Oscillation Criteria of Fourth-Order Differential Equations with Delay Terms

The aim of this paper is to derive oscillation criteria of the following fourth-order di ﬀ erential equation with delay term ð r ð x Þð z ′′′ ð x ÞÞ γ Þ ′ + ∑ ni =1 q i ð x Þ f ð z ð η i ð x ÞÞÞ = 0 , under the assumption Ð ∞ x 0 r − 1/ γ ð s Þ d s = ∞ : The results are based on comparison with the oscillatory behaviour of second-order delay equations and the generalised Riccati transformation. Not only do the provided theorems provide an entirely new technique but also they vastly improve on a number of previously published conclusions. We give three examples to illustrate our ﬁ ndings.


Introduction
Higher-order neutral differential equations have recently been recognized as being sufficient to describe a variety of real applications [1][2][3][4]. As a result, many researchers have studied the qualitative behaviour of solutions of these equations (see [5][6][7][8]). The research of oscillation and oscillatory behaviour of these equations, which has been investigated using multiple approaches and techniques, has received special attention (see [9][10][11]). The attempt to improve the work and obtain a generalised platform that covers all special cases inspires the investigation of fourth-and higher-order equations.
In this work, we are concerned with oscillation of fourthorder delay differential equations of the form where x ≥ x 0 . Throughout this work, we suppose the following: (i) r ∈ C 1 ð½x 0 ,∞Þ, RÞ and γ is a quotient of odd positive integers (ii) The following condition holds: for rðxÞ > 0, r ′ ðxÞ > 0, and By a solution of (1), we mean a function z ∈ C 3 ½x z ,∞Þ, x z ≥ x 0 , that has the property rðxÞðz ′ ′′ðxÞÞ γ ∈ C 1 ½x z ,∞Þ and fulfills (1) on ½x z , ∞Þ. If a solution of (1) has arbitrarily large zeros on ½x z , ∞Þ, then it is considered oscillatory; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory. Next, we give some previous findings in the literature that are relevant to the present work. Grace [12] has studied the equation in addition to Agarwal et al. [13] and Xu and Xia [14] who have studied the equation subject to condition (2). Zhang et al. [15] obtained oscillatory criteria of the equation with the condition ð ∞ Baculikova et al. [16] used the comparison theory to prove that if is oscillatory, then is oscillatory for even n. Grace et al. [7] presented oscillation criteria for fourth-order delay differential equations of the form Using the Riccati transformation, an oscillation criterion for fourth-order neutral delay differential equation of the form was obtained by Chatzarakis et al. [17]. By using the technique of the Riccati transformation and the theory of comparison with first-order delay equations, Bazighifan and Abdeljawad [18] established some new oscillation criteria for fourth-order advanced differential equations with p-Laplacian-like operator of the form Very recently, Bazighifan et al. [19] established new criteria for the oscillatory behaviour of the following fourthorder differential equations with middle term by the comparison technique and employing the Riccati transformation under the condition For convenience, in the present work, we denote where π, τ ∈ C 1 ððx 0 ,∞Þ, ð0,∞ÞÞ. The generalised Riccati transformation is defined as We remark that in the study of the asymptotic behaviour of the positive solutions of (1), there are only two cases: In this work, using the Riccati approach and a comparison with a second-order equation, we shall obtain oscillation criteria for (1).

Some Significant Auxiliary Lemmas
The following lemmas serve as a basis for our findings.

Oscillation Criteria
In this section, we shall obtain some oscillation criteria for equation (1).

Journal of Function Spaces
This implies that Thus, The proof is completed.
Proof. Let z be a solution of (1) where z > 0 and z ðjÞ ðxÞ > 0 for j = 1, 3 and z ′ ′ðxÞ < 0. From Lemma 2, we have that zð xÞ ≥ xz ′ ðxÞ. Integrating this inequality from ηðxÞ to x, we obtain Hence, from (3) we have By integrating (1) from x to u and since z′ðxÞ > 0, we get Now letting u ⟶ ∞ yields and so Integrating this from x to ∞ gives From (18), we have that ϑðxÞ > 0 for x ≥ x 1 and by differentiating, we get Now, using Lemma 1 with P = ϑðxÞ/τðxÞ, Q = 1/δðxÞ, and α = 1 yields From (1), (40), and (41), we have the following: This implies that Thus, The proof is completed.