Oscillatory Solutions to Neutral Delay Differential Equations

: This article aims to mark out new conditions for oscillation of the even-order Emden–Fowler neutral delay differential equations with neutral term (cid:16) β 1 ( ı ) Φ α [ ζ ( r − 1 ) ( ı )] (cid:17) (cid:48) + β 3 ( ı ) Φ α [ ς ( ξ ( ı ))] = 0. The obtained results extend, and simplify known conditions in the literature. The results are illus-trated with examples.


Introduction
Over the past few years, oscillation of Emden-Fowler-Type neutral delay differential equations with are attracting a lot of attention. As a matter of fact, natural of differential equation appear in the study of several real world problems such as biological systems, population dynamics, pharmacoki-netics, theoretical physics, biotechnology processes, chemistry, engineering, control, see [1][2][3][4][5][6][7].
The following relations are satisfied and In recent years, and in context of oscillation theory, many studies have been devoted to the oscillation conditions for non-linear delay differential equations; the reader can refer to [8][9][10][11][12][13][14][15][16].
These are some of the important Lemmas:
Then, we have these cases:

Lemma 5. Let (10) hold and
Then Proof. Let (10) hold. From the definition of ζ(ı), we have that Repeating the same process, we obtain Thus, (12) holds. This completes the proof.
Proof. The proof of this theorem is the same as that of Theorem 1.
So, we see that the conditions (20) and (21) holds. By Theorem 1, all solution of (37) is oscillatory.

Conclusions
In this article, we give several oscillatory properties of differential equation of evenorder with neutral term. The criteria obtained in this article complements the results in [19][20][21][22]. In our future work, and to supplement our results, we will present and discuss some oscillation theorems for differential equations of this type by using comparing technique with first/second-order delay differential equation.