Amended oscillation criteria for second-order neutral differential equations with damping term

The aim of this work is to improve the oscillation results for second-order neutral differential equations with damping term. We consider the noncanonical case which always leads to two independent conditions for oscillation. We are working to improve related results by simplifying the conditions, based on taking a different approach that leads to one condition. Moreover, we obtain different forms of conditions to expand the application area. An example is also given to demonstrate the applicability and strength of the obtained conditions over known ones.


Introduction
This work is concerned with studying the oscillation properties of the second-order neutral delay differential equations with damping term of the form where Throughout this work, we assume α ∈ Q + odd := {a/b : a, b ∈ Z + are odd}, r, c, τ , σ , p, q ∈ C([t 0 , ∞)), r is positive, c, p and q are nonnegative, τ (t) ≤ t, σ (t) ≤ t, σ (t) > 0, lim t→∞ τ (t) = lim t→∞ σ (t) = ∞, f ∈ C(R, R), and it satisfies the following property: where k > 0 is a constant and β ∈ Q + odd . Furthermore, this study requires that and c(t) < min 1, A real-valued function x ∈ C([t x , ∞), R), t x ≥ t 0 is a solution of (1) if x satisfies (1) on [t x , ∞), and it has the property that z(t) and r(t)(z (t)) α are continuously differentiable for t ∈ [t x , ∞). A nontrivial solution x is said to be oscillatory, if it has arbitrary large zeros. Otherwise, it is said to be non-oscillatory. The set of all eventually positive solutions of (1) is denoted by , that is, if x ∈ , then there exists a t 1 ≥ t 0 large enough such that x(t) > 0 for all t ≥ t 1 . We only focus on solutions of (1), which exist on [t 0 , ∞) and satisfy sup{|x(t)| : t x ≤ t} > 0 for every t ≥ t x .
Half-linear differential equations arise in real problems; for instance, in the study of non-Newtonian fluid theory and the turbulent flow of a polytropic gas in a porous medium; see [1,2].
In the past few years, there has been research studying the asymptotic properties and the oscillatory behavior of solutions of differential equations with different order. This research focused on developing and improving the oscillation criteria for differential equations. References [3][4][5][6][7][8][9] improved the oscillation criteria for noncanonical second-order equations with delay and advanced argument. For canonical second-order delay equations, Refs. [10,11] developed the oscillation criteria. The results in [12][13][14][15][16][17][18][19][20][21] dealt with the issue of oscillation of equations of higher order. For differential equations with damping, we present the following results that are closely related to this paper.
Tunc and Kaymaz [22] studied the oscillatory behavior of equations with damping term They are extended to more general second-order linear and/or nonlinear neutral differential equations with damping in [23][24][25]. Saker et al. [26] established Kamenev-type and Philos-type theorems for oscillation of equation with damping term Results in [26] are extended and improved results in [27][28][29].
In this paper, we try to improve the oscillation criteria for solutions of (1) by creating new and more effective criteria. In a noncanonical case, we always have two cases of derivatives signs for corresponding function z, which often leads to two independent conditions to ensure oscillatory. As an extension of the results of [4], we create a new criterion for oscillation of (1), which in turn is a simplification of the previous results in [26].
Although the theoretical advantage lies in reducing the number of conditions that are sufficient to verify the oscillation of solutions of differential equations, but sometimes a single condition is less effective in practical applications. So, we also follow the usual approach to creating two independent criteria of oscillation.
Remark 1 The functional inequalities in this paper are supposed to hold eventually, that is, they are satisfied for all t large enough.

Preliminaries
In the following, we provide some notations which help us to easily display the results. Moreover, we present the auxiliary lemmas which help in validating the main results.

Notation 2
For the sake of brevity, we define the functions Lemma 2 Assume that x ∈ . Then: (i) The function z satisfies z(t) > 0, (μ(t)r(t)(z (t)) α ) ≤ 0 and one of the next cases: and Proof Since x ∈ , it is obvious that there exists a t 1 ≥ t 0 such that x(t), x(τ (t)) and x(σ (t)) are positive functions. As a direct conclusion from the definition (2), we find that z(t) > 0. Next, from (1) and (3), we obtain which means that μ(t)r(t)(z (t)) α is a nonincreasing function and has a fixed sign. Since μ(t)r(t) > 0, we get either z (t) > 0 or z (t) < 0. Now, let z (t) < 0 for all t ≥ t 2 ≥ t 1 , where t 2 is large enough. Then there exists a constant K 1 > 0 such that On the other hand, using the fact that (μ(t)r(t)(z (t)) α ) ≤ 0, we get and therefore Integrating (10) from t to ∞ we obtain -z(t) ≤ -K 1/α 2 η(t). Thus, i.e., (6) holds. Moreover, we have and so d dt Hence, which, in view of (8), gives (7). The proof of the lemma is complete.

Lemma 3 Let x ∈ , the function z satisfy (D 2 ) and
Then Proof Suppose that x ∈ and the function z satisfies (D 2 ). Since z is a positive decreasing function, we have lim t→∞ z(t) = ≥ 0. Let > 0. From Lemma 2, there exists a t 1 ≥ t 0 such that Integrating this inequality from t 1 to t, we have Integrating (14) from t 1 to t, we get Taking lim t→∞ of this inequality and using (12), we are led to a contradiction with positivity of z. Therefore, = 0 and (13) hold. This completes the proof. Lemma 4 Let x ∈ , the function z satisfy (D 2 ) and (12) hold. If, moreover there exists a constant δ ∈ [0, 1) such that then d dt Proof Suppose that x ∈ and the function z satisfies (D 2 ). From Lemma 2, we see that (7) holds. Integrating (7) from t 1 to t, we obtain From Lemma 3, clearly (13) holds. Thus, for t ≥ t 2 , where t 2 is large enough. Combining (17) and (18) and using the fact that z β-α (t) ≥ (t), we find which, in view of (15), gives Thus, d dt This completes the proof.

The one criterion theorems
In the following theorems, we obtain a criterion which ensure oscillation of (1) without verifying the extra condition.
for any t 1 ≥ t 0 , then all solutions of (1) are oscillatory.
Proof Assuming that the required result is not fulfilled. We assume that (1) has a solution x ∈ . From Lemma 2, we see that z satisfies either (D 1 ) or (D 2 ) for all t ≥ t 1 . Let (D 2 ) hold. As in the proof of Lemma 2, we arrive at (19). Integrating (19) from t to ∞, we find for all t ≥ t 2 ≥ t 1 , where t 2 is large enough. From (7) in Lemma 2, we have Integrating this inequality from t 2 to t, we get Integrating again from t 2 to t, we obtain Taking lim t→∞ of this inequality and using (20), we are led to a contradiction with positivity of z.
for any t 1 ≥ t 0 , then all solutions of (1) are oscillatory.
Proof Assuming that the required result is not fulfilled. We assume that (1) has a solution x ∈ . From Lemma 2, we see that z satisfies either (D 1 ) or (D 2 ) for all t ≥ t 1 . Let (D 2 ) hold. From the proofs of Lemmas 2 and 4, we arrive at (11) and (17), respectively. Combining (11) and (17), we get and so which contradicts (25). Next, we let z (t) > 0 for t ≥ t 1 . We note that (25) along with (4) imply (24). The rest of the proof is similar to the proof of Theorem 1. The proof of the theorem is complete.

Theorems of two independent criteria
Theorem 3 Assume that (12) holds. If then all solutions of (1) are oscillatory.
Proof For the proof of this lemma, it suffices to use (19) [from the proof of Lemma 4] instead of (17) in the proof of Theorem 2.
Using Lemma 4, we obtain the new results which improve the previous theorems when Theorem 4 Assume that (12) and there exists a δ ∈ [0, 1) such that (15) holds. If then all solutions of (1) are oscillatory.
Proof Assuming that the required result is not fulfilled. We assume that (1) has a solution x ∈ . From Lemma 2, we see that z satisfies either (D 1 ) or (D 2 ) for all t ≥ t 1 . Let (D 2 ) hold. Using Lemma 4, we arrive at (16) and (19). From (16), we conclude that z σ (t) ≥ η δ (σ (t)) η δ (t) z(t), which with (19) gives As in the proof of Lemma 2, we arrive at (11). Combining (11) and (29), we obtain This contradicts (27). Next, we let (D 1 ) hold. We note that (12) along with (4) imply (24). The rest of the proof is similar to the proof of Theorem 1. The proof of the theorem is complete.