New oscillation criteria for second-order neutral differential equations with distributed deviating arguments

In this work, we create new oscillation conditions for solutions of second-order differential equations with continuous delay. The new criteria were created based on Riccati transformation technique and comparison principles. Furthermore, we obtain iterative criteria that can be applied even when the other criteria fail. The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.


Introduction
The importance of studying delay differential equations DDEs is not limited to the theoretical side only, but the applications of this type of equations extend to many branches of applied science. In fact, the neutral DDEs arise in the examination of vibrating masses attached to an elastic bar, in the solution of variational problems with time delays, and in problems concerning electric networks containing lossless transmission lines (as in high speed computers), see [1,2].
In this work, we discuss the oscillation properties of the second-order NDDE with distributed deviating arguments Throughout this work, we assume that and the following hypotheses hold: (H 1 ) α is a ratio of odd natural numbers, r ∈ C([t 0 , ∞), (0, ∞)), and ∞ t 0 lim t→∞ g(t, s) = ∞ for s ∈ [a, b], and g is strictly increasing with respect to t and s for all s ∈ (a, b).
For a solution of (1.1), we mean a function x ∈ C([t x , ∞)), t x ≥ t 0 , which has the property ℵ(t) and r(t)(ℵ (t)) α are continuously differentiable for t ∈ [t x , ∞) and satisfies (1.1) on [t x , ∞). We focus only on the solutions of (1.1) which satisfy sup{|x(t)| : t x ≤ t} > 0 for t ≥ t x . A solution x of (1.1) is called nonoscillatory if it is either eventually positive or eventually negative; otherwise it is called oscillatory.
In the next part of the introduction, we provide some related work that contributed to the development of the study of oscillatory behavior of NDDEs.
In the previous decade, under the hypothesis τ • g = g • τ , Han et al. [10] presented the oscillation criteria for the NDDE In 2012, by using the Riccati transformation technique, Liu et al. [11] and Wu et al. [12] obtained the oscillation conditions for the NDDE , α ≥ β, r (t) > 0, and g (t) > 0. Based on establishing new comparison theorems that compare the second-order equation with a first-order DDE, Baculikova and Dzurina [13] studied the NDDE Of interesting works recently, Moaaz et al. in [14,15] studied the oscillatory properties of (1.4) and improved the results in [13].
For NDDE with distributed deviating arguments (1.1), Candan [16] studied the sufficient conditions for the oscillation of solutions.
In this work, we are creating an improved relationship between the corresponding function ℵ and its first derivative. This new relationship helps us to get sharp criteria for testing the oscillation. Based on the Riccati transformation and comparison principles, we obtain new and different criteria for the oscillation of solutions of (1.1). The results obtained in this paper improve and extend the relevant previous results as illustrated by examples.
To prove our main results, we need the following auxiliary lemmas. The proof of the first lemma is similar to that of [13, Lemma 3] and hence we omit it.
for t ≥ t 1 .

Main results
For convenience, we denote the class of all eventually positive solutions by S + . Moreover, we assume the following notations: The following theorem gives a criterion for the oscillation of (1.1), depending on the comparison with a first-order DDE.
is oscillatory.
Proof Assume the contrary that there is a nonoscillatory solution x of (1.1). Then we can assume x ∈ S + , and so x(t), x(τ (t)), and x(g(t, s)) are positive for t ≥ t 1 ≥ t 0 and s ∈ [a, b]. It follows from Lemma 1.1 that (1.5) holds. Using the fact that r(t)(ℵ (t)) α is a nonincreasing function, we get It follows from (1.1) and (H 1 ) that From the definition of ℵ, we have which is a direct result of the facts that ℵ (t) > 0 and ∂ s g(t, s) > 0. Combining Thus, from (2.4), we find Integrating (2.5) from t 1 → t, we have Now, we set φ(t) := r(t)(ℵ (t)) α . Then, from (2.2) and (2.6), we obtain Combining (2.4) and (2.7), we have that φ is a positive solution of the first-order DD inequality From [24, Theorem 1], DDE (2.1) also has a positive solution, which is a contradiction. This contradiction completes the proof.
Applying a well-known condition [25, Theorem 2.1.1] for oscillation of first-order DDE (2.1), we get immediately the following criteria for oscillation of (1.1).

Corollary 2.1 Every solution of (1.1) is oscillatory if one of the following conditions is sat-
The next theorem gives another criterion for the oscillation of (1.1), depending on the Riccati transformation technique.

10)
where t 1 is sufficiently large.
It is easy to see that Corollary 2.1 cannot be applied in the case where t g a (t) However, if x ∈ S + and (2.16) holds, then we can get a sharp estimate of z(g(t))/z(t). Thus, we can obtain a sharp criteria for the oscillation of (1.1).

Further results
It is easy to notice that (2.7) is a sharper estimate than (2.2) for the relationship between ℵ and ℵ . By repeating the same steps that improved (2.2), we obtain iterative criteria that can be applied even when the other criteria fail.

Lemma 3.1
Assume that x ∈ S + . Then Proof Assume that x ∈ S + . Then x(t), x(τ (t)), and x(g(t, s)) are positive for t ≥ t 1 ≥ t 0 and s ∈ [a, b]. It follows from Lemma 1.1 that (1.5) holds. By induction, we will prove (3.1). Now, as in the proof of Theorem 2.1, we obtain (2.4) and (2.7). From (2.7), we obtain Next, for k = n, we suppose that ℵ ≥ U n r 1/α ℵ . Hence, we get ℵ(g a ) ≥ U n (g a )r 1/α (g a )ℵ (g a ) ≥ U n (g a )r 1/α ℵ , which with (2.4) gives Applying the Grönwall inequality in (3.4), we find for t ≥ s ≥ t 1 , and so Integrating (3.5) from t 1 → t, we see that This completes the proof.

Conclusion
The oscillation theory of DDEs has many applications in applied sciences. Thus, studying the oscillation of the solutions of these equations has practical importance besides the theoretical importance. In this study, we obtained different oscillation criteria with different techniques. These new criteria enable us to test the oscillation of a class of NDDEs with continuous delay. Our results extended to recently published works [14,15], and also improved [13,17].
Modeling by fractional-order differential equations has more advantages than by classical integer-order ones as it considers the effects of existence of time memory or long-range space interactions. So, it would be interesting to extend the results of this paper to the fractional delay differential equations. Moreover, it is interesting to study the periodicity behavior of solutions of the studied equation as an extension of the works [19,23].