More Effective Criteria for Oscillation of Second-Order Differential Equations with Neutral Arguments

: The motivation for this paper is to create new criteria for oscillation of solutions of second-order nonlinear neutral differential equations. In more than one respect, our results improve several related ones in the literature. As proof of the effectiveness of the new criteria, we offer more than one practical example.

On the other hand, there are many studies to improve the criteria for oscillation of solutions of non-canonical equations θ ζ 0 (∞) < ∞, some of which we will refer to below.

Oscillation Theorems in Canonical Case
In this section, we establish new criteria for oscillation of solution of (1) in canonical case θ ζ 0 (∞) = ∞. For convenience, we denote that: where c 1 and c 2 are positive constants.
Proof. Suppose the contrary; that (1) has an eventually non-oscillatory solution. Without loss of generality, we assume that x (ζ) > 0, x (τ (ζ)) > 0 and x (σ (ζ)) > 0 for ζ ≥ ζ 1 , where ζ 1 is sufficiently large. By Lemma 1, we have that (8) holds. Taking (8) into account, we obtain (1) gives Using the chain rule and simple computation, we see that From (10) and (11), we obtain Integrating this inequality from ζ 1 to ζ, we arrive at From the monotonicity of Proceeding as in the proof of Theorem 1 in [5] and using (13) instead of ((2.10) in [5]), we arrive at Thus, ψ is a positive solution of From Theorem 1 in [26], the equation also has a positive solution. It is well-known (see, e.g., [27], Theorem 2) that condition (9) implies oscillation of (14). This contradiction completes the proof.
then all solutions of (1) are oscillatory.
Proof. Proceeding as in the proof of Theorem 1, we arrive at (13). From (13), we see that Define the function Then ω (ζ) > 0 for ζ ≥ ζ 1 . From (1) and (17), we get Using Lemma 1 and (16), we obtain By integrating (18) from ζ to s, we conclude that Since w is positive decreasing function, we see that Hence, Set From (19), δ ≥ 1. Taking (15) and (19) into account, we find which is not possible with the permissible value α > 0 and δ ≥ 1. This contradiction completes the proof.
In the following, we give an example to illustrate our main results.

Conclusions
During this work, we highlighted the oscillatory properties of solutions of differential Equation (1). By using many techniques, we have created new criteria that are more effective than the relevant criteria in the literature. Moreover, we discussed oscillatory behavior in both canonical and non-canonical cases. Through the examples, it turns out that our results improve and complete some of the results in [4,5,7,11,16]. Finally, we can try to extend our results to differential equations with a damping term, in the future.

Conflicts of Interest:
The authors declare no conflict of interest.