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

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

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.

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

where

and

If

where

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

for is large enough. We consider each of three cases separately. Suppose first that has the property (1). We define the function by

Then, for Using we have

Since

we have that

Thus, we get

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

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

Define function by

Then for Hence, by (12) and (13), we obtain

Differentiating (13), we get

Using we have (8). From (E) and (8), we have

In view of (3), we see that

Hence,

which implies that

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

The following examples illustrate applications of our result in this paper.

Example 3.1. For and consider the third-order differential equation

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

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

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

Elabbasy, E.M., Moaaz, O. and Almehabresh, E.Sh. (2016) Oscillation Properties of Third Order Neutral Delay Differential Equations. Applied Mathe- matics, 7, 1780-1788. http://dx.doi.org/10.4236/am.2016.715149