New Results for Kneser Solutions of Third-Order Nonlinear Neutral Differential Equations

In this paper, we consider a certain class of third-order nonlinear delay differential equations r w ″ α ′ v + q v x β ς v = 0 , for v ≥ v 0 , where w v = x v + p v x ϑ v . We obtain new criteria for oscillation of all solutions of this nonlinear equation. Our results complement and improve some previous results in the literature. An example is considered to illustrate our main results.


Definition 2.
If the nontrivial solution x is neither positive nor negative eventually, then x is called an oscillatory solution. Otherwise, it is a non-oscillatory solution.
When studying the oscillating properties of neutral differential equations with odd-order, most of the previous studies have been concerned with creating a sufficient condition to ensure that the solutions are oscillatory or tend to zero; see [11][12][13][14][15][16][17][18][19][20]. For example, Baculikova and Dzurina [11,12], Candan [13], Dzurina et al. [15], Li et al. [18] and Su et al. [19] studied the oscillatory properties of (1) in the case where α = β and 0 ≤ p (v) ≤ p 0 < 1. Elabbasy et al. [16] studied the oscillatory behavior of general differential equation For an odd-order, Karpuz at al. [17] and Xing at al. [20] established several oscillation theorems for equation As an improvement and completion of the previous studies, Dzurina et al. [14], established standards to ensure that all solutions of linear equation by comparison with first-order delay equations.
The main objective of this paper is to obtain new criteria for oscillation of all solution of nonlinear Equation (1). Our results complement and improve the results in [11][12][13][14][15][16][17][18][19] which only ensure that non-oscillating solutions tend to zero.
Next, we state the following lemmas, which will be useful in the proof of our results.

Criteria for Nonexistence of Decreasing Solutions
Through this paper, we will be using the following notation:

Lemma 4.
Assume that x ∈ S 2 . Then for u ≤ , and Proof. Let x be an eventually positive solution of (1). Then, we can assume that where v 1 is sufficiently large. From Lemma 1, (1) and (I 2 ), we obtain Since Integrating this inequality from u to , we get Now, from (1) and (I 3 ), we obtain (£w (ϑ (v))) 1 Using (1), (5) and (8), we have Thus, The proof of the lemma is complete.
is oscillatory, then S 2 is an empty set.
Proof. Assume the contrary that x is a positive solution of (1) and which satisfies case (ii). Then, we assume that Using Lemma 4, we get (3) and (4). Combining (4) and (3) with Since £w (v) is non-increasing, we see that £w (v) ≤ £w (ς (v)) , and hence Using (11) along with (12), we have that By Theorem 1 [21], the associated delay Equation (10) also has a positive solution, which is a contradiction. The proof is complete.
then S 2 is an empty set.
Proof. As in the proof of Theorem 1, we obtain (12). Using Lemma 4, we get (3) and (4). Integrating (4) from θ (v) to v, we get which together with (12) gives Since w (v) < 0, there exists a constant M > 0 such that w (v) ≥ M for v ≥ v 2 , and hence (14) becomes From above inequality, taking the lim sup on both sides, we obtain a contradiction to (13). The proof is complete.

Corollary 1. Assume that there exists a function
Then S 2 is an empty set, if one of the statements is hold: and Proof. It is well-known from [22,23] that conditions (15)- (17) imply the oscillation of (10).

Criteria for Nonexistence of Increasing Solutions
then S 1 is an empty set.
Proof. Let x be a positive solution of (1) and which satisfies case (i). In view of case (i), we can define a positive function by Hence, by differentiating (19), we get Substituting (19) into (20), we have Now, define another positive function by By differentiating (22), we get Substituting (22) into (23) implies We can write the inequalities (21) and (25) in the form Taking into account Lemma 1,(4) and (26), we obtain Applying the following inequality Integrating last inequality from v 1 to v, we arrive at The proof is complete.
In this section we state and prove some results by considering Proof. Since w (v) is a non-increasing positive function, there exists a constant w 0 ≥ 0 such that lim v→∞ w (v) = w 0 ≥ 0. We claim that w 0 = 0. Otherwise, using Lemma 2, we conclude that From (1) and (29), we see that Integrating above inequality from v to ∞, we have It follows that (30) Integrating again from v 2 to ∞, we obtain which contradicts with (27). Therefore, lim v→∞ w (v) = 0, and from the inequality 0 < x (v) ≤ w (v) , we have property (28). The proof is complete. Theorem 6. Let condition (27) be satisfied and suppose that there exists a function ∈ C (I, R) such that If the first-order delay differential equation is oscillatory, then every solution x (v) of Equation (1) is either oscillatory or satisfies (28).

Proof.
Assume that x (v) is positive solution of (1), eventually. This implies that there exists v 1 ≥ v o such that either (i) or (ii) hold for all v ≥ v 1 . For (ii), by lemma 5, we see that (28) holds.
For (i), since w (v) is a non-decreasing positive function, there exists a constant c 0 such that lim v→∞ w (v) = c 0 > 0 (or c 0 = ∞). By Lemma 2, we have which implies that x (v) is a non-decreasing function and taking into account δ 0 ≥ 0, we get , and By substitution in (1), we have (£w (v)) + q (v) Using (7) and (31), we get (£w (v)) + q (v) Therefore, we have y = £w (v) is positive solution of a the first order delay equation The proof is complete.

Theorem 7.
If the first-order delay differential equation is oscillatory, eventually. Then, every solution x (v) of Equation (1) is either oscillatory or satisfies (28).