Oscillatory behavior of solutions of odd-order nonlinear delay differential equations

The objective of this study is to establish new sufficient criteria for oscillation of solutions of odd-order nonlinear delay differential equations. Based on creating comparison theorems that compare the odd-order equation with a couple of first-order equations, we improve and complement a number of related ones in the literature. To show the importance of our results, we provide an example.


Definition 2 A nontrivial solution
x of (1.1) is said to be oscillatory if there exists a sequence of zeros {t n } ∞ n=0 (i.e., x(t n ) = 0) of x such that lim n→∞ t n = ∞; otherwise, it is said to be nonoscillatory.
Although differential equations of even-order have been studied extensively, the study of qualitative behavior of odd-order differential equations has received considerably less attention in the literature, especially the third-order DDEs. However, certain results for third-order equations have been known for a long time and have some applications in mathematical modeling in biology and physics, see [17,23,25]. As a matter of fact, equation (1.1) under study is a so-called odd-order half-linear DDE, which has numerous applications in the research area of porous medium, see [13].
Different techniques have been used in studying the asymptotic behavior of DDEs. The articles [1, 3-9, 14-16, 27] were concerned with (in the canonical case and noncanonical case) the oscillation and asymptotic behavior of equation (1.1) and its special cases.
Based on creating comparison theorems that compare the odd-order DDEs with one or a couple of first-order DDEs, Agarwal et al. [1], Baculikova and Dzurina [3,4] and Chatzarakis et al. [8] studied the oscillatory and asymptotic behavior of special cases of the third-order DDE . By using the integral averaging technique, Bohner et al. [6] and Moaaz et al. [20] studied the asymptotic behavior of DDE with damping On the other hand, [5] used the Riccati transformation to study the asymptotic properties of the odd-order advanced equation where g(t) > t. The results concerned with the asymptotic properties and oscillation of the higher-order neutral DDEs were presented in [11,18,19,21,22,26]. In this paper, by using an iterative method, we create sharper estimates for increasing and decreasing positive solutions of (1.1). Thus, we create sharper criteria for oscillation of (1.1). Moreover, iterative technique allows us to test the oscillation, even when the related results fail to apply. The results reported in this paper generalize, complement, and improve those in [7-9, 14-16, 27]. To show the importance of our results, we provide an example.
Remark 1.1 We restrict our discussion to those solutions x of (1.1) which satisfy sup{|x(t)| : t ≥ T} > 0 for every T ∈ [t 0 , ∞). Remark 1.2 All functional inequalities and properties, such as increasing, decreasing, positive, and so on, are assumed to hold eventually, that is, they are satisfied for all t large enough.

Definition 3
The set of all positive solutions of (1.1) with property (2.1) or (2.2) is denoted by X + I or X + D , respectively.
Proof Let x ∈ X + I . Then there exists t 1 ≥ t 0 such that x(t) > 0 and x(σ (t)) > 0 for all t ≥ t 1 . Next, we will prove (2.3) using induction. For k = 0, using Lemma 2.1, we see that Then, from (1.1) and (2.4), we get If we set w := r(t)(x (n-1) (t)) α , then (2.5) becomes Applying the Grönwall inequality, we find Using Lemma 2.1 with F := x > 0, we see that By integrating this inequality from t 1 to t and taking into account (2.6), we see that Therefore, we have that The proof is complete.

Lemma 2.4
Assume that x ∈ X + D . Then Proof Let x ∈ X + D . Then there exists t 1 ≥ t 0 such that x(t) > 0 and x(σ (t)) > 0 for all t ≥ t 1 . Next, we will prove (2.7) using induction. For l = 0, since (r(z (n-1) )) ≤ 0, we get that Integrating (2.9) n -3 times from u to v, we get which, with (1.1), gives If we set ψ := r(t)(x (n-1) (t)) α , then (2.10) becomes Applying the Grönwall inequality, we find Thus, from (2.8), we see that Integrating this inequality n -2 times from u to v, we get Thus, the proof is complete.

Theorem 2.1
Assume that x is a positive solution of (1.1) and η k is defined as in Lemma 2.3. If the delay differential equation is oscillatory for some δ k ∈ (0, 1) and some k ∈ N, then X + I is empty.
Proof Assume to the contrary that x ∈ X + I . Then there exists t 1 ≥ t 0 such that x(t) > 0 and x(σ (t)) > 0 for all t ≥ t 1 If we set w := r(x (n-1) ) α , then (2.12) becomes In view of [24, Theorem 1], we have that (2.11) also has a positive solution, a contradiction. Thus, the proof is complete.

Corollary 2.1
Assume that x is a positive solution of (1.1) and η k is defined as in for some δ k ∈ (0, 1) and some k ∈ N, then X + I is empty.
Proof Assume to the contrary that x ∈ X + D . Then there exists t 1 ≥ t 0 such that x(t) > 0 and x(σ (t)) > 0 for all t ≥ t 1 . From Lemma 2.4, we have that (2.7) holds. Integrating (1.1) from σ (t) to t, we obtain and so Using (2.7) with u = σ (u) and v = σ (t), we get that which contradicts condition (2.14). This completes the proof.

Theorem 2.3
Assume that x is a positive solution of (1.1) and μ l,k is defined as in Lemma 2.4. If there exists a function θ ∈ C([t 0 , ∞), (0, ∞)) satisfying θ (t) < t and σ (t) < θ (t) such that the delay differential equation is oscillatory for some l ∈ N, then X + D is empty.
Proof Assume to the contrary that x ∈ X + D . Then there exists t 1 ≥ t 0 such that x(t) > 0 and x(σ (t)) > 0 for all t ≥ t 1 . From Lemma 2.4, we have that (2.7) holds. Using (2.7) with u = σ (t) and v = θ (t), we get that Thus, from (1.1), we obtain If we set ϕ := r(x (n-1) ) α , then (2.17) becomes In view of [24, Theorem 1], we have that (2.16) also has a positive solution, a contradiction. Thus, the proof is complete.
Applying a well-known criterion [12, Theorem 2] for delay equations (2.16) and (2.18) to be oscillatory, we obtain the following two corollaries.

Corollary 2.2
Assume that x is a positive solution of (1.1) and μ l,k is defined as in Lemma 2.4. If there exists a function θ ∈ C([t 0 , ∞), (0, ∞)) satisfying θ (t) < t and σ (t) < θ (t) such that for some l ∈ N, then X + D is empty.