OSCILLATION AND NONOSCILLATION PROPERTIES FOR A KIND OF NONLINEAR NEUTRAL IMPULSIVE DELAY DIFFERENTIAL SYSTEMS

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. This paper is concerned with oscillation and nonoscillation of a kind of nonlinear neutral impulsive differential systems with constant coefficients and constant delays by using the pulsatile constant. The sufficient and necessary conditions for oscillation in the case δ ∈ R\{0} are obtained. Two examples are included using the main results.


INTRODUCTION
Consider a class of first-order nonlinear neutral delay differential equations of the form y(t) − δ y(t − τ) + β e γy(t−σ ) − 1 = 0, (1) where γ, τ > 0, σ ≥ 0 and β are real constants.Let τ k , k ∈ N with τ 1 < τ 2 < ... < τ k < ... and lim k→∞ τ k = +∞ are fixed moments of impulsive effect with the property max{τ k+1 −τ k } < +∞, and satisfying where δ ∈ R\{0}, α ∈ R are constants.For (2), ∆ is the difference operator defined by The main aim of this work is to study oscillation and nonoscillation properties governing the impulse operators acting on (1) which we denote as the impulsive systems Impulsive differential equations are now recognized as an excellent source of models to simulate processes and phenomena observed in theoritical physics, chemical technology, population dynamics, industrial robotic, economics, rhythmical beating, merging of solutions and noncontinuity of solutions.Moreover, the theory of impulsive differential equations is emerging as an important area of investigation, since it is much richer than the corresponding theory of differential equations without impulse effect.We mention the monographs ( [1], [14], [18]), and [19]), where the various properties of their solutions are studied.
In the present years much effort has been devoted to study the oscillatory and asymptotic behaviour of solutions of various classes of functional differential equations of neutral type (see for e.g [16], [17]).However, the impulsive differential equations of neutral type is not well studied.Hence in this work, the author have made an attempt to study the oscillation and nonoscillation properties of solutions of a class of nonlinear neutral first order impulsive differential systems of the form (E).
The motivation of the present work come from the works [25]- [27].In [25] and [27], Tripathy, Santra and Pinelas have studied first order neutral impulsive delay differential systems with variable coefficients of the form and established sufficient and necessary conditions for oscillation of all solutions of (E 1 ) for different ranges of δ (t).In this direction, we refer to the reader some of the related works [9]- [13].The objective of this paper is to study (E) and establish conditions for oscillation and nonoscillation of solutions of (E) subject to its associated characteristic equation under We may expect the possible solutions of (E) as where i(t 0 ,t) = k = number of impulses τ k , k ∈ N, and A = 0 is a real number which is called as the pulsatile constant.A close observation reveals that y(t) = C 1 e −µt is a possible solution of (1) when (E) is without impulses and y(n) = C 2 A n is a possible solution of (2) when i(t 0 ,t) = n and the impulses are the discrete values only (∵ in case (2), µ = 0).Therefore, (4) seems to be the possible choice of solution of (E).

A function y
and y(t) satisfies (E) for all sufficiently large t ≥ 0, where ρ = max{τ, σ } and PC R + , R is the set of all functions U : R + → R which are continuous for t ∈ R + , t = τ k , k ∈ N, continuous from the left-side for t ∈ R + , and have discontinuity of the first kind at the A solution y(t) of (E) is said to be regular, if it is defined on some interval [T y , +∞) ⊂ [t 0 , +∞) and sup{|y(t)| : t ≥ T y } > 0 for every T y ≥ T .A regular solution y(t) of (E) is said to be eventually positive (eventually negative), if there exists t 1 > 0 such that y(t) > 0 (y(t) < 0) for t ≥ t 1 .
A regular nontrivial solution y(t) of (E) is said to be nonoscillatory, if there exists a point t 0 ≥ 0 such that y(t) has a constant sign for t ≥ t 0 ; otherwise, the regular solution y(t) is said to be oscillatory.

OSCILLATION AND NONOSCILLATION PROPERTIES
In this section, we study the oscillatory and nonoscillatory behaviour of solutions of (E) through its associated characteristic equation provided ( 3) and ( 4) holds.
Throughout the discussion, we assume that i(t − σ ,t) = n 1 > 0 and i(t − τ,t) = n 2 > 0 are the number of impulses between t − σ and t, and t − τ and t respectively.Theorem 2.1.Let τ > σ > 0, δ ∈ R\{0} and α = 0 = β .Then the system (E) admits an (i) oscillatory solution in the exponential impulsive form (4) if and only if the algebraic equation has at least one real root µ with µ > β α for αβ > 0 or µ < β α for αβ < 0; (ii) eventually positive solution in the form of (4) if and only if the algebraic equation (5) has at least one real root µ with Proof.
(i) Let y(t) be a regular nontrivial solution of the system (E) such that y(t) = e −µt A i(t 0 ,t) , t > t 0 > ρ.Due to (3), (1) becomes that is, due to (6).Once again we use ( 4) in ( 2) to obtain a relation of the form We may note that i(t 0 , τ k + 0) − i(t 0 , τ k − 0) = 1.Hence, the last inequality becomes Using the fact we obtain from (8) that If we choose A = 1 − α β µ, then it is easy to verify that (9) reduces to (7).Consequently, ( 7) is same as the algebraic equation (5).Moreover, (5) is the required characteristic equation for (E).Ultimately, if y(t) is an oscillatory solution of (E) with the pulsatile constant A = 1− α β µ < 0, where µ > β α for αβ > 0 or µ < β α for αβ < 0, then µ satisfies the characteristic equation (5).Conversely, consider the characteristic equation ( 5) and assume that µ = µ * is the real root of ( 5) with µ * > β α for αβ > 0 and µ * < β α for αβ < 0. Then (E) admits an oscillatory solution y(t) = e −µ * t A i(t 0 ,t) with the pulsatile The proof of (ii) follows from the proof of (i) and hence the details are omitted.This completes the proof of the theorem.(ii) eventually positive solution in the form of (4) if and only if the algebraic equation (10) has at least one real root µ with (ii) eventually positive solution in the form of (4) if and only if the algebraic equation (11) has at least one real root µ with Corollary 2.3.In Theorem 2.1, let α = β = 0. Then the system (E) admits an (i) oscillatory solution in the exponential impulsive form (4) if and only if the algebraic equation has at least one real root µ with µ > 1; (ii) eventually positive solution if and only if the algebraic equation ( 12) has at least one real root µ with µ < 1.
Remark 2.1.Following to Corollary 2.3, we may note that µ = 1 if and only if A = 0, that is, (E) has the trivial solution.