1. Introduction

2. Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order

We begin with the following notations: $U^{+}$ is the set of all eventually positive solutions of (1), $V\left(t\right):=a_{0{1/\beta {}_{}^{}\left(t\right)v^{\prime }\left(t\right)}_{}^{}}^{}$ ,

Lemma 1. Assume that $v\in U^{+}$ and there exists a  ${\delta }_0\in \left(0,1\right)$ such that:

(3)

Then, v eventually satisfies:

and:

Proof.  Let $$ u ∈ U + . Then, we have that $$ u g 0 t , and $$ u g 1 t are positive for $$ t ≥ t 1 , for some $$ t 1 ≥ t 0 . Therefore, it follows from (1) that:

Using (1) and Lemma 1 in ^[1], we see that:

and so:

(4)

Integrating this inequality from $$ t 1 to t and using the fact  $$ V β t ′ ≤ 0 , we find:

(5)

$$ C 1 Assume the contrary, that $$ v ′ t > 0 for $$ t ≥ t 1 . Thus, from (5), we have:

This, from (3), implies:

Letting $$ t → ∞ and taking the fact that $$ η t → 0 as $$ t → ∞ ,  we obtain $$ V β t → − ∞ , which contradicts the positivity of  $$ V t .

Next, since v is positive decreasing, we have that $$ lim t → ∞ v t = v 0 ≥ 0 . Assume the contrary, that $$ v 0 > 0 . Then, $$ v t ≥ v 0 for all $$ t ≥ t 2 , for some $$ t 2 ≥ t 1 . Thus, from (3) and (5), we have:

or

and so,

(6)

Using the fact that $$ η ′ t < 0 , we obtain that $$ η t < η ′ t 2 < η ′ t 1 for all $$ t ≥ t 2 ≥ t 1 . Hence, by integrating (6) from $$ t 1 to t, we obtain:

Letting $$ t → ∞ and taking the fact that $$ η t → 0 as $$ t → ∞ , we obtain $$ v t → − ∞ , which contradicts the positivity of $$ v t . Therefore, $$ v 0 = 0 .

$$ C 2 Since $$ V t is decreasing, we obtain:

and:

(7)

Then, $$ v / η ′ ≥ 0 .

$$ C 3 From (7), we obtain:

Thus, from (4) and the fact  $$ V ′ t ≤ 0 , we obtain:

and then:

The proof is complete.

Lemma 2. Assume that $$ u ∈ U + and there exists a $$ δ 0 ∈ 0 , 1 such that (3) holds. Then:

Proof.  Let $$ u ∈ U + . From Lemma 1, we have that $$ C 1 – $$ C 3 hold for $$ t ≥ t 1 .

Integrating $$ C 3 from $$ t 1 to t, we arrive at:

From (3), we obtain:

and:

(8)

Using $$ C 1 , we eventually have:

Hence, (8) becomes:

This implies that $$ v / η γ 0 δ 0 is a decreasing function.

The proof is complete.

2.2. Oscillation Theorems

is oscillatory, then every solution of (1) is oscillatory. Proof.  Assume the contrary, that (1) has a solution $$ u ∈ U + . Then, we have that $$ u t , and $$ u g 1 t are positive for $$ t ≥ t 1 , for some $$ t 1 ≥ t 0 . From Lemmas 1 and 2, we have that $$ C 1 – $$ C 4 hold for $$ t ≥ t 1 . Next, we define: From $$ C 1 , $$ w t > 0 for $$ t ≥ t 1 . Thus, Thus, it follows from $$ C 3 that:

(10)

Using $$ C 4 , we obtain that: which with (10) gives:

(11)

Now, we set: Then, $$ W t ≤ c ˜ 0 w g 0 t , and so,  (11) becomes: which has a positive solution. In view of ^[2] (Theorem 1), (9) also has a positive solution, which is a contradiction. The proof is complete. Corollary 1. Assume that $$ g 1 t ≤ g 0 t and there exists a $$ δ 0 ∈ 0 , 1 such that (3) holds. If:

(12)

then every solution of (1) is oscillatory. Proof.  It follows from Theorem 2 in ^[3] that the condition (12) implies the oscillation of (9). Next, by introducing two Riccati substitution, we obtain a new oscillation criterion for (1). Theorem 7. Assume that $$ g 1 t ≤ g 0 t and there exists a $$ δ 0 ∈ 0 , 1 such that (3) holds. If:

(13)

then every solution of (1) is oscillatory. Proof.  Assume the contrary, that (1) has a solution $$ u ∈ U + . Then, we have that $$ u t , and $$ u g 1 t are positive for $$ t ≥ t 1 , for some $$ t 1 ≥ t 0 . From Lemmas 1 and 2, we have that $$ C 1 – $$ C 4 hold for $$ t ≥ t 1 . Now, we define the functions: $$$$ Θ 1 : = V v , and: $$$$ Θ 2 : = V ∘ g 0 v ∘ g 0 . Then, $$ Θ 1 and $$ Θ 2 are negative for $$ t ≥ t 1 . From $$ C 4 , we obtain: Hence, and: Then:

(14)

and:

(15)

Combining (14) and (15), we obtain: Integrating this inequality from $$ t 1  to t, we have: From $$ C 2 , we obtain $$ η t Θ 1 t ≥ − 1 . Therefore, where: Since $$ η ′ t < 0 and $$ a ′ t ≥ 0 , we find: Taking $$ lim sup t → ∞ and using (13), we arrive at a contradiction. The proof is complete.

2.3. Applications and Discussion

Remark 1. It is easy to see that the previous works that dealt with the noncanonical case required either $$ a 1 t < 1 or $$ a 1 t < η t / η g 0 t . Since η is decreasing and $$ g 0 t ≤ t , we have that $$ η g 0 t ≥ η t . Then, the results of these works only apply when $$ a 1 t ∈ 0 , 1 .

Example 1. Consider the DDE:

(16)

where $$ t ≥ 1 , $$ a 1 * > 0 , and $$ κ < λ ∈ 0 , 1 . By choosing $$ δ 0 = a 2 * , the condition (12) becomes:

(17)

Using Corollary 1, Equation (16) is oscillatory if (17) holds. Remark 2. To apply Theorems 3 and 4 on (16), we must stipulate that $$ a 1 * < 1 . Let a special case of (16), namely, A simple computation shows that (16) is oscillatory if:

(18)

or:

(19)

or:

(20)

Consider the following most specific special case:

(21)

Note that (18)–(20) fail to apply. However, (17) reduces to: which ensures the oscillation of (21).