Abstract
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.
Highlights
The dynamic of an evolving system is most often subjected to abrupt changes such as shocks, harvesting, and natural disasters
We consider an initial value problem of a nonhomogeneous retarded functional equation coupled with the impulsive term
The integral equivalent equation with impulses satisfying the Caratheodory 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
Summary
The dynamic of an evolving system is most often subjected to abrupt changes such as shocks, harvesting, and natural disasters. 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. 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
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