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
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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
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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 ![]()