Oscillation Criteria of Solutions of Fourth-Order Neutral Differential Equations

: In this paper, we study the oscillation of solutions of fourth-order neutral delay differential equations in non-canonical form. By using Riccati transformation, we establish some new oscillation conditions. We provide some examples to examine the applicability of our results.

As a result, there is an ongoing interest in obtaining several sufficient conditions for the oscillatory behavior of the solutions of different kinds of differential equations, especially their the oscillation and asymptotic. Baculikova [10], Dzurina and Jadlovska [11], and Bohner et al. [12] developed approaches and techniques for studying oscillatory properties in order to improve the oscillation criteria of second-order differential equations with delay/advanced terms. Xing et al. [13] and Moaaz et al. [14] also extended this evolution to differential equations of the neutral type. Therefore, there are many studies on the oscillatory properties of different orders of some differential equations in noncanonical form, see [15][16][17][18][19][20][21][22][23][24][25].
The qualitative theory of differential equations as well as analytical methods for qualitative behavior of solutions have contributed to the development of many new mathematical ideas and methodologies for solving ordinary and fractional differential equations as well as systems of differential equations. From the viewpoint of applications, differential equations are crucially important for modeling any kind of dynamical systems or processes in real life. So, in this work, we study the oscillatory behavior of solutions of the fourth-order neutral delay differential equations in noncanonical form. However, to the best of our knowledge, only a few papers have studied the oscillation and qualitative behavior of fourth-order neutral delay differential equations in noncanonical form.

Mathematical Background
In this section, we collect some relevant facts and auxiliary results from the existing literature. Furthermore, we fix the notations.

Definition 1.
A solution of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. For convenience, we denote: Furthermore, and The motivation for this article is to complement the results reported in [13,26], which discussed the oscillatory properties of equation in a canonical form.
Xing et al. [13] discussed the equation Moreover, the authors used the comparison method to obtain oscillation conditions for this equation.
Agarwal et al. [26] investigated the oscillation of equation The authors used the integral averaging technique to obtain oscillatory properties for this equation.
Moaaz et al. [14] established some criteria of (1) under condition Tang et al. [27] presented oscillation results for (1) under In [18], the authors established asymptotic behavior for neutral equation The authors in [13,26] used the comparison technique that differs from the one we used in this article. Their approach is based on using these mentioned methods to reduce Equation (1) into a first-order equation, while in our article, we discuss the oscillation and asymptotic properties of differential equations in a noncanonical form of the neutral-type, and we employ a different approach based on using the Riccati technique to reduce the main equation into a first-order inequality to obtain more effective oscillation conditions for Equation (1).
Motivated by these reasons mentioned above, in this paper, we extend the results using Riccati transformation under (3). These results contribute to adding some important conditions that were previously studied in the subject of oscillation of differential equations with neutral term. To prove our main results, we give some examples.

Lemma 4. Let
x be a positive solution of (1), Then, b(z)(y (z)) α is non-increasing. Furthermore, the following cases are possible:
Using Lemma 2, we find From definition of y, we get that Hence, from (10), if we set w := b(y ) α > 0, then From [19] (Corollary 1), we find (9) also has a positive solution. Thus, Lemma 5 is proved.
Proof. Let (8) hold with property (S 3 ). Using Lemma 2, we obtain As in the proof of Lemma 6, we find (11). Next, if we set G := b(y /y ) α < 0, then we find Hence, from the fact that y < 0 and (13), we get Thus, we get that (14) holds. It follow from [19] that (12) has a non-oscillatory solution. Lemma 6 is proved. Theorem 1. Let (9) and (12) be oscillatory. If then every non-oscillatory solution of (1) tends to zero.

Lemma 7. Assume that (8) holds and
Then Proof. Let (8) hold. From the definition of y(t), we get Repeating the same process, we find which yields Thus, (20) holds. Lemma 7 is proved.
Proof. Let (8) hold. From Lemma 4, we have cases (S 1 )-(S 4 ). Let (S 3 ) holds. From Lemma 9, we find (27) holds. differential equation. Our technique essentially simplifies the process of investigation and reduces the number of conditions required in previously known results. We may say that, in future work, we will study oscillatory properties of Equation (1) with p-Laplacian like operators and under the condition An interesting problem is to extend our results to even-order damped differential equations with p-Laplacian like operators b(t) y (n−1) (t) p−1 + q(t) y (n−1) (t) under the condition