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Oscillation criteria are established for third-order neutral delay differential equations with deviating arguments. These criteria extend and generalize those results in the literature. Moreover, some illustrating examples are also provided to show the importance of our results.

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

(H)

A function

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 [

under the condition

Li et al. [

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

where

and

If

where

then every solution

Proof. Assume that

for

Then,

Since

we have that

Thus, we get

for

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

which contradicts (2). Assume now that

sume that

Dividing the above inequality by

Letting

that is

Define function

Then

Differentiating (13), we get

Using

In view of (3), we see that

Hence,

which implies that

By (13) and (15)-(17), we get

Multiplying the last inequality by

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

Let

and

Furthermore

such that

Using our result, every solution of (18) is either oscillatory or converges to zero as

Example 3.2. For

Let

and

Furthermore

such that

Using our result, every solution of (19) is either oscillatory or converges to zero as

Example 3.3. For

Let

and

Furthermore

such that

Using our result, every solution of (20) is either oscillatory or converges to zero as

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