was embedded in the space of generalized ordinary differential equations (GODEs) and, using similar argument as presented in Federson and Taboas [6] and Federson and Schwabik [5], we showed the relationship between the generalized ordinary differential equation and the nonhomogeneous retarded functional equation coupled with impulsive term by the construction of a local flow using topological dynamic satisfying certain technical conditions.The uniqueness of the equation solution was proved.