EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 6545 ISSN 1307-5543 – ejpam.com Published by New York Business Global Investigation of the Asymptotic and Monotonic Properties of Solutions of Functional Differential Equations with Multiple Delays and a Damping Term Sarah Aloraini1, Ahmed S. Almohaimeed1,∗, Osama Moaaz1 1 Department of Mathematics, College of Science, Qassim University, P. O. Box 6644, Buraydah, 51452 Saudi Arabia Abstract. Our objective in this work is to establish new asymptotic and monotonic properties of second-order differential equations involving multiple delay terms and a damping component. These properties are instrumental in advancing the understanding of the qualitative behavior of their solutions. This study employs a refined comparison technique utilizing first-order differential equations. The resulting criteria broaden and enhance previously established findings in the litera- ture. A natural progression and forthcoming challenge of this research is examining the oscillatory behavior and asymptotic characteristics of solutions for higher-order neutral delay differential equa- tions, where the interplay between delay and neutral components becomes progressively intricate. 2020 Mathematics Subject Classifications: 34C10, 34K11 Key Words and Phrases: Differential equations, oscillatory characteristics, deviating argument, damping term 1. Introduction Second-order differential equations (DEs) are used in engineering as well as science to examine a variety of events. Particularly, second-order nonlinear equations play a signif- icant role in this field. The nonlinear terms in these equations grow more quickly than the linear terms when the dependent variable rises. There are many applications for these equations, such as population dynamics models, nonlinear oscillatory systems, and me- chanical systems with nonlinear damping [1–4]. Since second-order nonlinear differential equations display a wide range of phenomena, they are an interesting topic for applied mathematical study (see [5–8]). The neutral delay differential equation (NDDE) is a type of delay differential equation (DDE) in which the highest-order derivative of the unknown function occurs both in ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.6545 Email addresses: 441212392@qu.edu.sa (S. Aloraini), ahs.almohaimeed@qu.edu.sa (A. S. Almohaimeed), o.refaei@qu.edu.sa, o_moaaz@mans.edu.eg (O. Moaaz) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 2 of 15 delayed and non-delayed forms. This class of equations commonly appears in the dynamic modeling of various real-world systems, including pharmacokinetics, frequency-modulated processes, bursting rhythm phenomena in biological and medical sciences, and optimal control models within economic theory [9, 10]. In particular, second-order NDDE hold significant interest in fields such as robotics, where they are employed in the development of bipedal locomotion systems, and in biology, where they contribute to understanding the mechanisms underlying human postural balance [11, 12]. Within the present study, we analyze the oscillatory characteristics of solutions result- ing from the following second-order NDDEs with multiple delays and a damping term: ( r (s) ( z′ (s) )κ)′ + h (s) ( z′ (s) )κ + n∑ i=1 qi (s)x κ (ηi (s)) = 0, s ≥ s0, (1) in which z (s) := x (s) + p (s)x (τ (s)) . Our investigation proceeds under the following hypotheses: (H1) κ ≥ 1 is a fraction formed by dividing odd positive integers, and n ∈ Z+; (H2) r ∈ C1 ([s0,∞),R+) , and∫ ∞ s0 1 r1/κ (u) exp ( −1 κ ∫ u h (ρ) r (ρ) dρ ) du = ∞; (2) (H3) p, h, qi ∈ C ([s0,∞), [0,∞)) , for i = 1, 2, ..., n, 0 ≤ p(s) ≤ p0 < 1, where p0 is a constant, and qi does not vanish identically on any half-line of the form [t∗,∞), t∗ ≥ t0; (H4) τ, ηi ∈ C([s0,∞),R), τ(s) ≤ s, ηi(s) < s, lims→∞ τ (s) = ∞, and lims→∞ ηi (s) = ∞, for i = 1, 2, ..., n. Based on a solution to (1), we consider a real-valued function x ∈ C ([s∗,∞) ,R) , s∗ ≥ s0, where z, r (z′)κ ∈ C1 ([s∗,∞) ,R), and x fulfills (1) on [s∗,∞). Our focus is confined to solutions x to (1) that meet sup {|x (s)| : s ≥ s1} > 0 for all s1 ≥ s∗. A solution x to (1) is called oscillatory if it possesses zeros that are arbitrarily large; otherwise, it is referred to as nonoscillatory. Condition (2) is called the canonical condition. It has a fundamental effect on the study because it ensures that there are no positive solutions with corresponding decreasing functions. Remark 1. All the relationships and inequalities in the study are eventually satisfied, i.e. for t ≥ t∗ where t∗ ≥ t0. S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 3 of 15 2. Literature review Since NDDE appears in the modeling of many systems in our lives, see [13–15]. The oscillation of DEs has been a major focus of interest among researchers. For example, [16, 17] was interested in studying first-order NDDEs but [18–21] was interested in the second-order equations. Conversely, the even/nth order equations were studied in works [22–25]. In 1978, Brands [26] proved that, in the case of bounded delays, the solutions of x′′ (s) + q (s)x (s− η) = 0, are oscillatory if the solutions of x′′ (s) + q (s)x (s) = 0 are also oscillatory. Then, based on the well-known results obtained by Ladas et al. [27] for the oscillation of the first-order DDE, Wei [28] provided the famous oscillation criteria lim sup s→∞ ∫ s η(s) η (ρ) q (ρ) dρ > 1 or lim inf s→∞ ∫ s η(s) η (ρ) q (ρ) dρ > 1 e , for the second order DDE x′′ (s) + q (s)x (η (s)) = 0. Then Koplatadze et al. [29] improved these criteria to the form lim sup s→∞ ∫ s η(s) q (ρ) [ η (ρ) + ∫ η(ρ) s1 νη (ν) q (ν) dν ] dρ > 1, for η′ ≥ 0, or lim inf s→∞ ∫ s η(s) q (ρ) [ η (ρ) + ∫ η(ρ) s1 νη (ν) q (ν) dν ] dρ > 1 e . Investigating the oscillation behavior by using the Riccati technique was a very useful tool in literature, since this technique aims to diminish the order of the DE to the first-order DE and allows us to benefit from the wide range of oscillation research regarding these first-order ones. Dzurina and Stavroulakis [30] used the Riccati technique to provide the oscillation criteria∫ ∞ Rκ (η (ρ)) q (ρ)− κη′ (ρ) 4L1R (η (ρ)) r1/κ (η (ρ)) dρ = ∞, (3) where κ ≥ 1, η′ > 0, and 0 < L1 < 1, for the half linear DDE( r (s) ( x′ (s) )κ)′ + q (s)xκ (η (s)) = 0. (4) S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 4 of 15 After that, Sun and Meng [31] developed (3) to∫ ∞ Rκ (η (ρ)) q (ρ)− ( κ κ+ 1 )κ+1 η′ (ρ) R (η (ρ)) r1/κ (η (ρ)) dρ = ∞. One may find some oscillation results for the noncanonical case of (4), i.e.,∫ ∞ s0 r−1/κ (ρ) dρ < ∞, in the works of Chatzarakis et al. [32, 33]. Wong [34] found the conditions that guarantee the oscillation of solutions to the NDDE: (x (s) + px (s− τ))′′ + q (s) f (x (s− τ)) = 0, where the delays and the neutral coefficient τ, η are constants. Xu and Meng [35] gener- alized the previous criteria to the NDDE( r (s) ( [x (s) + p (s)x (τ (s))]′ )κ)′ + q (s)xκ (η (s)) = 0. (5) In 2018, Grace et al. [36] established criteria that are sufficient for the solutions of (5) to oscillate when 0 ≤ p < 1, and showed that the oscillation can be ensured provided that at least one of the next conditions is satisfied lim sup s→∞ ∫ s η(s) Q (ρ) R̃κ (η (ρ)) dρ > 1, for η′ > 0, or lim inf s→∞ ∫ s η(s) Q (ρ) R̃κ (η (ρ)) dρ > 1 e . On the other hand, Baculikova and Džurina in [37, 38] investigated the second-order Emden–Fowler NDDE (1). Employing the comparison theorem in order to formulate adequate criteria for the oscillation of the solutions to (1) when κ = β = 1, 0 ≤ p < ∞, and ∫ ∞ s0 r−1/κ (ρ) dρ = ∞. Bohner et al. [39] considered the noncanonical case of (1) for κ = β and 0 ≤ p < 1. Then, Baculikova and Dzurina [40] extended the previous results for κ ≥ β and η, τ (s) > 0 of (1). Liu et al. in [41] studied (1) by applying the Riccati technique under the conditions κ ≥ β, r′ (s) > 0, and η′ (s) > 0, they provided new oscillation criteria for (1) for both canonical and noncanonical cases. S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 5 of 15 3. Asymptotic and Monotonic Properties For convenience, we define the following notations: ϱ (s) := exp (∫ s h (ρ) r (ρ) dρ ) , η (s) := min {ηi (s) , i = 1, 2, ..., n} , Q (s) := ϱ (s) n∑ i=1 qi (s) (1− p (ηi (s))) κ , R (s) := ∫ s s1 1 ϱ1/κ (ρ) r1/κ (ρ) dρ, The following lemmas provide some asymptotic and monotonic properties of the cor- responding function to positive solutions. They also provide the necessary relationships for proving oscillation theorems. Lemma 1. Grant that x is a positive solution to (1). Hence, eventually, z(s) > 0, z′(s) > 0, (r (s) ( z′ (s) )κ )′ ≤ 0, (6) and [ ϱ (s) r (s) ( z′ (s) )κ]′ + ϱ (s) n∑ i=1 qi (s)x κ (ηi (s)) ≤ 0. (7) Proof. Presume that x is a positive solution to (1). Equation (1) transforms into 1 ϱ (s) [ ϱ (s) r (s) ( z′ (s) )κ]′ = ( r (s) ( z′ (s) )κ)′ + h (s) ( z′ (s) )κ = − n∑ i=1 qi (s)x κ (ηi (s)) , or [ ϱ (s) r (s) ( z′ (s) )κ]′ ≤ −ϱ (s) n∑ i=1 qi (s)x κ (ηi (s)) ≤ 0. So, r (z′)κ is a non-increasing function, and either z′ (s) > 0 or z′ (s) < 0, eventually. Grant that z′ (s) < 0 for s ≥ s1. Then, there is a K1 > 0 such that ϱ (s) r (s) ( z′ (s) )κ ≤ −K1, and thus, z′ (s) ≤ −K 1/κ 1 1 ϱ1/κ (s) r1/κ (s) . (8) S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 6 of 15 Integrating (8) over [s1, s], we arrive at z (s)− z (s1) ≤ −K 1/κ 1 ∫ s s1 1 ϱ1/κ (ρ) r1/κ (ρ) dρ. Hence, using (2), we obtain lims→∞ z (s) = −∞, a contradiction. So, z′ (s) > 0, eventually. Lemma 2. Presume that x is a positive solution to (1). Therefore the following relations hold eventually: (P1) x (s) ≥ (1− p (s)) z (s) ; (P2) [ϱ (s) r (s) (z ′ (s))κ] ′ +Q (s) zκ (η (s)) ≤ 0; (P3) z (s) ≥ ϱ1/κ (s) r1/κ (s) z′ (s)R (s) ; (P4) z (s) /R (s) is non-increasing function. Proof. Grant that x is a positive solution to (1). Based on the definition of z, we get x (s) ≥ z (s)− p (s) z (τ (s)) ≥ (1− p (s)) z (s) . Combined this with (7), we obtain [ ϱ (s) r (s) ( z′ (s) )κ]′ ≤ −ϱ (s) n∑ i=1 qi (s) (1− p (ηi (s))) κ zκ (ηi (s)) ≤ −zκ (η (s)) ϱ (s) n∑ i=1 qi (s) (1− p (ηi (s))) κ = −Q (s) zκ (η (s)) . Now, we have z (s) ≥ ∫ s s1 1 ϱ1/κ (ρ) r1/κ (ρ) [ ϱ1/κ (ρ) r1/κ (ρ) z′ (ρ) ] dρ ≥ ϱ1/κ (s) r1/κ (s) z′ (s)R (s) , and hence ( z (s) R (s) )′ = R (s) z′ (s)− ϱ−1/κ (s) r−1/κ (s) z (s) R2 (s) ≤ 0. This completes the proof. Lemma 3. Grant that x is a positive solution to (1), provide∫ ∞ s1 Rκ (η (ρ))Q (ρ) dρ = ∞. (9) Then lim s→∞ z (s) R (s) = 0. S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 7 of 15 Proof. Let’s say that x is a positive solution to (1). From (P3), (P4), we get z (η (s)) R (η (s)) ≥ z (s) R (s) ≥ ϱ1/κ (s) r1/κ (s) z′ (s) . Using (P2), we have −Q (s) zκ (η (s)) ≥ [ ϱ (s) r (s) ( z′ (s) )κ]′ = [( ϱ1/κ (s) r1/κ (s) z′ (s) )κ]′ = κ ( ϱ1/κ (s) r1/κ (s) z′ (s) )κ−1 [ ϱ1/κ (s) r1/κ (s) z′ (s) ]′ ≥ κ ( z (η (s)) R (η (s)) )κ−1 [ ϱ1/κ (s) r1/κ (s) z′ (s) ]′ , and so [ ϱ1/κ (s) r1/κ (s) z′ (s) ]′ ≤ −1 κ z1−κ (η (s)) R1−κ (η (s)) Q (s) zκ (η (s)) = −1 κ Rκ−1 (η (s))Q (s) z (η (s)) . (10) Next, we see that[ ϱ1/κ (s) r1/κ (s)R2 (s) ( z (s) R (s) )′]′ = [ R (s) ϱ1/κ (s) r1/κ (s) z′ (s)− z (s) ]′ = R (s) [ ϱ1/κ (s) r1/κ (s) z′ (s) ]′ ≤ −1 κ R (s)Rκ−1 (η (s))Q (s) z (η (s)) . (11) Integration of (11) for s from s1 to s yields ϱ1/κ (s) r1/κ (s)R2 (s) ( z (s) R (s) )′ ≤ −L− 1 κ ∫ s s1 R (ρ)Rκ−1 (η (ρ))Q (ρ) z (η (ρ)) dρ (12) where L := −ϱ1/κ (s1) r 1/κ (s1)R 2 (s1) ( z (s) R (s) )′∣∣∣∣ s=s1 > 0. Let w := z/R. From (6) and (12), w is positive and decreasing. So, lim s→∞ w (s) = w0 ≥ 0. Assume the contrary that w0 > 0. Hence, (12) becomes w′ (s) ≤ − 1 κ 1 ϱ1/κ (s) r1/κ (s)R2 (s) ∫ s s1 R (ρ)Rκ−1 (η (ρ))Q (ρ) z (η (ρ)) dρ. (13) S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 8 of 15 Integration of (13) over [s1,∞) gets := z (η (ρ)) w0 − w (s1) ≤ −1 κ ∫ ∞ s1 1 ϱ1/κ (u) r1/κ (u)R2 (u) ∫ u s1 R (ρ)Rκ−1 (η (ρ))Q (ρ) z (η (ρ)) dρdu = −1 κ ∫ ∞ s1 1 ϱ1/κ (u) r1/κ (u)R2 (u) ∫ u s1 R (ρ)Rκ (η (ρ))Q (ρ)w (η (ρ)) dρdu ≤ −w0 κ ∫ ∞ s1 1 ϱ1/κ (u) r1/κ (u)R2 (u) ∫ u s1 R (ρ)Rκ (η (ρ))Q (ρ) dρdu = −w0 κ ∫ ∞ s1 Rκ (η (ρ))Q (ρ) dρ, which contradicts (9), and thus w0 = 0. 4. Oscillation Theorems The following theorem uses a comparison approach with a first-order equation to study the oscillatory behavior of solutions to equation (1). Theorem 1. Assume that (9) holds. If the DDE w′ (s) + ( 1 κϱ1/κ (s) r1/κ (s)R2 (s) ∫ s s0 R (ρ)Rκ (η (ρ))Q (ρ) dρ ) w (η (s)) = 0 (14) is oscillatory, then equation (1) oscillates. Proof. Assume the contrary that x is a positive solution to (1). Since in the demon- stration of Lemma 3, we have that w = z/R satisfies (12). Then, w′ (s) ≤ −1 ϱ1/κ (s) r1/κ (s)R2 (s) [ L+ 1 κ ∫ s s1 R (ρ)Rκ (η (ρ))Q (ρ)w (η (ρ)) dρ ] ≤ −1 ϱ1/κ (s) r1/κ (s)R2 (s) [ L+ 1 κ w (η (s)) ∫ s s1 R (ρ)Rκ (η (ρ))Q (ρ) dρ ] = −1 ϱ1/κ (s) r1/κ (s)R2 (s) [ L− 1 κ w (η (s)) ∫ s1 s0 R (ρ)Rκ (η (ρ))Q (ρ) dρ + 1 κ w (η (s)) ∫ s s0 R (ρ)Rκ (η (ρ))Q (ρ) dρ ] . (15) Since w → 0 as s → ∞, there is a s2 ≥ s1 such that L− 1 κ w (η (s)) ∫ s1 s0 R (ρ)Rκ (η (ρ))Q (ρ) dρ > 0 for s ≥ s2. (16) S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 9 of 15 Combining (15) and (16), we obtain w′ (s) + ( 1 κϱ1/κ (s) r1/κ (s)R2 (s) ∫ s s0 R (ρ)Rκ (η (ρ))Q (ρ) dρ ) w (η (s)) ≤ 0. (17) Now, we have w is a positive solution to (17). According to Theorem 1 in [42], equation (14) holds as well a positive solution, a contradiction. Using the oscillation criteria of solutions of first-order differential equations, which are known in the literature, we present the following corollary: Corollary 1. Assume that (9) holds. If lim inf s→∞ ∫ s η(s) 1 ϱ1/κ (u) r1/κ (u)R2 (u) ∫ u s0 R (ρ)Rκ (η (ρ))Q (ρ) dρdu > κ e , (18) hence equation (1) oscillates. Proof. Using Theorem 2 in [43], condition (18) implies oscillation of equation (14). Thus, from Theorem 1, equation (1) oscillates. Using a different approach, we can establish another comparison criterion by relating the oscillation of the solutions of equation (1) to a first-order equation, as in the following theorem: Theorem 2. Equation (1) oscillates if condition (9) holds and the DDE H ′ (s) + 1 κ Rκ (η (s))Q (s)H (η (s)) ≤ 0 (19) is oscillatory. Proof. Assume the contrary that x is a positive solution to (1). As in the demonstration of Lemma 3, we arrive at (10). Using (P3), (10) reduces to[ ϱ1/κ (s) r1/κ (s) z′ (s) ]′ ≤ − 1 κ Rκ (η (s))Q (s) ϱ1/κ (η (s)) r1/κ (η (s)) z′ (η (s)) . Then, H := ϱ1/κr1/κz′ is a positive solution to H ′ (s) + 1 κ Rκ (η (s))Q (s)H (η (s)) ≤ 0. According to Theorem 1 in [42], equation (19) holds as well a positive solution, a contra- diction. Corollary 2. If lim inf s→∞ ∫ s η(s) Rκ (η (ρ))Q (ρ) > κ e , (20) then equation (1) oscillates. S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 10 of 15 Proof. First, we note that (9) is necessary for the validity of (20). Using Theorem 2 in [43], condition (20) implies oscillation of equation (19). Thus, from Theorem 2, equation (1) oscillates. Example 1. Consider the NDDE( x (s) + 1 2 x (λs) )′′ + ℓ s ( x (s) + 1 2 x (λs) )′ + n∑ i=1 ki s2 x (µis) = 0, (21) where ℓ ∈ [0, 1), λ, µi ∈ (0, 1], and ki > 0 for i = 1, 2, ..., n. Then, ϱ (s) = sℓ, R (s) = 1 1− ℓ s1−ℓ, and Q (s) = 1 2 sℓ n∑ i=1 ki s2 . Assume that µ0 := min {µi, i = 1, 2, ..., n} , and K := n∑ i=1 ki. Then, using Corollary 2, equation (21) oscillates if 1 1− ℓ µ1−ℓ 0 K ln 1 µ0 > 2 e . 5. Numerical Solutions In this section, we include figures that show the numerical solution of the following example Example 2. Consider the NDDE d ds ( (s+ 1) d ds [x(s) + 0.4x(s− 1)] ) + 1 s+ 1 d ds [x(s) + 0.4x(s− 1)] + 2x(s− 2) + 2x(s− 3) = 0 (22) with history x(s) = cos(s) for s ≤ 3. We solve equation (22) numerically by transforming it into a first-order system based on defining x1(s) = x(s), x2(s) = x′1(s) in the following form x′1(s) = x2(s), S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 11 of 15 x′2(s) = −0.4x′2(s− 1)− s+ 2 (s+ 1)2 (x2 (s) + 0.4x (s− 1)) (23) − 2 s+ 1 (x1 (s− 2) + x1 (s− 3)) . We then use Matlab 2024 to find the numerical solutions of the transformed system (23). The numerical solution of the given equation (22) is shown in Figure 1. The oscillatory behavior is present and can be seen in the solutions. The numerical solution shown in this figure is consistent with the analytical investigations discussed in the previous section. Figure 1 shows the numerical solution of the given equation (22), whereas Figure 2 illustrates the numerical solution of the same equation in the absence of the damping term. The effect of damping can be clearly seen when comparing the two figures. Figure 1: Numerical solution of equation (22) 6. Conclusion Analyzing the monotonic, asymptotic, and oscillatory characteristics of solutions to DEs is fundamental for understanding the behavior of the models described by these equations. As it is shown in our findings, we derive some new monotonic and asymptotic properties for positive solutions to NDEs with multiple delays and a damping term (1). We then used these properties to derive sufficient criteria for the oscillation of all solutions of the equation under consideration. We also applied our outcomes to the special case in Example 21. The improved methodology employed in the analysis and investigation of the oscillatory behavior of equations with numerous delays and a middle term is what S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 12 of 15 Figure 2: Numerical solution of equation (22) without damping makes our results distinctive. A natural progression and forthcoming challenge of this research is examining the oscillatory behavior and asymptotic characteristics of solutions for higher-order neutral delay differential equations, where the interplay between delay and neutral components becomes progressively intricate. Acknowledgements The authors gratefully acknowledge Qassim University, represented by the Deanship of Graduate Studies and Scientific Research, on the financial support for this research under the number (QU-J-PG-2-2025-53684) during the academic year 1446AH / 2024 AD. Conflict of interest The authors declare there is no conflicts of interest. Data availability statement No data resulting from this study. References [1] R. P. Agarwal and P. J. Y. Wong. Advanced Topics in Differential Equations. Marcel Dekker, New York, 1995. S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 13 of 15 [2] R. M. Hafez and Y. H. Youssri. Legendre-collocation spectral solver for variable-order fractional functional differential equations. Computational Methods for Differential Equations, 8:99–110, 2020. [3] R. M. Hafez and Y. H. Youssri. Shifted gegenbauer-gauss collocation method for solving fractional neutral functional-differential equations with proportional delays. Kragujevac Journal of Mathematics, 46:981–996, 2022. [4] C. G. Philos. An oscillation criterion for superlinear differential equations of second order. Journal of Mathematical Analysis and Applications, 148(2):306–316, 1990. [5] W. M. Abd-Elhameed, J. A. T. Machado, and Y. H. Youssri. Hypergeometric frac- tional derivatives formula of shifted chebyshev polynomials: tau algorithm for a type of fractional delay differential equations. International Journal of Nonlinear Sciences and Numerical Simulation, 23:1253–1268, 2021. [6] W. M. Abd-Elhameed, Y. H. Youssri, and A. G. Atta. Tau algorithm for fractional delay differential equations utilizing seventh-kind chebyshev polynomials. Journal of Mathematical Modeling, 12(2):277–299, 2024. [7] I. T. Kiguradze and T. A. Chanturia. Asymptotic Properties of Solutions of Nonau- tonomous Ordinary Differential Equations. Kluwer Academic Publishers, Dordrecht, 1993. [8] W. T. Reid. Ordinary Differential Equations. John Wiley & Sons, New York, 1971. [9] G. Ballinger and X. Liu. Permanence of population growth models with impulsive effects. Mathematical and Computer Modelling, 26(12):59–72, 1997. [10] S. Tang and L. Chen. Global attractivity in a ”food-limited” population model with impulsive effects. Journal of Mathematical Analysis and Applications, 292(1):211–221, 2004. [11] N. MacDonald. Biological Delay Systems: Linear Stability Theory. Cambridge Uni- versity Press, Cambridge, 1989. [12] J. K. Hale. Partial neutral functional differential equations. Revue Roumaine de Mathématiques Pures et Appliquées, 39(4):339–344, 1994. [13] Y. Kuang, editor. Delay Differential Equations. Academic Press, New York, 1993. [14] V. Kolmanovskii and A. Myshkis. Applied Theory of Functional Differential Equa- tions, volume 85. Springer Science & Business Media, 2012. [15] N. MacDonald and N. MacDonald. Biological Delay Systems: Linear Stability Theory. Cambridge University Press, 2008. [16] K. Gopalsamy and B. G. Zhang. Oscillation and nonoscillation in first order neutral differential equations. Journal of Mathematical Analysis and Applications, 151(1):42– 57, 1990. [17] S. Pinelas and S. S. Santra. Necessary and sufficient condition for oscillation of nonlinear neutral first-order differential equations with several delays. Journal of Fixed Point Theory and Applications, 20:1–13, 2018. [18] S.-Y. Zhang and Q.-R. Wang. Oscillation of second-order nonlinear neutral dynamic equations on time scales. Applied Mathematics and Computation, 216(10):2837–2848, 2010. [19] M. Bohner, S. Grace, and I. Jadlovska. Sharp results for oscillation of second-order S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 14 of 15 neutral delay differential equations. Electronic Journal of Qualitative Theory of Dif- ferential Equations, (4):1–23, 2023. [20] B. Almarri, O. Moaaz, A. E. Abouelregal, and A. Essam. New comparison theorems to investigate the asymptotic behavior of even-order neutral differential equations. Symmetry, 15(5):1126, 2023. [21] O. Moaaz, H. Ramos, and J. Awrejcewicz. Second-order emden–fowler neutral differ- ential equations: A new precise criterion for oscillation. Applied Mathematics Letters, 118:107172, 2021. [22] R. P. Agarwal, M. Bohner, T. Li, and C. Zhang. A new approach in the study of oscillatory behavior of even-order neutral delay differential equations. Applied Mathematics and Computation, 225:787–794, 2013. [23] O. Bazighifan, O. Moaaz, R. A. El-Nabulsi, and A. Muhib. Some new oscillation results for fourth-order neutral differential equations with delay argument. Symmetry, 12(8):1248, 2020. [24] O. Moaaz, S. Furuichi, and A. Muhib. New comparison theorems for the nth order neutral differential equations with delay inequalities. Mathematics, 8(3):454, 2020. [25] T. X. Li, Z. L. Han, P. Zhao, and S. R. Sun. Oscillation of even-order neutral delay differential equations. Advances in Difference Equations, pages 1–9, 2010. [26] J. J. A. M. Brands. Oscillation theorems for second-order functional differential equations. Journal of Mathematical Analysis and Applications, 63:54–64, 1978. [27] G. Ladas, V. Lakshmikantham, and J. S. Papadakis. Oscillations of higher-order retarded differential equations generated by the retarded argument. In Delay and Functional Differential Equations and Their Applications, pages 219–231. Academic Press, 1972. [28] J. J. Wei. Oscillation of second order delay differential equation. Annals of Differential Equations, 4(4):473–478, 1988. [29] R. Koplatadze, G. Kvinikadze, and I. P. Stavroulakis. Oscillation of second order linear delay differential equations. Functional Differential Equations, 7(1-2):121–145, 2000. [30] J. Džurina and I. P. Stavroulakis. Oscillation criteria for second-order delay differ- ential equations. Applied Mathematics and Computation, 140(2-3):445–453, 2003. [31] Y. G. Sun and F. W. Meng. Note on the paper of džurina and stavroulakis: ”oscilla- tion criteria for second-order delay differential equations”. Applied Mathematics and Computation, 174(2):1634–1641, 2006. [32] G. E. Chatzarakis, J. Džurina, and I. Jadlovská. New oscillation criteria for second- order halflinear advanced differential equations. Applied Mathematics and Computa- tion, 347:404–416, 2019. [33] G. E. Chatzarakis, S. R. Grace, I. Jadlovská, T. Li, and E. Tunç. Oscillation criteria for third-order emden–fowler differential equations with unbounded neutral coeffi- cients. Complexity, 2019(1):5691758, 2019. [34] J. S. W. Wong. Necessary and sufficient conditions for oscillation of second order neutral differential equations. Journal of Mathematical Analysis and Applications, 252:342–352, 2000. S. Aloraini, A. S. Almohaimeed, O. Moaaz / Eur. J. Pure Appl. Math, 18 (4) (2025), 6545 15 of 15 [35] R. Xu and F. Meng. Some new oscillation criteria for second order quasi-linear neutral delay differential equations. Applied Mathematics and Computation, 182(1):797–803, 2006. [36] S. R. Grace, J. Džurina, I. Jadlovska, and T. Li. An improved approach for studying oscillation of second-order neutral delay differential equations. Journal of Inequalities and Applications, pages 1–13, 2018. [37] B. Baculikova and J. Džurina. Oscillation theorems for second order neutral differ- ential equations. Computers & Mathematics with Applications, 61:94–99, 2011. [38] J. Džurina. Oscillation theorems for second order advanced neutral differential equa- tions. Tatra Mountains Mathematical Publications, 48:61–71, 2011. [39] M. Bohner, S. Grace, and I. Jadlovska. Oscillation criteria for second-order neutral delay differential equations. Electronic Journal of Qualitative Theory of Differential Equations, (60):1–12, 2017. [40] B. Baculikova and J. Dzurina. Oscillation theorems for second-order nonlinear neutral differential equations. Computers & Mathematics with Applications, 62(12):4472– 4478, 2011. [41] H. Liu, F. Meng, and P. Liu. Oscillation and asymptotic analysis on a new generalized emden–fowler equation. Applied Mathematics and Computation, 219:2739–2748, 2012. [42] Ch. G. Philos. On the existence of nonoscillatory solutions tending to zero at infinity for differential equations with positive delays. Archiv der Mathematik (Basel), 36:168– 178, 1981. [43] Y. Kitamura and T. Kusano. Oscillation of first-order nonlinear differential equa- tions with deviating arguments. Proceedings of the American Mathematical Society, 78(1):64–68, 1980.