New Results for Oscillation of Solutions of Odd-Order Neutral Differential Equations

: Differential equations with delay arguments are one of the branches of functional differential equations which take into account the system’s past, allowing for more accurate and efﬁcient future prediction. The symmetry of the equations in terms of positive and negative solutions plays a fundamental and important role in the study of oscillation. In this paper, we study the oscillatory behavior of a class of odd-order neutral delay differential equations. We establish new sufﬁcient conditions for all solutions of such equations to be oscillatory. The obtained results improve, simplify and complement many existing results.

By a solution of (1), we mean a continuous real-valued function x(t) for t ≥ t x ≥ t 0 , which has the property: Υ is continuously differentiable n times for t ≥ t x , r Υ (n−1) α is continuously differentiable for t ≥ t x , and x satisfies (1) on [t x , ∞). We consider only the nontrivial solutions of (1) is present on some half-line [t x , ∞) and satisfying the condition sup{|x(t)| : t ≤ t < ∞} > 0 for any t ≥ t x . On many occasions, symmetries have appeared in mathematical formulations that have become essential for solving problems or delving further into research. High quality studies that use nontrivial mathematics and their symmetries applied to relevant problems from all areas were presented. In fact, in recent years, many monographs and a lot of research papers have been devoted to the behavior of solutions of delay differential equations. This is due to its relevance for different life science applications and its effectiveness in finding solutions of real world problems such as natural sciences, technology, population dynamics, medicine dynamics, social sciences and genetic engineering. For some of these applications, we refer to [1][2][3]. A study of the behavior of solutions to higher order differential equations yield much fewer results than for the least order equations although they are of the utmost importance in a lot of applications, especially neutral delay differential equations. In the literature, there are many papers and books which study the oscillatory and asymptotic behavior of solutions of neutral delay differential equations by using different technique in order to establish some sufficient conditions which ensure oscillatory behavior of the solutions of (1), see [4][5][6].
The authors in [1,3,7] have studied the oscillatory behavior of the higher-order differential equation And the author of [8] extended the results to the following equation Agarwal, Li and Rath [9][10][11][12] investigated the oscillatory behavior of quasi-linear neutral differential equation under the condition 0 ≤ p(t) < 1.
The latter differential equation was studied by Xing et al. in [13] under the condition The aim of this paper is to study the oscillatory behavior of the solutions of odd-order NDDE (1). By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of (1) is either oscillatory or tends to zero.

Auxiliary Results
In order to prove our main results, we will employ the following lemmas.
Then G attains its maximum value on R at v * = (αC/(α + 1)D) α and Lemma 2 ([15]). Assume that c 1 , c 2 ∈ [0, ∞) and γ > 0. Then where . Assume that f (n) (t) is of fixed sign and not identically zero on [t 0 , ∞) and that there exists a t 1 ≥ t 0 such that f (n−1 and there occur two cases for the derivatives of the function Υ: The rest of the proof is similar to proof of ([3] Lemma 2). Thus, the proof completed.
Proof. Let x be a positive solution of (1). Using (I 4 ) in (1), we have thus, This implies that where = c−p 0 ( +c) +c > 0. Using (6) in (5), we obtain Integrating the above inequality from t to ∞, we obtain Integrating (7) twice from t to ∞, we have Repeating this procedure, we arrive at Now, integrating from t 1 to ∞, we see that which contradicts (4), and so we have verified that lim t→∞ Υ(t) = 0.

Main Results
In the following lemma, we will use the notation Lemma 6. Let x be a positive solution of the equation in (1). If (8) and the equality h • ζ = ζ • h hold, then the following inequality is valid Moreover, if (8) and (9) hold, then Proof. Let x be a positive solution of (1). Then, there exists t 1 ≥ t 0 such that x(t) > 0, x(h(t)) > 0 and x(ζ(t)) > 0 for t ≥ t 1 . By the equality Υ(t) = x(t) + p(t)x(ζ(t)) together with Lemma 2, we obtain the inequality From (5) and the properties h • ζ = ζ • h and ζ ≥ ζ 0 , we obtain Using the latter inequalities and taking those in (5) and (13) into account as well, we obtain which with (12) gives This proves the inequality in (10). In order to show inequality (11) we proceed as follows. From (8) and (9), we obtain Moreover, . (15) Combining (14) with (15) and taking into account (12), we have This proves (11) and completes the proof of Lemma 6. (8) hold. Morever, assume that (4) is satisfied and that there exists a function δ ∈ C 1 ([t 0 , ∞), (0, ∞)) with the property that for all sufficiently large t 1 ≥ t 0 ,there exists t 2 ≥ t 1 such that lim sup Then, a solution x(t) to (1) either oscillates or else tends to zero when t → ∞.

Theorem 2.
Suppose that the functions h and ζ satisfy (8), (9) and h(t) ≤ ζ(t) for t 0 . In addition, suppose that (4) is satisfied. If there exists a function δ ∈ C 1 ([t 0 , ∞), (0, ∞)) with the property that for all sufficiently large t 1 ≥ t 0 ,there exists t 2 ≥ t 1 such that lim sup is valid. Then a solution x(t) of Equation (1) oscillates or tends to zero when t → ∞.
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