EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5729 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fourth Order Functional Differential Equations of Neutral Type: Enhanced Oscillation Theorems Waed Muhsin1, Barakah Almarri2, Mohammad S. Jazmati3, Osama Moaaz1,3,∗, Elmetwally M. Elabbasy1 1 Department of Mathematics, Faculty of Science, Mansoura University, 35516 Mansoura, Egypt 2 Department of General Studies, Jubail Industrial College, 8244, Rd Number 6, Al Huwaylat, 35718, Al Jubail, Saudi Arabia 3 Department of Mathematics, College of Science, Qassim University, P. O. Box 6644, Buraydah, 51452 Saudi Arabia Abstract. In this paper, we investigate the oscillatory properties of fourth-order neutral delay differential equation solutions in the canonical situation. To our knowledge, this equation has received minimal research. We prove new improved features and relationships for the solution and the accompanying function. Based on these relationships, oscillation theorems were developed that guarantee oscillation of all solutions to the considered equation. The comparison principle used in our results is one of the most significant methods for investigating the oscillatory behavior of delay differential equations. The findings of our study extend and develop a number of previous findings in the literature. 2020 Mathematics Subject Classifications: 34C10, 34K11 Key Words and Phrases: differential equation; oscillation criteria; fourth-order differential equations; neutral delay argument 1. Introduction The objective of this paper is to investigate the oscillation of the fourth-order neutral delay differential equations (NDDEs)( a3 (u) (a2 (u) (a1 (u) Ω ′ (u))′)′ )′ + q (u)x (ρ (u)) = 0, u ≥ u0, (1) where Ω (u) = x (u) + h (u)x (κ (u)). Based on the following : ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5729 Email addresses: waed.zarebah@gmail.com (W. Muhsin), marribj@rcjy.edu.sa (B. Almarri), Jzmaty@qu.edu.sa (M. S. Jazmati), o moaaz@mans.edu.eg (O. Moaaz), emelabbasy@mans.edu.eg (E. M. Elabbasy) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 2 of 17 (A1) ai ∈ C(4−i) ([u0,∞) , (0,∞)) , i = 1, 2, 3 and satisfies Ii (u) := ∫ u u0 1 ai (ξ) dξ → ∞ as u → ∞; (2) (A2) κ, ρ ∈ C ([u0,∞) , (0,∞)) satisfy κ (u) < u, ρ (u) < u, ρ′ (u) > 0, limu→∞ κ (u) = ∞ and limu→∞ ρ (u) = ∞; (A3) h, q ∈ C ([u0,∞) , (0,∞)) . A studied solution to (1) is defined as a real-valued function x that is four times differentiable, that has the property a3 (u) (a2 (u) (a1 (u) Ω ′ (u))′)′ ∈ C1 ([u0,∞) , (0,∞)) and fulfills (1) for any suitably large u on [u0,∞). We concentrate only on those solutions of (1) that satisfy sup{|x (u) | : u ≥ U} > 0, for all U ≥ u0, i.e., it is a nontrivial solution. A solution x to (1) is said to be oscillating or non-oscillating based on whether it is positive, negative, or neither positive nor negative eventually in the first place. That is, what interests us is the behavior of the solution in the neighborhood of the infinite points. If all solutions of the equation oscillate (if it has arbitrarily large zeros), then the equation is oscillatory; otherwise, it is said to be non-oscillatory [4]. Understanding, studying, and then analyzing functional differential equations is the basis of many diverse disciplines in mathematics (both pure and applied), engineering, and physics, all of which focus on the properties of differential equations (DEs) in their various forms. As we know, many technological, physical, or biological processes are modeled using dynamic differentials. The existence and uniqueness of solutions of these equations is a key focus when studying the differential equations of these models, as a closed-form solu- tion may not be found for nonlinear dynamic differentials obtained when modeling many phenomena. Numerical techniques can be used as an alternative method to approximate the results [12]. Due to the difficulty of finding these solutions, researchers have focused on knowing the properties of these solutions under the name of qualitative theory of equations, of which oscillation theory is one of its main subfields. Asymptotic and oscillatory characteristics of solutions are the focus of this theory; see [27]. Fite produced an essay [16] on differential equation oscillation theory, written in the early 20th century. Many inquiries into the oscillation theory of delay differential equations began with his publication. Delay and neutral delay differential equations have a very diverse history and are used in many applications in the natural sciences and engineering; for further details, see Hale [18]. In a differential equation, when the highest order derivative of the function, whether known or unknown, is present in both the delay-containing and delay-free portions of this differential equation, the equation is said to include a neutral delay (ND). Neutral delay differential equations (NDDEs) are equations that depend on the present and delayed values of a function and its derivatives and are of significant importance due to W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 3 of 17 their widespread applications in various fields, including engineering, physics, and biology. They are complex because they include delays in the derivatives and are fundamental to the study of dynamical systems that contain time delays. Unlike ordinary delay differen- tial equations (ODDEs). Neutral differential equations are characterized by the presence of both the derivative of the dependent variable and its delayed term in the highest deriva- tive. This distinguishes them from other functional differential equations, such as delay differential equations, which involve delays only in the dependent variable without its derivatives. There has long been a lot of research into the oscillatory behavior and asymptotic features of the solutions to many different types of functional differential equations. Here, we recall the pioneering works of [14, 26, 46, 48]. In the last few years, there has been a major expansion and evolution of oscillation theory, which currently includes the study of oscillation for fractional and ordinary DEs solutions with delay, neutral, mixed, or damping components. For instance, [43–45] con- tains mixed equations; [5, 11, 23, 32, 33, 35, 39] contains neutral equations; and [7, 47] demonstrates the advancement in the investigation of higher-order equations. Additionally, the oscillation of damping equations can be traced in [6, 8]. On the other hand, refs.[19] dealt with fractional DEs, while [21, 22] is concerned with dynamic equations. Oscillation theory is interesting both theoretically and practically. It is known that homogeneous first-order ordinary differential equations do not have oscillatory solutions. However, the presence of deviant arguments can cause oscillation of solutions; see [15]. Therefore, the focus has been on studying and understanding the behavior of second- and higher-order differential equation solutions. We should not fail to include the ancient papers that were and probably still are a source of inspiration for many studies that were interested in discovering oscillation criterion when studying second-order DE (linear, half-linear, superlinear, and sublinear); see [3, 20]. The extensive applications of fourth-order DDEs in civil, aeronautical engineering, and mechanical make them extremely valuable in daily life. Because this kind of equation is so important in so many different domains, research on it is still ongoing (see references [10, 17, 34]). It also has biological applications, such as analyzing oscillations in neuromuscular systems [38]. Most of the recently published results on fourth-order DEs in the canonical case deal with equation (1) when a1 (u) = a2 (u) = 1. Therefore, the goal of this work is to extend the results obtained by studying this form of fourth-order DEs involving the neutral de- lay. Additionally, some similar findings served as inspiration for our study in particular: Masood et al. [29] took into account the oscillatory characteristics of solutions of( a3 (u) (x ′′′ (u))α )′ + q (u)xα (ρ (u)) = 0, u ≥ u0, where α is a ratio of two odd integers. They found some properties for a class of positive solutions of the latter quasi-linear DDE and then created an oscillation criterion. W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 4 of 17 By using some inequalities and the theory of comparison, Bazighifan et al. [9] studied the following equation( a3 (u) (Ω ′′′ (u))α )′ + q (u)xβ (ρ (u)) = 0, u ≥ u0, where α and β are quotients of two odd positive integers. They presented some oscillation criteria for the above equation and improved some previously published results. In 2013, Agarwal [1] investigated the oscillatory behavior of the n-order equation Ω(n) (u) + q (u)x (ρ (u)) = 0, (3) and arrived at criteria that enhance findings reported in the literature. In what follows, the oscillation for (3) was explored by Li and Rogovchenko [28]. They employed first-order delay equation comparison as their method. In addition, Salah et al. [42] studied equation (3) when n = 4, i.e. Ω4 (u) + q (u)x (ρ (u)) = 0, u ≥ u0. For the oscillation of every solution to their equation, aptly, they derived a new single- oscillation criterion. For some instances of positive solutions to the investigated equation, they established additional monotonic properties. Furthermore, they employed an iterative process to enhance these attributes. The evolution of these monotonic features helps to produce new and more effective standards for confirming the equation’s oscillation. The results obtained, utilizing an example Euler-type equation an, extend and improve upon earlier findings in the literature. On the other hand, Jadlovska [24] used an iteratively enhanced monotonicity charac- teristic of non-oscillatory solutions approach to offer a single-condition sharp criterion for the oscillation of the spacial case of the above equation when h (u) = 0, i.e x4 (u) + q (u)x (ρ (u)) = 0, u ≥ u0. Perhaps one of the oldest studies in the literature, [25], that presented some oscillation criteria for equation [ a3 (u)x ′′ (u) ]′′ + x (u)F ( x2, u ) = 0, u ≥ 0, (4) the function F in this equation is given as follows: xF ( x2, u ) is continuous for |x| < ∞, u ≥ 0, and F (y, u) is positive for y > 0, u ≥ 0. Following Nehari [37] and Wong [13], where equations of the form (4) are classified based on the nonlinearity of xF ( x2, u ) with respect to x. They determined the effect that a3 (u) exercises upon the oscillatory character of (4) in conjunction with its nonlinearity. New monotonic properties of the second-order NDE( a3 (u) ( Ω′ (u) )α)′ + q (u)xα (ρ (u)) = 0, W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 5 of 17 were derived by Moaaz et al. [30]. Subsequently, they employed these characteristics to derive optimal oscillation parameters, employing various approaches to achieve this objective. Additionally, they provided new standards that guarantee the oscillation of the fourth-order NDE ( a3 (u) ( Ω′′′ (u) )α)′ + q (u)xα (ρ (u)) = 0. (5) Earlier, Agarwal et al.[2] evaluated the oscillatory behavior of the solutions of the fourth-order functional DE[ a−1 3 (u) ([ a−1 2 (u) ([ a−1 1 (u) ( x′ (u) )α1 ]′)α2 ]′)α3 ]′ + λq (u) f (x (ρ (u))) = 0, where ∫∞ u0 a −1/αi i (ξ) dξ → ∞ as u → ∞, i = 1, 2, 3, λ = ±1, f ∈ C ((0,∞) , (0,∞)) , xf (x) > 0 and f ′ (x) ≥ 0 for x ̸= 0. Very recently, Nabih et al. [36] concentrated on examining the oscillation of (5); they discovered novel characteristics that allow them to employ more efficient terms. Using the comparison approach and the generic form of Riccati, they were able to derive criteria that eliminated positive decreasing solutions. Lemma 1. [40] Let Ψ ∈ Cn ([u0,∞) ,R). If Ψ(n) (u) is eventually of fixed sign for all large u, then there exist a ux ≥ u0 and a µ, 0 ≤ µ ≤ n, with n+ µ even for the derivative Ψ(n) (u) ≥ 0, or n+ µ odd for the derivative Ψ(n) (u) ≤ 0 such that (i) µ > 0 implies Ψ(κ) (u) > 0 for u ≥ ux, κ = 0, 1, ..., µ− 1; (ii) µ ≤ n− 1 implies (−1)µ+κΨ(κ) (u) > 0 for u ≥ ux, κ = µ, µ+ 1, ..., n− 1. Lemma 2. [31, Lemma 2.1] Suppose that there is an eventual positive solution to (1) for x (u). Then, eventually x (u) > γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[Ω (κ(2t) (u)) h ( κ(2t) (u) ) − Ω ( κ(2t+1) (u) )] , (6) for any integer γ ≥ 0. This search is intended to look into the oscillatory behavior of solutions to NDDEs of the fourth order. Any functional differential equation has three types of solutions: oscillatory, negative, and positive. As we know, negative solutions are thus disregarded due to the symmetry between the solutions (the positive and the negative) of the investigated equation. The novelty of the paper lies in the demonstrated improved relationships and features between the corresponding function and the solution; these relationships affect the oscillation criteria, and improving them leads to improving these criteria. The new criteria ensure that all solutions of the studied equation are oscillatory based on the comparison principle with first-order differential equations. It should be noted that if all the solutions of the equation are oscillatory, this means that the equation itself is an oscillatory equation. The findings discovered complement several well-known in the literature. W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 6 of 17 This is how our paper will be structured: In Section 2.1, by using the comparison principle with first-order equations, we introduce oscillation criteria, where the classical relationship that links the solution x (u) to its corresponding function Ω (u) has been applied. In Section 2.2, we present the improved criteria, which apply the new improved relationship between the corresponding function and the solution. 2. Comparison with first order equations 2.1. Oscillation criteria For the purpose of clarity, let Ii (u) := ∫ u u0 1 ai (ξ) dξ, Iij (u) := ∫ u u0 1 ai (ξ) Ij (ξ) dξ, Iijk (u) := ∫ u u0 1 ai (ξ) Ijk (ξ) dξ where i, j, k ∈ {1, 2, 3} , Q (u) := (1− h (ρ (u))) q (u) , R1 (u) := ( 1 a2 (u) ∫ ∞ u 1 a3 (ξ) ∫ ∞ v Q (v) dvdξ )∫ ρ(u) u1 1 a1 (v) dv, and R2 (u) := Q (u) ∫ u u1 1 a1 (v1) ∫ v1 u1 1 a2 (ξ) ∫ v u1 1 a3 (v) dvdξdv1, for any integer u ≥ u1, where u1 ≥ u0. Lemma 3. Let condition (2) hold and x (u) be an eventually positive to (1). Then, (a3(a2(a1Ω ′)′)′)′ (u) ≤ 0 and either Ω (u) ∈ L− ⇐⇒ Ω (u) > 0 Ω′ (u) > 0 (a1Ω ′)′ (u) < 0 (a2(a1Ω ′)′)′ (u) > 0; Ω (u) ∈ L+ ⇐⇒ Ω (u) > 0 Ω′ (u) > 0 (a1Ω ′)′ (u) > 0 (a2(a1Ω ′)′)′ (u) > 0. Proof. Suppose (1) has an eventually positive solution at x (u). From (1) we get (a3(a2(a1Ω ′)′)′)′ (u) ≤ 0. Lemma 1 must be used to deduce the cases L− and L+ for the corresponding function Ω (u) to the solution x (u) and its derivatives. Remark 1. The decomposition of (1) is found in the set L of all positive solutions L = L− ∪ L+. Now, we can derive the next theorem using the comparison principle: W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 7 of 17 Theorem 1. Assume that both first-order DDEs g′ (u) +R1 (u) g (ρ (u)) = 0 (7) and g′ (u) +R2 (u) g (ρ (u)) = 0 (8) are oscillatory. Then equation (1) is oscillatory. Proof. Assume that x is an eventually positive solution of (1), say for u ≥ u1. Through Lemma 3, we can deduce that Ω (u) ∈ L+ or Ω (u) ∈ L−. It follows from the monotonicity of a1 (u) Ω ′ (u) that Ω (u) ≥ ∫ u u1 1 a1 (v) a1 (v) Ω ′ (v) dv ≥ a1 (u) Ω ′ (u) ∫ u u1 1 a1 (v) dv, (9) or Ω (u) ≥ a1 (u) Ω ′ (u) I1 (u) . By the definition of Ω (u) , we obtain x (u) ≥ Ω (u)− h (u)x (ρ (u)) ≥ Ω (u)− h (u) Ω (ρ (u)) ≥ (1− h (u)) Ω (u) , which together with (1), implies( a3(a2(a1Ω ′)′)′ )′ (u) ≤ −q (u) (1− h (ρ (u))) Ω (ρ (u)) ≤ −Q (u) Ω (ρ (u)) . (10) Assume first that Ω (u) ∈ L−. Integrating (10) from u to ∞, we get a3 (u) (a2(a1Ω ′)′)′ (u) ≥ ∫ ∞ u Q (v) Ω (ρ (v)) dv. Using this fact Ω (ρ (u)) is increasing, the last inequality becomes (a2(a1Ω ′)′)′ (u) ≥ Ω (ρ (u)) 1 a3 (u) ∫ ∞ u Q (v) dv. Integrating once more, we are led to −(a1Ω ′)′ (u) ≥ Ω (ρ (u)) 1 a2 (u) ∫ ∞ u 1 a3 (v) ∫ ∞ v Q (ξ) dξdv. Combining the last inequality with (9), gives −(a1Ω ′)′ (u) ≥ ( Ω′ (ρ (u)) a1 (ρ (u)) a2 (u) ∫ ∞ u 1 a3 (v) ∫ ∞ v Q (ξ) dξdv )∫ ρ(u) u1 1 a1 (v) dv W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 8 of 17 = a1 (ρ (u)) Ω ′ (ρ (u))R1 (u) . Thus, the function g (u) := a1 (u) Ω ′ (u) is a positive solution of the first-order DD in- equality g′ (u) +R1 (u) g (ρ (u)) ≤ 0. Therefore, we conclude that the related DE (7) as well has a positive solution by applying the Philos theorem [41]; this contradicts the assumptions made at the beginning of the theorem. Now, we shall assume that Ω (u) ∈ L+. Since a3 (u) (a2(a1Ω ′)′)′ (u) is decreasing, one gets a2 (u) (a1Ω ′)′ (u) ≥ ∫ u u1 1 a3 (v) a3 (v) ( a2(a1Ω ′)′ )′ (v) dv ≥ a3 (u) ( a2(a1Ω ′)′ )′ (u) ∫ u u1 1 a3 (v) dv, or a2 (u) (a1Ω ′)′ (u) ≥ a3 (u) ( a2(a1Ω ′)′ )′ (u) I3 (u) . Integrating from u1 to u, we get Ω′ (u) ≥ 1 a1 (u) ∫ u u1 1 a2 (ξ) ∫ v u1 1 a3 (v) a3 (v) ( a2(a1Ω ′)′ )′ (v) dvdξ ≥ a3 (u) ( a2(a1Ω ′)′ )′ (u) 1 a1 (u) ∫ u u1 1 a2 (ξ) ∫ v u1 1 a3 (v) dvdξ. Integrating once more, we arrive at Ω (u) ≥ a3 (u) ( a2(a1Ω ′)′ )′ (u) ∫ u u1 1 a1 (v1) ∫ v1 u1 1 a2 (ξ) ∫ v u1 1 a3 (v) dvdξdv1 We see that g (u) = a3 (u) (a2(a1Ω ′)′)′ (u) satisfies Ω (u) ≥ g (u) ∫ u u1 1 a1 (v1) ∫ v1 u1 1 a2 (ξ) ∫ v u1 1 a3 (v) dvdξdv1. Setting the last estimate into( a3(a2(a1Ω ′)′)′ )′ (u) +Q (u) Ω (ρ (u)) ≤ 0. We note that g (u) is a positive solution of the first-order DD inequality g′ (u) +R2 (u) g (ρ (u)) ≤ 0. Therefore, we conclude that the related DE (7) as well has a positive solution by applying the Philos theorem [41]; this contradicts the assumptions made at the beginning of the theorem. This ends the proof. Corollary 1. Let (2) hold. If lim inf u→∞ ∫ u κ(u) Ri (ξ) dξ > 1 e , (11) for i = 1, 2, then (1) is oscillatory. W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 9 of 17 2.2. Improved criteria To be clear, let U1 (u) := γ∑ t=0 ( 2t∏ m=0 h ( κ[m] (u) ))[ 1 h ( κ(2t) (u) ) − 1 ] I123 ( κ[2t] (u) ) I123 (u) q (u) , U2 (u) := γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[ 1 h ( κ(2t) (u) ) − 1 ] I1 ( κ(2t) (u) ) I1 (u) q (u) , S1 (u) := ( 1 a2 (u) ∫ ∞ u 1 a3 (ξ) ∫ ∞ v U1 (v) dvdξ )∫ ρ(u) u1 1 a1 (v) dv, and S2 (u) := U2 (u) ∫ u u1 1 a1 (v1) ∫ v1 u1 1 a2 (ξ) ∫ v u1 1 a3 (v) dvdξdv1, for any integer γ ≥ 0. We prove the following results, which provide details about the behavior of the positive solutions L+. Lemma 4. Assume that x ∈ L+. Then, eventually, ( a3 (u) (a2 (u) (a1 (u) Ω ′ (u))′)′ )′ + U1 (ρ (u)) Ω (ρ (u)) ≤ 0. (12) Proof. Assume that x ∈ L+. We obtain a2 (u) (a1Ω ′)′ (u) ≥ ∫ u u1 1 a3 (v) a3 (v) ( a2(a1Ω ′)′ )′ (v) dv ≥ a3 (u) ( a2(a1Ω ′)′ )′ (u) ∫ u u1 1 a3 (v) dv ≥ a3 (u) ( a2(a1Ω ′)′ )′ (u) I3 (u) , (13) hence, a3 (u) ( a2(a1Ω ′)′ )′ (u) I3 (u)− a2 (u) (a1Ω ′)′ (u) ≤ 0, this implies( a2 (u) (a1Ω ′)′ (u) I3 (u) )′ = I3 (u) a3 (u) (a2(a1Ω ′)′)′ (u)− a2 (u) (a1Ω ′)′ (u) a3 (u) I23 (u) = 1 a3 (u) I23 (u) [ I3 (u) a3 (u) ( a2(a1Ω ′)′ )′ (u)− a2 (u) (a1Ω ′)′ (u) ] ≤ 0. (14) Applying this information, we determine that a1 (u) Ω ′ (u) ≥ ∫ u u1 I3 (v) a2 (v) (a1Ω ′)′ (v) a2 (v) I3 (v) dv ≥ a2 (u) (a1Ω ′)′ (u) I3 (u) I23 (u) , W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 10 of 17 or Ω′ (u) ≥ 1 a1 (u) a3 (u) ( a2(a1Ω ′)′ )′ (u) I23 (u) . (15) Yields,( a1 (u) Ω ′ (u) I23 (u) )′ = I23 (u) a2 (u) (a1Ω ′)′ (u)− I3 (u) (a1Ω ′) (u) a2 (u) I223 (u) = 1 a2 (u) I223 (u) [ I23 (u) a2 (u) (a1Ω ′)′ (u)− I3 (u) (a1Ω ′) (u) ] ≤ 0. Hence, Ω (t) ≥ ∫ u u1 I23 (v) a1 (v) Ω ′ (v) a1 (v) I23 (v) dv ≥ a1 (u) Ω ′ (u) I23 (u) ∫ u u1 I23 (v) 1 a1 (v) dv ≥ a2 (u) (a1Ω ′)′ (u) I3 (u) I123 (u) , (16) we get, ( Ω (u) I123 (u) )′ = I123 (u) a1 (u) Ω ′ (u)− I23 (u) Ω (u) a1 (u) I2123 (u) = [ I123 (u) a1 (u) Ω ′ (u)− I23 (u) Ω (u) ] 1 a1 (u) I2123 (u) ≤ 0. (17) Now, we determine that Ω ( κ(2t) (u) ) ≥ Ω ( κ(2t+1) (u) ) , based on the information that κ(2t+1) (u) ≤ κ(2t) (u) ≤ u and Ω′ (u) > 0. Which, along with Lemma 2, implies that x (u) > γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[Ω (κ(2t) (u)) h ( κ(2t) (u) ) − Ω ( κ(2t+1) (u) )] , ≥ γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[ 1 h ( κ(2t) (u) ) − 1 ] Ω ( κ(2t) (u) ) . (18) Moreover, as (Ω (u) /I123 (u)) ′ ≤ 0 and κ(2t) (u) ≤ u, we have Ω ( κ(2t) (u) ) I123 ( κ(2t) (u) ) ≥ Ω (u) I123 (u) , and Ω ( κ(2t) (u) ) ≥ I123 ( κ(2t) (u) ) I123 (u) Ω (u) . W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 11 of 17 Thus, by applying the subsequent inequality and putting into (18), yields x (u) > Ω (u) γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[ 1 h ( κ(2t) (u) ) − 1 ] I123 ( κ(2t) (u) ) I123 (u) , this produces (12), when combined with (1). This ends the proof. The results that we present below shed light on the behavior of the positive solutions L−. Lemma 5. Assume that x ∈ L−. Then, eventually,( a3 (u) (a2 (u) (a1 (u) Ω ′ (u))′)′ )′ + U2 (ρ (u)) Ω (ρ (u)) ≤ 0. (19) Proof. Assume that x ∈ L−. We derive, for u ≥ u1, Ω (u) ≥ ∫ u u1 1 a1 (ξ) a1 (ξ) Ω ′ (ξ) dξ ≥ a1 (u) Ω ′ (u) I1 (u) . Then, ( Ω (u) I1 (u) )′ = a−1 1 (u) I21 (u) [ a1 (u) Ω ′ (u) I1 (u)− Ω (u) ] ≤ 0, which, with the fact that κ(2t) (u) ≤ u, gives Ω ( κ(2t) (u) ) ≥ I1 ( κ(2t) (u) ) I1 (u) Ω (u) . (20) Using the facts Ω′ (u) > 0 and the inequality (20), relation (6) reduces to x (u) > γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[ 1 h ( κ(2t) (u) ) − 1 ] Ω ( κ(2t) (u) ) > Ω (u) γ∑ t=0 ( 2t∏ m=0 h ( κ(m) (u) ))[ 1 h ( κ(2t) (u) ) − 1 ] I1 ( κ(2t) (u) ) I1 (u) . Combining this inequality with (1), we get (19). This ends the proof. Theorem 2. Suppose that the two first-order DDEs g′ (u) + S1 (u) g (ρ (u)) = 0 (21) and g′ (u) + S2 (u) g (ρ (u)) = 0 (22) are oscillatory, then equation (1) is oscillatory. W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 12 of 17 Proof. Assume that (1) has an eventually positive solution at x. One solution to Lemma 3 meets any of the the possibilities of L+ or L−. By replacing the relationship (10) by (12) in the case L+ and by (19) in the case L−. We get (21) and (22). This ends the proof. Example 1. Consider (x (u) + h0x (κ0u)) (4) + q0 u4 x (ρ0u) = 0, (23) where u > 0, h0 ∈ [0, 1) , q0 > 0, ρ, κ ∈ (0, 1), and a1 (u) = a2 (u) = a3 (u) = 1. For our equation it is easy to verify that Q (u) := (1− h (ρ (u))) q (u) = q0 (1− h0) 1 u4 , and R1 (u) = ( 1 a2 (u) ∫ ∞ u 1 a3 (ξ) ∫ ∞ ξ Q (v) dvdξ )∫ ρ(u) u1 1 a1 (v) dv = (∫ ∞ u ∫ ∞ ξ q0 (1− h0) 1 v4 dvdξ )∫ ρ0u u1 dv = q0 (1− h0) (∫ ∞ u ∫ ∞ ξ 1 v4 dvdξ ) ρ0u = q0 (1− h0) (∫ ∞ u 1 3ξ3 dξ ) ρ0u = q0 (1− h0) ( 1 6u2 ) ρ0u = ρ0q0 (1− h0) 6 1 u . Thus, condition (11) for i = 1 becomes lim inf u→∞ ∫ u κ(u) R1 (ξ) dξ = lim inf u→∞ ∫ u κ0u ρ0q0 (1− h0) 6 1 ξ dξ = ρ0q0 (1− h0) 6 ln 1 κ0 , which is satisfied when ρ0q0 (1− h0) 6 ln 1 κ0 > 1 e . (24) Similarly, we have R2 (u) = Q (u) ∫ u u1 1 a1 (v1) ∫ v1 u1 1 a2 (ξ) ∫ ξ u1 1 a3 (v) dvdξdv1 W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 13 of 17 = q0 (1− h0) 1 u4 ∫ u u1 ∫ v1 u1 ∫ ξ u1 dvdξdv1 = q0 (1− h0) 1 u4 u3 6 = q0 (1− h0) 6 1 u . Thus, condition (11) for i = 2 becomes lim inf u→∞ ∫ u κ(u) R2 (ξ) dξ = lim inf u→∞ ∫ u κ0u q0 (1− h0) 6 1 ξ dξ = q0 (1− h0) 6 ln 1 κ0 , which is satisfied when q0 (1− h0) 6 ln 1 κ0 > 1 e . (25) Using Corollary 1, we conclude that equation (23) is oscillatory if both conditions (24) and (25) are satisfied. 3. conclusion In the canonical case, this work focuses on a fundamental matter, namely understand- ing the relationships between the corresponding function Ω (u) and the solution x (u) for the fourth-order NDDE. We have presented oscillation theories through which we have es- tablished oscillation criteria that guarantee the oscillation of the solutions to the equation under study., first after using the classical relation that links the solution to its corre- sponding function in the two cases (L− and L+) for positive solutions of our equation. Second, we have improved this classical relation in the two cases (L− and L+). Notably, the main distinction between L+ and L− is the difference in the second derivative of the corresponding function Ω (u); however, this change affects several of the monotonic and asymptotic characteristics. The results of this study complement many of previously published findings in the liter- ature. To our knowledge, this equation has not been studied by many researchers, so it would be a good idea to apply these results to non-linear higher-order NDEs in the future. Competing interests There are no competing interests W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 14 of 17 References [1] Ravi P. Agarwal, Martin Bohner, Tongxing Li, and Chenghui Zhang. A new approach in the study of oscillatory behavior of even-order neutral delay differential equations. Applied Mathematics and Computation, 225:787–794, November 2013. [2] Ravi P. Agarwal, Said R. Grace, and Jelena V. Manojlovic. Oscillation criteria for certain fourth order nonlinear functional differential equations. Mathematical and Computer Modelling, 44(1–2):163–187, March 2006. [3] Ravi P. Agarwal, Said R. Grace, and Donal O’Regan. Oscillation Theory for Second Order Linear, Half-Linear, Superlinear and Sublinear Dynamic Equations. Springer eBooks, January 2002. [4] Ravi P. Agarwal, Said R. Grace, and Donal O’Regan. Oscillation theory for second order dynamic equations. CRC Press eBooks, November 2002. [5] Barakah Almarri, Belal Batiha, Omar Bazighifan, and Fahd Masood. Third-order neutral differential equations with non-canonical forms: Novel oscillation theorems. Axioms, 13(11):755, October 2024. [6] Barakah Almarri, S. Janaki, V. Ganesan, Ali Hasan Ali, Kamsing Nonlaopon, and Omar Bazighifan. Novel oscillation theorems and symmetric properties of nonlinear delay differential equations of fourth-order with a middle term. Symmetry, 14(3):585, March 2022. [7] Ghada AlNemer, Waed Muhsin, Osama Moaaz, and Elmetwally M. Elabbasy. On the positive decreasing solutions of half-linear delay differential equations of even order. Mathematics, 11(6):1282, March 2023. [8] Saad Althobati, Jehad Alzabut, and Omar Bazighifan. Non-linear neutral differential equations with damping: Oscillation of solutions. Symmetry, 13(2):285, February 2021. [9] Omar Bazighifan, Osama Moaaz, Rami El-Nabulsi, and Ali Muhib. Some new os- cillation results for fourth-order neutral differential equations with delay argument. Symmetry, 12(8):1248, July 2020. [10] Omar Bazighifan, Marianna Ruggieri, and Andrea Scapellato. An improved criterion for the oscillation of fourth-order differential equations. Mathematics, 8(4):610, April 2020. [11] Martin Bohner, Said R. Grace, and Irena Jadlovska. Sharp results for oscillation of second-order neutral delay differential equations. Electronic Journal of Qualitative Theory of Differential Equations, 2023(4):1–23, January 2023. [12] M. Braun. Differential Equations and Their Applications: An Introduction to Applied Mathematics. Springer Science & Business Media, June 2013. [13] CV Coffman and JSW Wong. Oscillation and nonoscillation theorems for second order. Funkcialaj Ekvacioj, 15:119–130, 1972. [14] Elmetwally M. Elabbasy, Amany Nabih, Taher A. Nofal, Wedad R. Alharbi, and Osama Moaaz. Neutral differential equations with noncanonical operator: Oscillation behavior of solutions. AIMS Mathematics, 6(4):3272–3287, January 2021. [15] L. H. Erbe, Qingkai Kong, and B. G. Zhang. Oscillation Theory for Functional W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 15 of 17 Differential Equations. Routledge eBooks, October 2017. [16] William Benjamin Fite. Concerning the zeros of the solutions of certain differen- tial equations. Transactions of the American Mathematical Society, 19(4):341–352, January 1918. [17] Said R. Grace, Jozef Džurina, Irena Jadlovska, and Tongxing Li. On the oscillation of fourth-order delay differential equations. Advances in Difference Equations, 2019(1), March 2019. [18] Jack K Hale and Sjoerd M Verduyn Lunel. Introduction to functional differential equations, volume 99. Springer Science & Business Media, 2013. [19] Muhammad Hamid, Muhammad Usman, Rizwan Ul Haq, and Zhenfu Tian. A spec- tral approach to analyze the nonlinear oscillatory fractional-order differential equa- tions. Chaos Solitons & Fractals, 146:110921, April 2021. [20] Mustafa Hasanbulli and Yuri V. Rogovchenko. Oscillation criteria for second or- der nonlinear neutral differential equations. Applied Mathematics and Computation, 215(12):4392–4399, January 2010. [21] Ahmed M. Hassan, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, and Samy E. Affan. Enhanced oscillation criteria for non-canonical second-order advanced dynamic equations on time scales. Symmetry, 16(11):1457, November 2024. [22] Taher S. Hassan, Rami Ahmad El-Nabulsi, Naveed Iqbal, and Amir Abdel Menaem. New criteria for oscillation of advanced noncanonical nonlinear dynamic equations. Mathematics, 12(6):824, March 2024. [23] Irena Jadlovska. New criteria for sharp oscillation of second-order neutral delay differential equations. Mathematics, 9(17):2089, August 2021. [24] Irena Jadlovska, Jozef Džurina, John R. Graef, and Said R. Grace. Sharp oscillation theorem for fourth-order linear delay differential equations. Journal of Inequalities and Applications, 2022(1), September 2022. [25] Takaŝi Kusano and Manabu Naito. Nonlinear oscillation of fourth order differential equations. Canadian Journal of Mathematics, 28(4):840–852, August 1976. [26] G. Ladas, V. Lakshmikantham, and J.S. Papadakis. Oscillations of higher-order retarded differential equations generated by the retarded argument, pages 219–231. Elsevier eBooks, January 1972. [27] Gangaram S Ladde, Vangipuram Lakshmikantham, and Bing-Gen Zhang. Oscillation theory of differential equations with deviating arguments. M. Dekker New York, NY, USA, 1987. [28] Tongxing Li and Yuriy V. Rogovchenko. Oscillation criteria for even-order neutral differential equations. Applied Mathematics Letters, 61:35–41, May 2016. [29] Fahd Masood, Osama Moaaz, Shyam Sundar Santra, U. Fernandez-Gamiz, and Hamdy A. El-Metwally. Oscillation theorems for fourth-order quasi-linear delay dif- ferential equations. AIMS Mathematics, 8(7):16291–16307, January 2023. [30] Osama Moaaz and Wedad Albalawi. Differential equations of the neutral delay type: More efficient conditions for oscillation. AIMS Mathematics, 8(6):12729–12750, Jan- uary 2023. [31] Osama Moaaz, Clemente Cesarano, and Barakah Almarri. An improved relationship W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 16 of 17 between the solution and its corresponding function in fourth-order neutral differential equations and its applications. Mathematics, 11(7):1708, April 2023. [32] Osama Moaaz, Ioannis Dassios, Waad Muhsin, and Ali Muhib. Oscillation theory for non-linear neutral delay differential equations of third order. Applied Sciences, 10(14):4855, July 2020. [33] Osama Moaaz, Rami Ahmad El-Nabulsi, Waad Muhsin, and Omar Bazighifan. Im- proved oscillation criteria for 2nd-order neutral differential equations with distributed deviating arguments. Mathematics, 8(5):849, May 2020. [34] Ali Muhib, Osama Moaaz, Clemente Cesarano, and Sameh S. Askar. New conditions for testing the oscillation of fourth-order differential equations with several delays. Symmetry, 14(5):1068, May 2022. [35] Waed Muhsin, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, and Elmet- wally M. Elabbasy. Delay differential equations with several sublinear neutral terms: Investigation of oscillatory behavior. Symmetry, 15(12):2105, November 2023. [36] Amany Nabih, Osama Moaaz, Ghada AlNemer, and Elmetwally M. Elabbasy. New conditions for testing the asymptotic and oscillatory behavior of solutions of neutral differential equations of the fourth order. Axioms, 12(2):219, February 2023. [37] Zeev Nehari. On a class of nonlinear second-order differential equations. Transactions of the American Mathematical Society, 95(1):101–123, January 1960. [38] MN O¡ guztöreli and RB Stein. An analysis of oscillations in neuro-muscular systems. Journal of Mathematical Biology, 2(2):87–105, 1975. [39] Najiyah Omar, Osama Moaaz, Ghada AlNemer, and Elmetwally M. Elabbasy. New results on the oscillation of solutions of third-order differential equations with multiple delays. Symmetry, 15(10):1920, October 2023. [40] C Philos. A new criterion for the oscillatory and asymptotic behavior of delay differ- ential equations. Sci., Ser. Sci. Math, January 1981. [41] Ch G Philos. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Archiv der Mathematik, 36:168–178, 1981. [42] Hend Salah, Osama Moaaz, Sameh S. Askar, Ahmad M. Alshamrani, and Elmet- wally M. Elabbasy. Optimizing the monotonic properties of fourth-order neutral differential equations and their applications. Symmetry, 15(9):1744, September 2023. [43] Shyam Sundar Santra, Apurba Ghosh, and Ioannis Dassios. Second-order impulsive differential systems with mixed delays: Oscillation theorems. Mathematical Methods in the Applied Sciences, 45(18):12184–12195, October 2021. [44] Shyam Sundar Santra, Khaled Mohamed Khedher, and Shao-Wen Yao. New aspects for oscillation of differential systems with mixed delays and impulses. Symmetry, 13(5):780, May 2021. [45] Ercan Tunç and Orhan Dzdemir. Comparison theorems on the oscillation of even order nonlinear mixed neutral differential equations. Mathematical Methods in the Applied Sciences, 46(1):631–640, June 2022. [46] Guojing Xing, Tongxing Li, and Chenghui Zhang. Oscillation of higher-order quasi- linear neutral differential equations. Advances in Difference Equations, 2011(1), Oc- tober 2011. W. Muhsin et al. / Eur. J. Pure Appl. Math, 18 (1) (2025), 5729 17 of 17 [47] Chenghui Zhang, Tongxing Li, Bo Sun, and E. Thandapani. On the oscillation of higher-order half-linear delay differential equations. Applied Mathematics Letters, 24(9):1618–1621, April 2011. [48] Quanxin Zhang, Jurang Yan, and Li Gao. Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients. Computers & Mathematics With Applications, 59(1):426–430, July 2009.