Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations

By a solution of (1) we mean a function y ∈ C3[ty,∞), ty ≥ t0, satisfying (1) on [ty,∞) and such that r(t)(z′′′(t))γ ∈ C1[ty,∞). We consider only those solutions y of (1) that satisfy sup{|y(t)| : t ≥ T} > 0 for all T ≥ ty. A solution y of (1) is said to be nonoscillatory if it is ultimately positive or negative; otherwise, it is said to be oscillatory. The equation itself is called oscillatory if all its solutions are oscillatory. Delay differential equations play an important role in applications of real-world life. One area of active research in recent years is studying the sufficient conditions for oscillation of delay differential equations, see [1–23] and the references therein.

A solution y of (1) is said to be nonoscillatory if it is ultimately positive or negative; otherwise, it is said to be oscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
Delay differential equations play an important role in applications of real-world life. One area of active research in recent years is studying the sufficient conditions for oscillation of delay differential equations, see  and the references therein.
Let us briefly comment on a number of related results, which motivated our study. The authors in [28,29] were concerned with oscillatory behavior of solutions of fourth-order neutral differential equations and established some new oscillation criteria.
In [30,31] the authors considered the equation and established the criteria for the solutions to be oscillatory when 0 ≤ p(t) < 1.
Our aim in the present paper is employing the Riccati technique to establish some new Kamenev-type and Philos-type conditions for the oscillation of all solutions of equation (1) under condition (2).
The paper is organized as follows. In Sect. 2, we give four lemmas to prove the main results. In Sect. 3, we establish new oscillation results for (1) by using Riccati transformation. In Sect. 4, we establish some new Kamenev-type oscillation criteria for (1). In Sect. 5, we use the integral averaging technique to establish some new Philos-type conditions for the oscillation of all solutions of equation (1). Finally, we present an example and some conclusions to illustrate the main results.
Remark 1.1 All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough.
Remark 1.2 Without loss of generality, we can deal only with the positive solutions of (1).
Notation For convenience, we use the following notation: and

Some auxiliary lemmas
We will employ the following lemmas: , Lemma 2.1) Let γ ≥ 1 be the ratio of two odd numbers, and let V > 0 and U be constants. Then . Assume that y (n) (t) is of fixed sign and not identically zero on [t 0 , ∞) and that there exists t 1 ≥ t 0 such that y (n-1) (t)y (n) (t) ≤ 0 for all t ≥ t 1 . If lim t→∞ y(t) = 0, then for every μ ∈ (0, 1), there exists t μ ≥ t 1 such that

Lemma 2.3 ([35]) Let y(t) be a positive and n-times differentiable function on an interval
[T, ∞) with its nth derivative y (n) (t) nonpositive on [T, ∞), not identically zero on any interval of the form [T , ∞), T ≥ T, and such that y (n-1) (t)y (n) (t) ≤ 0, t ≥ t y . Then there exist for all sufficient large t.

Lemma 2.4
Assume that y is an eventually positive solution of (1). Then Proof Let y be an eventually positive solution of (1). Then there exists t 1 ≥ t 0 such that y(t) > 0, y(τ (t)) > 0 and y(δ(t)) > 0 for t ≥ t 1 . Since r (t) > 0, we have for t ≥ t 1 . From the definition of z we get which, together with (1), gives The proof is complete.

Oscillation criteria
In this section, we establish new oscillation results for (1) by using the Riccati transformation.
Lemma 3.1 Let y be an eventually positive solution of (1). If there exist constants ε ∈ (0, 1) and ζ > 0such that then Proof Let y be an eventually positive solution of (1). Using Lemma 2.4, we obtain that (14) holds. From (17) we see that ϕ(t) > 0 for t ≥ t 1 , and using (14), we obtain From Lemma 2.3 we have which is Using (17) we have Since z (t) > 0, there exist t 2 ≥ t 1 and a constant M > 0 such that Then (22) turns into that is, The proof is complete.

Lemma 4.1 Let y be an eventually positive solution of
Proof Let y be an eventually positive solution of (1). Using Lemma 2.4, we obtain that (14) holds. From (32) we see that (t) > 0 for t ≥ t 1 , and using (14), we obtain which is ζ δ(t)) .
and so lim sup which contradicts (36), and this completes the proof.

Philos-type oscillation result
In the section, we employ the integral averaging technique to establish a Philos-type oscillation criterion for (1).
Remark 5. 1 We can easily see that the results obtained in [32,33] cannot be applied to (36), so our results are new.
Remark 5.2 We can generalize our results by studying the equation q i (t)y β δ i (t) = 0, t ≥ t 0 , j ≥ 1.
For this, we leave the results to interested researchers.

Conclusions
The aim of this paper was to provide a study of asymptotic nature for a class of fourthorder neutral delay differential equations. We used a Riccati substitution and the integral averaging technique to ensure that every solution of the studied equation is oscillatory.
The results presented complement some of the known results reported in the literature. A further extension of this paper is using our results to study a class of systems of higherorder neutral differential equations, including those of fractional order. Some research in this area is in progress.