Oscillation Properties of Third Order Neutral Delay Differential Equations ()
1. Introduction
This article is concerned with the oscillation and the asymptotic behavior of solutions of the third-order neutral delay differential equations with deviating argument of the form
(E)
where We assume that:
(H);, is a quotient of odd positive integers, and
A function is called a solution of (E), if it has the properties and satisfies (E) on We consider only those solutions of (E) which satisfy for all We assume that (E) possesses such solution. A solution of (E) is called oscillatory if it has arbitrarily large zeros on; otherwise, it is called nonoscillatory.
In the recent years, great attention in the oscillation theory has been devoted to the oscillatory and asymptotic properties of the third-order differential equations (see [1] - [14] ). Baculikova et al. [2] [3] , Dzurina et al. [4] and Mihalikova et al. [11] studied the oscillation of the third-order nonlinear differential equation
under the condition
Li et al. [10] considered the oscillation of
under the assumption
The aim of this paper is to discuss asymptotic behavior of solutions of class of third order neutral delay differential Equation (E) under the condition
(1)
By using Riccati transformation technique, we established sufficient conditions which insure that solution of class of third order neutral delay differential equation is oscillatory or tends to zero. The results of this study extend and generalize the previous results.
2. Main Results
In this section, we will establish some new oscillation criteria for solutions of (E).
Theorem 2.1. Assume that conditions (1) and (H) are satisfied. If for some function for all sufficiently large and for one has
(2)
where
(3)
and
(4)
If
(5)
where
(6)
then every solution of (E) is either oscillatory or converges to zero as
Proof. Assume that is a positive solution of (E). Based on the condition (1), there exist three possible cases
(1)
(2)
(3)
for is large enough. We consider each of three cases separately. Suppose first that has the property (1). We define the function by
(7)
Then, for Using we have
(8)
Since
we have that
(9)
Thus, we get
(10)
for Differentiating (7), we obtain
It follows from (E), (7) and (8) that
that is
which follows from (9) and (10) that
Hence, we have
Integrating the last inequality from to t, we get
(11)
which contradicts (2). Assume now that has the property (2). Using the similar proof ( [1] , Lemma 2), we can get due to condition (4). Thirdly, as-
sume that has the property (3). From is decreasing. Thus we get
Dividing the above inequality by and integrating it from t to l, we obtain
Letting we get
that is
(12)
Define function by
(13)
Then for Hence, by (12) and (13), we obtain
. (14)
Differentiating (13), we get
Using we have (8). From (E) and (8), we have
(15)
In view of (3), we see that
(16)
Hence,
which implies that
(17)
By (13) and (15)-(17), we get
Multiplying the last inequality by and integrating from to t, we obtain
which follows that
which contradicts (5). This completes the proof. W
3. Examples
The following examples illustrate applications of our result in this paper.
Example 3.1. For and consider the third-order differential equation
(18)
Let such that Note that,
and
Furthermore
such that, are defined as in (3) and,
Using our result, every solution of (18) is either oscillatory or converges to zero as if
Example 3.2. For and consider the third-order differential equation
(19)
Let such that Note that,
and
Furthermore
such that, are defined as in (3) and
Using our result, every solution of (19) is either oscillatory or converges to zero as if for some
Example 3.3. For and consider the third-order differential equation
(20)
Let such that Note that,
and
Furthermore
such that, are defined as in (3) and,
Using our result, every solution of (20) is either oscillatory or converges to zero as if