On the Oscillations of Nonlinear Third Order Neutral Differential Equations

This paper is concerned with the oscillation of solutions of a class of third order nonlinear neutral differential equations. New sufficient conditions guarantee that every solution is either oscillatory or tends to zero are given. The obtained results improve some recent published results in the literature. Some illustrative examples are given. Citation: El-Sheikh MMA, Sallam RA, Salem S (2017) On the Oscillations of Nonlinear Third Order Neutral Differential Equations. J Appl Computat Math 6: 361. doi: 10.4172/2168-9679.1000361


Introduction
In this paper, we are concerned with the oscillatory behavior of solutions of third order differential equations of the type In the sequel, we assume the following conditions: The study of the oscillatory behavior of solutions of third-order differential equations has recieved great interest in the last few decades. One of the reasons for that is because in the real life, during the study of some physical phenomena, the qualitative behavior of solutions of third-order differential equations can be succesfully used to predict dynamic behavior of solutions of third-order partial differential equations.
Following this trend we are concerned in this paper with the oscillatory behavior of the third-order neutral differential equation (E).
By a solution of (E), we mean a function ( ) x T ∞ . In this paper, we consider only those solutions x of (E) which satisfy A solution of (E) is called oscillatory if it has arbitrarily large zeros on [ , ) x T ∞ ; otherwise it is called nonoscillatory.
Recently, increasing attention has been devoted to the oscillation of differential equations of the form (E) and some of its exceptions; have been the subject of intensive researchs see for example the papers [1][2][3][4][5][6][7][8][9][10][11][12][13] and references cited in. In particular, we mention here the paper of Grace et al. [5] which studied the oscillation of the third order delay differential equation.
By comparing with the first order delay equation, where in their comparison principle it is always required that ( ) < t t τ . More recently, Baculková and Džurina [3] improved their results for the case when  However, the results [13] cannot be applied when  In this paper are cocerned with this gap for the more general equation (E) by applying a technique similar to that given by those of refs. [8] and [10].

Lemma 1
Let x(t) be a positive solution of eqns. (E) and (1.2) holds. Then there are only one of the following two cases:

Proof:
The proof is similar to the proof of Lemma 1 [1] and so it is omitted.

Lemma 2
Let x(t) be a positive solution of (E). Suppose further that (1.2) holds and the corresponding z(t) satisfies case (II) in Lemma 1. If Proof: Assume that x(t) is a positive solution of (E). It is clear that there exists a finite limit, say Integrating from t to ∞ and using the fact that ( ) > z t l , we obtain Again by integrating eqn. (2.4) from t to ∞ , we get . This completes the proof. Now we outline the following two lemmas [1].

Lemma 3
Assume Proof: Then for every, there exists a

Lemma 4
Assume Further, we give the following auxiliary result which is extracted from those [6]and [7].

Lemma 5
Let 1 γ ≥ be a ratio of two odd positive numbers. Then, and

Main Results
In this section, we establish new oscillation criteria for eqn. (E) by using a generalized Riccati transformation and integral averaging technique of Philos-type [12]. Let Note that for = 1 γ . X γ reduces to the class of functions X used [8]. For = 1 ρ and = 0 ϕ , X γ reduces to the class of functions W γ used [9].

Theorem 6
Suppose that the conditions (A 1 )-(A 3 holds for some (0,1) c ∈ and for some H X γ ∈ , where In view of (E), we have Now, consider a generalized Riccati substitution of the form Then by eqn. (3.6), we get Therefore from Lemma 4 and Lemma 5, it follows that, for any (0,1) Combining eqns. (3.8) and (3.9), we get

H t s H t T T h t s H t s s s ds a s s
Applying the inequality eqn. (2.7) of Lemma 3, we get ( )   Firstly if case (I) and case (II) hold, respectively, we can obtain the conclusion of Theorem 7 by applying the proof of theorem 6.  Proof: Assume that x(t) is a non-oscillatory solution of (E). Without loss of generality, we may assume that x(t) is eventually positive. Going through as in the proof of Theorem 6, we arrive eqn.