Oscillation Properties of Third Order Neutral Delay Differential Equations

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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

References

[1] Baculikova, B. and Dzurina, J. (2010) Oscillation of Third-Order Functional Differential Equations. EJQTDE, 43, 1-10.

http://dx.doi.org/10.14232/ejqtde.2010.1.43

[2] Baculikova, B. and Dzurina, J. (2010) Oscillation of Third-Order Neutral Differential Equations. Mathematical and Computer Modelling, 52, 215-226.

http://dx.doi.org/10.1016/j.mcm.2010.02.011

[3] Baculikova, B. and Dzurina, J. (2010) On the Asymptotic Behavior of a Class of Third Order Nonlinear Neutral Differential Equations. Central European Journal of Mathematics, 8, 1091-1103.

http://dx.doi.org/10.2478/s11533-010-0072-x

[4] Dzurina, J., Thandapani, E. and Tamilvanan, S. (2010) Oscillation of Solutions to Third-Order Half-Linear Neutral Differential Equations. EJDEs, 2012, 1-9.

[5] Elabbasy, E.M., Barsoum, M.Y. and Moaaz, O. (2015) Boundedness and Oscillation of Third Order Neutral Differential Equations with Deviating Arguments. Journal of Applied Mathematics and Physics, 3, 1367-1375.

[6] Elabbasy, E.M., Hassan, T.S. and Moaaz, O. (2012) Oscillation Behavior of Second-Order Nonlinear Neutral Differential Equations with Deviating Arguments. Opuscula Mathematica, 32, No. 4.

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[8] Elabbasy, E.M. and Moaaz, O. (2016) On the Oscillation of Third Order Neutral Differential Equations. Asian Journal of Mathematics and Applications, 2016, Article ID: 0274.

[9] Elabbasy, E.M. and Moaaz, O. (2016) Oscillation Criteria for Third Order Nonlinear Neutral Differential Equations with Deviating Arguments. International Journal of Science and Research, 5, No. 1.

[10] Li, T., Thandapani, E. and Graef, J.R. (2012) Oscillation of Third-Order Neutral Retarded Differential Equations. International Journal of Pure and Applied Mathematics, 75, 511-520.

[11] Mihalikova, B. and Kostikova, E. (2009) Boundedness and Oscillation of Third Order Neutral Differential Equations. Tatra Mountains Mathematical Publications, 43, 137-144.

http://dx.doi.org/10.2478/v10127-009-0033-6

[12] Moaaz, O. (2014) Oscillation Theorems for Cartain Second-Order Differential Equations. Lambert Academic Publishing, Germany.

[13] Saker, S.H. (2006) Oscillation Criteria of Third-Order Nonlinear Delay Differential Equation. Mathematica Slovaca, 56, 433-450.

[14] Saker, S.H. and Dzurina, J. (2010) On the Oscillation of Certain Class of Third-Order Nonlinear Delay Differential Equations. Mathematica Bohemica, 135, 225-237.