Oscillations of Second-Order Neutral Impulsive Differential Equations
© J.-F. Cheng and Y.-M. Chu. 2010
Received: 2 January 2010
Accepted: 28 February 2010
Published: 7 March 2010
Oscillation theory is one of the directions which initiated the investigations of the qualitative properties of differential equations. This theory started with the classical works of Sturm and Kneser, and still attracts the attention of many mathematicians as much for the interesting results obtained as for their various applications.
In 1989 the paper of Gopalsamy and Zhang  was published, where the first investigation on oscillatory properties of impulsive differential equations was carried out.
The monograph  is the first book to present systematically the results known up to 1998, and to demonstrate how well-know mathematical techniques and methods, after suitable modification, can be applied in proving oscillatory theorems for impulsive differential equations.
Recently, the oscillatory theory of differential equations has been the subject of intensive research [3–8]. In particular, many remarkable results for the oscillatory properties of various classes of impulsive differential equations can be found in the literature [2, 9–11].
The notion of characteristic system was first introduced by Bainov and Simeonov ; it can be used in obtaining of various necessary and sufficient conditions for oscillation of constant coefficients linear impulsive differential equations of first order with one or several deviating arguments.
As we know, on the case when we investigate constant coefficients linear neutral differential equations without impulse, it is very significant to obtain necessary and sufficient conditions for the oscillation corresponding to their characteristic equations; some necessary and sufficient conditions (in terms of the characteristic equation) for the oscillation of all solutions of first-or second-order neutral differential equations were established in [12–20]. However, the oscillation theory of the second order impulsive differential equations is not yet perfect compared to the second order differential equations with deviating argument (see ). For example, due to some obstacles of theoretical and technical character in handling with constant coefficients linear impulsive differential equations of second or higher order, there are no results which studied the necessary and sufficient conditions in monograph . How to establish the necessary and sufficient conditions for second order constant coefficients linear impulsive differential equations corresponding to their characteristic systems? This is also an important open problem (see monograph ). In this paper, we study and solve this problem for a class of linear impulsive differential equations of second order with advanced argument.
We shall restrict ourselves to the studying of impulsive differential equations for which the impulse effects take part at fixed moments . Considering the second order neutral impulsive differential equations with constant coefficients
This condition is equivalent to the next one.
As customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise the solution is called nonoscillatory. As usual, we use the term "finally" to mean "for sufficiently large ". For other related notions, see monographs [2, 9–11] for details.
2. Asymptotic Behavior of the Solutions
In the sequence, we assume that conditions (H1.1)–(H1.3) hold, and
is also a solution of (1.1).
It follows from condition (H1.2) that if is a moment of impulse effect, then is also a such moment. Thus, is also a solution of (1.1) since (1.1) is a linear one with constant coefficients. Therefore, is a solution of (1.1) which follows from the linear combination of solutions.
We clearly see that
3. Oscillation of the Unbounded Solutions
The present section is devoted to the characterizing of the oscillatory properties of solutions of the periodic neutral impulsive differential equation (1.1).
We are looking for a positive solution of (1.1) having the form
Substituting (3.1) in (1.1), and from condition (H1.3), we obtain the characteristic system of (1.1) as follows:
Equation (3.3) is called a characteristic equation, corresponding to the (1.1).
If the conditions (H1.1)–(H1.3) hold, then the following assertions are equivalent.
(a)Each unbounded regular solution of (1.1) is oscillatory.
(b)The characteristic equation (3.3) has no positive real roots.
The proof of (a) (b) is obvious. In fact, if the characteristic equation (3.3) has a real root , then we clearly see that and therefore the function
is an unbounded positive solution of (1.1).
If (1.1) has an eventually positive solution, then
which completes the proof of Lemma 3.2.
Combining (2.24), (3.19), and (3.22), we have
and the contradiction completes the proof of Lemma 3.3.
as defined in the proof of Lemma 3.3(c).
Therefore, the proof of Lemma 3.4 is completed.
Now, considering the sequence of functions
The repeated applications of Lemma 3.4 lead to
which contradicts with the fact proved in Lemma 3.3(b) that
Therefore, the proof of Theorem 3.1 is completed.
4. Oscillation of the Bounded Solutions
In Section 2, we complete the case of . Now we consider the conditions ensuring oscillation of the case of , that is, considering the conditions ensuring oscillation of the bounded solutions of (1.1). Then in view of Lemma 2.1(c), .
We are looking for a positive solution of (1.1) with the form
Substituting (4.1) in (1.1), just like in Section 2, we can obtain the characteristic equation of (1.1) as follows:
The proof of (a) (b) is obvious. In fact, if the characteristic equation (4.2) has a real root , then and , therefore the function
is a bounded positive solution of (1.1).
As in Section 3 we assume, without further mention, that (1.2) holds. We also assume that (4.2) has no real roots . For the sake of contradiction, we assume that (1.1) has a bounded finally positive solution .
This contradicts with(3.54); therefore, The proof of Lemma 4.3 is completed.
Therefore, the proof of Lemma 4.4 is completed.
Now, we consider the sequence of functions
The repeated applications of Lemma 4.4 lead to
From our Theorems 3.1 and 4.1, we get the following well-known results:
Each unbounded regular solution of the equation (1.5) is oscillatory.
(b)The characteristic equation (4.51) has no positive real roots.
The following assertions are equivalent.
(a)Each bounded regular solution of the equation (1.5) is oscillatory.
(b)The characteristic equation (4.51) has no real root .
The authors wish to thank the anonymous referee for the very careful reading of the manuscript and fruitful comments and suggestions.
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