Nonoscillatory and Oscillatory Criteria for First Order Nonlinear Neutral Impulsive Di ff erential Equations

Received: January 11, 2011 Accepted: January 26, 2011 doi:10.5539/jmr.v3n2p52 Abstract A survey of recent studies in neutral impulsive differential equations reveals that most of such works revolve around the quest for oscillatory conditions for linear impulsive differential equations. The development of oscillatory and nonoscillatory criteria for nonlinear impulsive differential equations has so far attracted very little attention. In this paper, we obtain sufficient conditions for the existence of oscillatory and nonoscillatory solutions for nonlinear first order neutral impulsive differential equations with constant delays.


Introduction
Research about oscillations for linear neutral impulsive differential equations with or without delays has enjoyed unprecedented patronage in recent times (Isaac and Lipcsey, 2009b, c;Isaac and Lipcsey, 2010a, b;Graef et al, 2002;Graef et al, 2004;El-Morshedy and Gopalsamy, 2000;Luo et al, 2000;Giang and Gyori, 1993).Unfortunately, there appear to be limited investigations about oscillations for nonlinear neutral impulsive differential equations which underline the foundation of modern applications.Even the limited studies are mainly concerned with linearization techniques (Isaac and Lipcsey, 2009a;Berezansky and Braverman, 1996).Worse still, the concept of nonoscillations for nonlinear neutral impulsive equations presently suffers almost complete neglect.In this study, we make a deliberate attempt to clear these obstacles and extend the concepts beyond the existing boundaries.
We begin with the discussion on the existence of nonoscillatory solutions for first order nonlinear neutral impulsive differential equations for t ≥ t 0 > 0 and k : t k ≥ t 0 > 0 identifying some essential sufficient criteria.Next, we study the oscillations of the nonlinear neutral impulsive equation obtaining some new conditions for all solutions of Equation (1.2) to be oscillatory.
Our conditions are "sharp" in the following sense.If Equations (1.1) and (1.2) are linear with constant coefficients, the conditions become both necessary and sufficient.In what follows, we recall some of the basic notions and definitions that will be of importance as we advance through the article.
The solution y(t) for t ∈ [t 0 , T ) of a given impulsive differential equation or its first derivative y (t) is a piece-wise continuous function with points of discontinuity t k ∈ [t 0 , T ), t k t, 0 ≤ k < ∞.Consequently, in order to simplify the statements of our assertions later, we introduce the set of functions PC and PC r which are defined as follows: Let r ∈ N, D := [T, ∞) ⊂ R and let the set S := {t k } k=N be fixed.Except stated otherwise, we will assume that the elements of S are moments of impulse effect and satisfy the property: We denote by PC (D, R) the set of all functions ϕ : D → R, which are continuous for all t ∈ D, t S .They are continuous from the left and have discontinuity of the first kind at the points for which t ∈ S , while by PC r (D, R), we denote the set of functions ϕ : (Bainov and Simeonov, 1998).To specify the points of discontinuity of functions belonging to PC or PC r , we shall sometimes use the symbols PC (D, R; S ) and PC r (D, R; S ), r ∈ N.

Definition 1.1
A solution y(t) of Equation (1.1) or (1.2) is said to be (i) Finally positive (finally negative) if there exists T ≥ 0 such that y(t) is defined and is strictly positive (strictly negative) for t ≥ T ; (ii) Oscillatory, if it is neither finally positive nor finally negative; and (iii) Nonoscillatory, if it is either finally positive or finally negative (Bainov and Simeonov, 1998;Isaac and Lipcsey, 2010b).

The Existence of Nonoscillatory Solutions
We return to Equation (1.1) and introduce Conditions C2.1 -C2.4: , k ∈ Z for all sufficiently large t and there exit p i (t) ≥ a 0 > 0 and p ik ≥ a 0k > 0, for some i ∈ I m , k ∈ Z.
where h 0 is a constant and there exists another constant for all sufficiently large t.Then Equation (1.1) has a nonoscillatory solution which converges to zero as t → ∞.
Proof.We return to the family of quasi-equicontinuous functions F and set Let us denote by A B , all bounded piece-wise continuous functions in PC([t 0 , T )) and define a norm in A B as follows: Endowed with this norm, A B is a Banach space and F is a bounded convex closed set in A B .
We define a mapping ϕ as follows: (ϕy where T is sufficiently large.Precisely, Clearly by virtue of the proposed value of T above, Inequality (2.3) holds and (2.5) At this point, we need to prove the following facts: (a) ϕF ⊂ F ; (b) If lim j→∞ y j − y = 0, then y ∈ F , where y j ∈ F is a sequence; (c) ϕF is relatively compact.
Let us now examine their verification one after the other.
(a) For t ≥ T and k: t k ≥ T , we obtain, for y ∈ F , (ϕy The first inequality is due to Equation (2.4) and the definition of F and the last inequality is because of Inequality (2.3).At the same time, using analogous reasoning, we obtain Consequently, Expressions (2.4) and (2.6) imply that (ϕy)(t) ∈ PC([t 0 , T )) and For t ≥ t ≥ T and k: where the first inequality is given using the Triangle inequality.The following is based on the definition of F and the Mean Value Theorem.The next step is due to Equation (2.2) and the last step from Equation (2.5).Additionally, for t 0 ≤ t ≤ t ≤ T , the Mean Value Theorem can be applied to Equation (2.4) leading to the result Thus, (b) By definition, ϕ is a piece-wise continuous mapping.Assume the existence of a sequence y j ∈ F such that lim j→∞ y j (t) − y(t) = 0, (2.7) then y ∈ F .
Indeed, for t ≥ T and k such that t k ≥ T , where The first inequality is obtained from Equation (2.4) and the last steps are because of the definition of a norm in A B . Obviously, However, expresion Therefore, in view of Equation (2.7) and Lebesgues dominated convergence theorem, we can assert that (2.9) Whenever t 0 ≤ t ≤ T , the following condition holds: (2.10) The combination of Equations (2.9) and ( 2 (c) In this final stage, we show that ϕF is relatively compact.Obviously from the proofs of (a) and (b) above, ϕF is uniformly bounded and quasi-equicontinuous in [t 0 , T ).This implies that for each y ∈ F , where b 0 > 0 and Hence, for any arbitrarily pre-assigned small positive number ε, there exists a sufficiently large T > t 0 such that whenever (2.12) On the other hand, if we set δ = ε L and assume that |t − t | < δ, then for all t 0 ≤ t ≤ t ≤ T and k: (2.13) Thus, from Conditions (2.12) and (2.13), we can affirm that ϕF is quasi-equicontinuous in [t 0 , T ) and hence, ϕF is relatively compact.By virtue of Schauder Tikhonov Fixed Point Theorem, the mapping ϕ has a fixed point y * (t) ∈ F which is a nonoscillatory solution of Equation (1.1) and converges to zero when t → ∞.This completes the proof of Theorem 2.1.
Corollary 2.1 Assuming that the function p i (t) satisfies Conditions C2.2 and C2.3 as well as q j (t) ∈ PC(R + , R + ) and q jk ≥ 0, the following two conditions hold.If p i (t) ≤ p i , q j (t) ≤ q j and there exists a positive λ such that has a nonoscillatory solution which converges to zero as t → ∞.
Remark 2.1 When p i (t) ≡ p i and q j (t) ≡ q j , Inequality (2.14) is equivalent to the characteristic system of Equation (2.15) which has no solutions in R + × [0, 1).Therefore, Inequality (2.14) is a necessary and sufficient condition for Equation (2.15) with constant coefficients to have a nonoscillatory solution (Bainov and Simeonov, 1998).

Oscillatory Conditions
We now consider the nonlinear neutral delay impulsive differential equation with variable coefficients for t ≥ t 0 > 0 and ∀k : t k ≥ t 0 > 0. We introduce Conditions C3.1 to C3.4: Next, we establish the following lemmas which will be useful in the proof of the main result.Then finally z(t) > 0 with z (t) < 0 and Δz(t k ) < 0.
Proof.From Equation (3.1) we can affirm that z (t) < 0 and Δz(t k ) < 0 finally.It remains to show that z(t) > 0 finally.By contradiction, we assume that z(t) is finally negative.This implies that there exist a sufficiently large T such that z(t) < −d < 0 for all t ≥ T , where d is a positive constant.Hence In particular, Hence, y(t) cannot be finally positive.This contradicts the initial assumption of the Lemma and hence completes the proof.
Lemma 3.2 Assume that Conditions C3.2 and C3.4 with the inequalities and the solution of Equation (3.1) be such that the solution (λ(t), λ k ) of the associated generalized characteristic system satisfies the inequalities (3.7) (3.8) and Q(t), Q(t k ) are strictly increasing.Then Q −1 (t) and Q −1 (t k ) are well defined, strictly increasing and (3.9) By virtue of Inequality (3.5), the ratios Λ(t−σ ) Λ(t) and Λ(t k −σ ) Λ(t k ) are bounded above under Inequality (3.6).This implies that Inequality (3.8) is valid and completes the proof.
Let us now prove the following result.
Proof.We first assume that Condition (3.11) is satisfied.Without loss of generality, assume that Equation (3.1) has a finally positive solution y(t).Let y(t) > 0, y(t − ζ) > 0, for t ≥ T 1 ≥ t 0 .Then, by Lemma 3.1, z(t) > 0, z < 0 and Δz(t k ) < 0 for t ≥ T 1 and ∀k : t k ≥ T 1 , where z(t) is defined by Equation (3.4).For t ≥ T 1 , t t k and from Equation (3.1), we have (3.13) Notice that the first equation is due to the definition of z(t) in Equation (3.4).The following inequality represents an upper estimate of the expansion on the left side and the last equation is based on Equation (3.1).Using analogous reasoning, we obtain the following result fot the corresponding impulsive part: for each t ≥ T 1 and k: t k ≥ T 1 .Therefore taking Inequality (3.7) into account, Equation (3.13) is reduced to and It is obvious that λ(t) > 0 and λ k > 0 for each t ≥ T 1 and for all k: t k ≥ T 1 .From Inequality (3.14), we have where σ * = min 1≤ ≤m j {σ } and α = min 1≤ ≤m j {α }.In view of Lemma 3.2, we have which implies that lim t→∞ inf λ(t) < ∞ and lim From Inequality (3.11), there exists δ ∈ (0, 1) such that (3.17) Substituting (3.17) into Inequality (3.14), we obtain which comes into contradiction with Inequality (3.16).This completes the proof of Theorem 3.1 under Inequality (3.11).