OSCILLATIONS IN SECOND ORDER LINEAR NEUTRAL IMPULSIVE DIFFERENTIAL EQUATIONS WITH CONSTANT DELAYS

Sufficient conditions are established for the oscillation of all solutions of a class of second order neutral differential equations with constant delay arguments and constant impulsive jumps for the cases where the coefficient p is a constant and when p is a variable. Examples are provided for clarity.

However, in spite of the large number of investigations of impulsive differential equations, their oscillation theory has not yet been fully elaborated, unlike the case of oscillation theory for delay differential equations. The monographs by Erbe etal, Gyori and Ladas, and Ladde etal [1,[2][3] contain excellent surveys of known results for delay and neutral delay differential equations.
Moreso, unlike oscillatory theory of first order neutral impulsive differential equations, not much has been done in the area of oscillations of second order neutral delay impulsive differential equations. This work, therefore, aims at establishing sufficient conditions for the oscillation of all solutions of a certain type of linear neutral impulsive differential equations of the second order with delay of the form (2) Second order differential equations in general, are most important in applications. Same also applies to neutral second order delay impulsive differential equations which have been developed to model impulsive problems in physics, population dynamics, biotechnology, pharmacokinetics, industrial robotics, and so forth. The introduction of oscillation and non-oscillation theory has further boosted the concept and particularly helped in identifying more areas of applications both within and outside differential equations. In particular, we use second order differential equations with impulse to understand the mathematical model for collision of viscoelastic bodies (see for e.g. [26]) and in impact theory in which an impact is an interaction of bodies which happens in a short period of time and can be considered as an impulse. Giving importance to such type of application, an attempt is made here to study the oscillation properties of (1) and (2).
A neutral delay impulsive differential equation of the second order is a differential system comprising a second-order differential equation and its impulsive conditions in which the highestorder derivative of the unknown function appears in the differential equation both with and without delays.

OSCILLATIONS IN IMPULSIVE DIFFERENTIAL EQUATIONS WITH CONSTANT DELAY
Now, the above definition becomes more meaningful if we define other related terms and concepts that will continue to be useful as we progress through the article.
It is known that the solution y( t ) for 0 t [ t ,T )  of a given impulsive differential equation or its first derivative y ( t )  is a piece-wise continuous function with points of discontinuity  (1) and (2) with initial function   ( ) (1) and (2) for all sufficiently large  t 0; ii) kN and satisfies equation (1); iii) and satisfy equation (2) (1) and (2) is said to be regular if it is defined in some half line As is customary, a nontrivial solution y( t ) of an impulsive differential equation is said to be finally positive (finally negative) if there exist T0  such that y( t ) is defined and is strictly positive (negative) for tT  ; non-oscillatory if it is either finally positive or finally negative, and oscillatory if it is neither finally positive nor finally negative [23,25].
An impulsive differential equation is said to be oscillatory if all its solutions are oscillatory.

STATEMENT OF THE PROBLEM
We now return to the linear neutral delay impulsive differential equation of the second order under investigation: where  =    and   This completes the proof of Lemma 2.1. . Assume that the second order impulsive ordinary differential equation

MAIN RESULTS
is oscillatory for some . Then every solution of equation (3) is oscillatory.

Consequently, for every
z t , for t t , t S t t z t z t , for t t , t S. t (9) By Lemma 2.1, inequality (5) is true. Combining inequalities (5) and (9), we obtain has a non-oscillatory solution. This contradicts our initial assumption and, thus, completes the proof of Theorem 3.1.

OSCILLATIONS IN IMPULSIVE DIFFERENTIAL EQUATIONS WITH CONSTANT DELAY
We now consider the case of the linear equation (3) with variable coefficient p as follows: Then every solution of equation (12) is oscillatory.
Proof: By contradiction, we assume that y( t ) is a finally positive solution of equation (12) and

y t p(t )y t
We see that z( t ) takes on non-negative values finally. From equation (10), we have that which implies that z ( t )  is a strictly decreasing function of t and so z( t ) is a strictly monotone function. From the above observations it follows that either tt lim z( t ) lim z ( t ) Let us assume that condition (14) holds. Integrating both sides of inequality (13) from 0 t to t and letting t → , we obtain Hence, and therefore 0 = . Finally, by conditions (14) and (15) It is easy to see that the assumptions of Theorem 3.2 are satisfied here. Therefore, every solution of equation (16)

CONCLUSION
It has become imperative in recent times to determine the properties of the solutions of certain mathematical equations from the knowledge of associated equations. In this work, we have made an effort to study the oscillation properties of a class of neutral delay differential equations with impulse of the form (3) by establishing a comparison theorem which compares the neutral impulsive differential equation (3) with the impulsive ordinary differential equation (6), in the sense that every oscillation criterion for the impulsive ordinary differential equation (6) becomes an oscillation criterion for the neutral impulsive differential equation (3). The formulated comparison theorem essentially simplifies the examination of the oscillatory properties of equation (3) and enables us also to eliminate some conditions imposed on the given problem.