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we consider the third-order neutral functional differential equations with deviating arguments. A new theorem is presented that improves a number of results reported in the literature. Examples are included to illustrate new results.

In this paper we consider third order neutral differential equations of the form

where

(A_{1})

(A_{2})

(A_{4})

(A_{5})

(A_{6})

(A_{7})

We mean by a solution of Equation (1) a function

Asymptotic properties of solutions of differential equations of the second and third order have been subject of intensive studying in the literature. This problem for neutral differential equations has received considerable attention in recent years (see [

Recently, in [

In [

[

and

In this paper, we establish some oscillation criteria for Equation (1), which complement and extend the results in [

We begin with analyzing of the asymptotic behavior of possible non-oscillatory solutions of the Equation (1) in the case when

or

for all sufficiently large t. Denote by

Lemma 1.1 Let_{1}) and (A_{2}) hold and x be continuous non-oscillatory solution of the functional inequality (a). Then

Lemma 1.2 Let_{1}) and (A_{2}) hold and x be continuous non-oscillatory solution of the functional inequality (b). If

These lemmas are modifications of the Lemma 1 in the paper [

In this part, for the sake of convenience, we introduce the following notation:

In this section, we will establish some oscillation criteria for Equation (1) in the case when

Lemma 2.1 Let x be a bounded positive solution of Equation (1) on the interval I. Then there exists a

Proof. Let x be a bounded positive solution of Equation (1) on the interval I. From (A_{1}), (A_{2}) and (A_{6}), there exists a

Hence, the function

By integrating the last inequality from

Letting_{3}), we have

By integrating this inequality from _{3}), we get

By integrating this inequality from _{3}), we get

Theorem 2.1 if

Then every bounded solution

Proof. Let x be a bounded non-oscillatory solution of Equation (1) on the interval I. Without loss of generality we may assume that

for all sufficiently large t. Repeating this procedure and the monotonicity of

integer

where

Thus, from Equation (1), we obtain

Now, since

If

Thus, form Lemma 2.1, we get

So, for

Hence, from (6), we get

where

We note that_{4}) and (A_{6}), we get

From (5), we have

By (A_{4}) and (A_{6}), we get

Thus, from (7), we have

Then, substituting (8) in (9), it follows that

By integrating this inequality from

where

Hence, from (10), we have

which contradicts (3). Therefore,

In the following Theorem, we establish some sufficient conditions for boundedness and oscillation of Equation (1) under the condition

Theorem 2.2 Let (11) holds. If there exist an integer

then every bounded solution

Proof. Let x be a bounded non-oscillatory solution of Equation (1) on the interval I. Without loss of generality we may assume that

and

Thus, Equation (1) implies that

By integrating this inequality from

where

where

Combining (13) and (14), we have

Hence, we get

for

Corollary 2.1 Let (11) holds. If

then every bounded solution

Example 2.1 Consider the differential equation

where

and

of the above equation are oscillatory.

Remark 2.1 If

In this section, we will present some oscillation criteria for Equation (1) under the case

Lemma 2.2 If

Proof. The proof of this lemma is similar to the proof Lemma 1 in [

Theorem 2.3 Let (16) holds. If

and there exist a positive real function

Then every solution of Equation (1) is either oscillatory or tends to zero.

Proof. Let x be a non-oscillatory solution of Equation (1) on the interval I. Without loss of generality we may assume that

where_{6}) and (16), we have

By integrating two times from t to

Integrating the last inequality from

This contradicts to the condition (17), then

since

for_{4}) and (A_{6}), we get

Also, we have

Since

Now, we define

By differentiating and using (19) and (20), we get

Hence, we obtain

By integrating the above inequality from

Taking the superior limit as

Remark 2.2 We can rewrite the condition (17) in the Theorem 2.3 as following

Remark 2.3 If

Example 2.2 Consider the differential equation

where

are satisfied then every solutions of this equation is either oscillatory or tends to zero.

Elmetwally M. Elabbasy,Magdy Y. Barsoum,Osama Moaaz, (2015) Boundedness and Oscillation of Third Order Neutral Differential Equations with Deviating Arguments. Journal of Applied Mathematics and Physics,03,1367-1375. doi: 10.4236/jamp.2015.311164