Results on Existence and Uniqueness of Solution Of Impulsive Neutral Integro-Differential System

In this work, the solution of the impulsive neutral integro-differential system is analysed. The Du Bois–Reymond’s assumptions on solution variation of piecewise smooth functions are used to establish the existence of only the impulsive term as a solution of the system at the points of discontinuity. The theories of infinitesimal generator of a strongly continuous compact semigroup is used to formulate theorems on existence and uniqueness of system solution, and proves are provided using an approximate piecewise continuous, compact operator and a continuous positive non decreasing function ) , 0 ( ) , 0 [ :     . Results obtained are improvement on the qualitative analysis of impulsive neutral integro-differential system 2010 Mathematical Subject Classification: 74G30, 93D20, 93B05

. Results obtained are improvement on the qualitative analysis of impulsive neutral integro-differential system

Introduction
An impulsive neutral integro-differential equation is an equation which involves both the integral and derivatives of the unknown function, with time lag incorporated in both the state and derivative of the system (which describe the historical value of the rate of change and the state) and a coupled impulsive term showing the abrupt changes of the state at certain moments of time between intervals of continuous evolution.This equation has wide application in various evolutionary processes including population dynamics, aeronautics, economics and engineering.There have been increasing interests in the analysis of the qualitative properties of the impulsive neutral integro-differential equation, mostly on the theories of existence and uniqueness of the system solution (see ;Bainov, Myshkis & Zahariev, 1987;Bainov & Simeonov, 1985;Benchohra & Ntouyas, 2006;Haddad, Chellaboina & Nersesor, 2008;Hale & Kato, 1987;Igobi, Eni, Eteng & Atsu, 2011;Isaac & Lipscey, 2010;Jiang & Shen, 2011).Some researchers have employed the approximation of the impulsive neutral integro-differential equation as an integro-differential equation coupled with a difference equation to be satisfied at certain fixed or variable impulse times.The resulting solutions are thereby piecewise continuous (with discontinuity at the impulse times).This approach ensures that the well-established results for integro-differential equations are utilised to developed theories on existence and uniqueness of the system solution (Agawal & Saker, 2001;Ballinger, 1999;Benchohra & Ntouyas, 2006;Bao & Hou, 2010;Diop, Ezzinbe & Zene, 2015;Diop, Ezzinbe & Lo, 2012;Kavilla, Arjunan & Ravichandran, 2014;Li & N'Guerekata, 2010).
Alternative approach used by Halanay and Wexler (1968) and Pandit and Deo (1982) involves defining a measure differential equation (incorporating Dirac delta functions) where the derivative involved is a distributional derivative.The points at which impulses occur are fixed, generalized functions are considered and the resulting solutions are of bounded variation.The disadvantage of this approached is that most classical theory cannot be applied to these types of systems.
In this research, the first approach is employed to analyse the solution of the impulsive neutral integro-differential system in the Banach space ) (X .The Du Bois-Reymond's assumptions on solution variation of piecewise smooth functions are used to establish the impulsive term as the only solution term of the system at the points of discontinuity.The theories of infinitesimal generator of a strongly continuous compact semigroup is used to formulate theorems on existence and uniqueness of system solution, and proves are provided using an approximate piecewise continuous, compact operator and a continuous positive non decreasing function is called a matrix solution of the homogeneous linear system (2.1) if each of its columns is a vector solution.A matrix solution  is called the fundamental matrix solution of (2.1) if its columns form a fundamental set of solutions whose columns are linearly independent, and ) ( 0 t  is invertible.

Definition 2.1
Let A be an infinitesimal generator of a strongly continuous compact semigroup (.) R satisfying the Chapman-Kolmogorov identities i.
Then, there exists a strongly continuous exponentially bounded family where

Hypothesis 2.1
Assume the following Du Bois-Reymond's assumptions on solution variation of piecewise smooth function hold

Proof
Assume hypothesis (2.1) hold so that,  ,  ( ( satisfying definition (2.1), then the mild solution ) , ( Consequently, for each possible mild solution By assumption (3.0) of H 1 , there exists a function (3.12) H 5 .Let there exists Let hypothesis (3.0) and (3.1) hold such that for each bounded Then, system (2.1) has at least one solution.

Proof
Consider the operator and a close nonempty and convex set Then, for any x , for a sufficiently small  , since

Conclusion
The analysis of the solution of the impulsive neutral integro-differential system in the Banach space X was considered.
Theorem and prove on the existence of only the impulsive term as a solution of the system equations at the points of discontinuity was presented using the Du Bois-Reymond's assumptions on solution variation of piecewise smooth functions.Theorems on existence and uniqueness of the system solution were formulated using the rich theories of an infinitesimal generator A of a strongly continuous compact semigroup (.) R , and proves were provided using an approximate piecewise continuous, compact operator and a continuous positive non decreasing function

.
Results obtained are improvement on the qualitative analysis of impulsive neutral integro-differential system t