An Oscillation Test for Solutions of Second-Order Neutral Differential Equations of Mixed Type

: It is easy to notice the great recent development in the oscillation theory of neutral differential equations. The primary aim of this work is to extend this development to neutral differential equations of mixed type (including both delay and advanced terms). In this work, we consider the second-order non-canonical neutral differential equations of mixed type and establish a new single-condition criterion for the oscillation of all solutions. By using a different approach and many techniques, we obtain improved oscillation criteria that are easy to apply on different models of equations.

Let x be a real-valued function defined for all s in a real interval [s x , ∞), s x ≥ s 0 , which has the properties and Then, x is called a solution of (1) on [s x , ∞) if x satisfies (1) for all s ≥ s x . We will consider only the solutions of (1) that exist on some half-line [s x , ∞) for s x ≥ s 0 and satisfy the condition sup{|x(s)| : s c ≤ s < ∞} > 0 for any s c ≥ s x .
A nontrivial solution x of any differential equation is said to be oscillatory if it has arbitrary large zeros; otherwise, it is said to be non-oscillatory.
The oscillation and asymptotic behavior of solutions to various classes of delay and advanced differential equations have been widely discussed in the literature. For second-order delay equations, the studies found in [1][2][3][4][5] were concerned with studying the oscillatory behavior of the equation: with the canonical operator π(s 0 ) = ∞, where One can find developments and comparisons of the oscillation criteria of (2) in the recently published paper by Moaaz et al. [4] for a non-canonical case, that is, Bohner et al. [6] simplified and improved the previous results found by Agarwal et al. [7] and Han et al. [8]. For more general equations and more accurate results, see [9,10].
For second-order advanced equations, Chatzarakis et al. [11,12] studied the asymptotic behavior of the equation: in the non-canonical case, and improved a number of pre-existing results.
Thandapani et al. [23] considered the equation where α, β, and γ are the ratios of odd positive integers, and established some sufficient conditions for the oscillation of all of the solutions. For more results, techniques, and approaches that deal with the oscillation of delay differential equations of higher orders, see [25][26][27][28][29][30][31][32][33]. The objective of this paper is to study the oscillatory and asymptotic properties of a class of delay differential equations of mixed neutral type with the non-canonical operator. The oscillation criteria are obtained via only one condition, and hence, they are easy to apply. Moreover, by using generalized Riccati substitution, we get new criteria that improve some of the results reported in the literature. An example is provided to illustrate the significance of the main results.

Preliminary Results
In the following, we present the notations used in this study: -For the continuous function r, we define the integral operator κ(u, v) for u < v as -For any solution x of (1), we define the corresponding function υ as -Briefly, we use the notations and , and C are real constants, B > 0, and α ∈ Q + odd . Then, the maximum value of Θ on R at v * = C + (αA/((α + 1)B)) α is Lemma 2. Let x be a positive solution of (1). If υ is decreasing, then eventually. Further, if υ is increasing, then for all s ≥ s 1 ≥ s 0 .

Main Results
for s 1 ≥ s 0 , then all solutions of (1) are oscillatory.
then all solutions of (1) are oscillatory.

Theorem 3.
Assume that H(s) ≥ G(s) > 0 and (15) hold. Further, if the differential equation is oscillatory, then all solutions of (1) are oscillatory.
Proof. Assume the contrary: that (1) has a non-oscillatory solution x on [s 0 , ∞). Without loss of generality (since the substitution y = −x transforms (1) into an equation of the same form), we suppose that x is an eventually positive solution. Then, there exists s 1 ≥ s 0 such that x( 1 (s)) > 0, x( 2 (s)) > 0, x(θ 1 (s)) > 0, and x(θ 2 (s)) > 0 for all s ≥ s 1 . As in the proof of Theorem 1, υ > 0 or υ < 0 eventually. Assume that υ < 0 on [s 1 , ∞). Since θ 2 (s) ≥ θ 1 (s), we get, from (3), that Now, we see that υ > 0 is a solution of the inequality Using [34], we find that (21) also has a positive solution-a contradiction. By proceeding as in the proof of Theorem 1, the proof of this theory is completed.
then all solutions of (1) are oscillatory.
and lim sup s→∞ s s 1 then all solutions of (1) are oscillatory.

Conclusions
Most works that studied the oscillatory behavior of mixed equations regarded the canonical case π(l 0 ) = ∞. Likewise, works that were concerned with the non-canonical case of neutral equations obtained two conditions for testing the oscillation. In this paper, we focused on studying the non-canonical case, and we created criteria with only one condition that is easy to verify. Therefore, our results are an extension, complement, and improvement to previous results in the literature. It is interesting to extend the results of this paper to higher-order equations. Funding: There was no external funding for this article.