EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 1, Article Number 5591 ISSN 1307-5543 – ejpam.com Published by New York Business Global Fourth-Order Differential Equations: Asymptotic and Oscillatory Behaviors of Solutions Alanoud Almutairi Department of Mathematics, Faculty of Science, University of Hafr Al Batin, P.O. Box 1803, Hafar Al Batin 31991, Saudi Arabia Abstract. Our aim in this work is to derive conditions and criteria for the oscillation of some dif- ferential equations of p-Laplace type with a delayed term. Therefore, we develop these criteria that confirm to us that the equations studied are oscillatory by applying comparison with lower-order equations and Riccati techniques . Finally, we can elucidate the meaning of the new inequalities by applying our findings to a few particular cases of the studied equation. Our findings build on earlier findings that looked at equations with a delay term and operators of the p-Laplace type. To demonstrate the importance of the acquired results, we provide an example. 2020 Mathematics Subject Classifications: 34C10, 34K11 Key Words and Phrases: Asymptotic behavior, p-Laplacian, Fourth-order, Delay differential equations 1. Introduction The study of systems influenced by their historical behavior requires the use of func- tional equations (FDEs), which are equations where the variables’ current values are de- pendent on their past or future states. Delay differential equations are important cate- gories of these equations. These formulas are essential for simulating intricate systems in disciplines like biology, engineering, and physics. For instance, in control theory, FDEs regulate feedback systems to maintain stability, and in ecology, they aid in the analysis of population dynamics based on historical states. Continuous research in DDEs is essential for creating theoretical underpinnings and some techniques to address real-world issues as contemporary systems become more complicated [12]-[19]. In our work, we focus on the oscillation of ( κ (η) ∣∣z′′′ (η)∣∣p−2 z′′′ (η) )′ + j∑ i=1 ai (η) f (z (bi (η))) = 0, η ≥ η0, (1) DOI: https://doi.org/10.29020/nybg.ejpam.v18i1.5591 Email address: amalmutairi@uhb.edu.sa (A. Almutairi) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 2 of 11 where κ ∈ C1 ([η0,∞),R) , κ (η) > 0, κ′ (η) ≥ 0, ai ∈ C[η0,∞), a (η) > 0, bi ∈ C[η0,∞), bi (η) ≤ η, limη→∞ bi (η) = ∞; i = 1, 2, .., j, f ∈ C (R,R) such that f (η) /ηp−1 ≥ ℓ > 0, for η ̸= 0, p > 1is a constant, (2) and under the condition ∫ ∞ η0 1 κ1/(p−1) (η) dη = ∞. (3) Definition 1. [6] If κ (η) (z′′′ (η))p−1 ∈ C1[ηz,∞), and z (η) satisfies (1) on [ηz,∞), then a function z ∈ C3[ηz,∞), ηz ≥ η0, is a solution of (1). If a solution of (1) con- tains arbitrarily large zeros on [ηz,∞), it is said to be oscillatory; if not, it is said to be nonoscillatory. If all of its solutions are oscillatory, the equations (1) are considered oscillatory. The oscillation criteria for equtions with p-Laplace type have drawn a lot of interest from scientists, engineers, and researchers in the study of the oscillation to DDEs, which has grown in importance and prominence in recent years. Some advanced models based on delays differential equations with fractional characteristics have been developed as a result of this interest and have shown value in a variety of sectors [7]-[2]. This technique is effective for researching the transmission of ultrasound, mimicking the behavior of proteins and polymers, and examining the mechanical behavior of human tissues under stress. We can better comprehend numerous biological and physical processes thanks to these mod- els, which also enable scientists come up with creative solutions for challenging practical problems (see [1]-[11]). In order to apply mathematical methods to practical or real-world issues, the issue must be stated in mathematical terms. This entails developing a model a mathematical descrip- tion of the issue. It is known mathematically that derivatives describe rates of change, so equations that link functions and their derivatives are often included in mathemati- cal models. These equations, also referred to as differential equations, are used in many scientific domains, including economics, chemistry, physics, and biology [20]-[21]. The qualitative theory of differential equations has an important place in the study of applied as well as theoretical mathematics. It introduces dynamical systems, a popular area of mathematics in recent years, and it is works as an expansion and generalization of some types of ordinary equations. It also comes in quite handy when dealing with complicated differential equations that are impossible to solve with traditional techniques. Making assumptions on the solutions behavior without actually solving them is the basic idea underlying qualitative analysis of differential equations [16, 23]. In mathematics, a delay differential equation is a kind of FDEs that expresses the deriva- tive of some function at a given time in types of the function’s values at previous times. These equations are called hereditary systems, dead time systems, aftereffects systems, time delay systems. Additionally, there are sophisticated differential equations that can be used in a variety of real-world scenarios where the rate at which a system’s state where these changes are based on the current and future situation. The equation can be changed A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 3 of 11 to highlight the effects of possible future actions. Population dynamics, mechanical control engineering and economic issues are a few domains where these equations are frequently applied. NDEs are used in many areas of natural and technological inquiry[3, 14]. One subfield of qualitative theory, oscillation theory, examines the qualitative character- istics of differential equation solutions, including stability, oscillation, and others, without actually solving the problems [8]-[9]. According to [? ]-[15], the solutions of the examined equation are divided into three distinct classes: oscillatory solutions, positive and negative eventually solutions. Researchers started studying the equations of fourth-order after the oscillation for the second-order equations developed, see [17, 22]. Bazighifan and colleagues [6], we employed some techniques to ontain the adequate and required criteria for the oscillation of( κ (η) ∣∣z′′′ (η)∣∣p−2 z′′′ (η) )′ + a (η) f (z (b (η))) = 0, (4) under the condition ∫ ∞ η0 1 κ1/p−1 (η) dη = ∞. (5) New standards were presented by Bazighifan and Thabet [5] to evaluate the oscillatory of fourth-order DEs. Li et al. [13] concentrated on the oscillation of(( z(u−1) (η) )p−1 )′ + a (η) f (z (b (η))) = 0, (6) by utilizing the integral averaging method with Riccati technique and found new criteria for oscillation. Theorem 1. ([1]) If lim sup η→∞ ∫ η η0 ( µ (s) a (s)− λφ (µ′ (s))(p−1)+1 (µ (s) bu−2 (s) b′ (s))p−1 ) ds = ∞, (7) where λ := (1/ ((p− 1) + 1))(p−1)+1 (2 (u− 1)!)p−1 , µ ∈ C1 ([η0,∞) , (0,∞)) and φ > 1, then every solution of (6) is oscillatory. Theorem 2. ([4]) Let f ( η1/p−1 ) /η ≥ 1 for 0 < η ≤ 1, h ∈ (0, 1) such that lim inf η→∞ ∫ η bi(η) a (s) f ( h (u− 1)! bu−1 (s) κ1/p−1 (b (s)) ) ds > 1 e , (8) then (6) is oscillatory. The goal of researching this work is to enhance and supplement the findings of [18]. The structure of the paper is as follows. We provide a few lemmas in Section 2 that will be helpful in demonstrating our findings. We provide a new standards of oscillation for (1) by the use of generalized Riccati transformations in Section 3. Lastly, a few examples are taken into consideration to highlight the main findings. A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 4 of 11 2. Preliminary Results The lemmas, and presumptions presented in this part are crucial for streamlining the mathematical computations utilized in this work. There are just two instances when examining the asymptotic behavior of the positive solutions of (1). Case (1) : z(m) (η) > 0 for m = 0, 1, 2, 3; Case (2) : z(m) (η) > 0 for m = 0, 1, 3 and z′′ (η) < 0. For convenience, we denote R (η) := ∫ ∞ η 1 κ1/p−1 (s) ds, F+ (η) := max {0, F (η)} , ϱ (η) := µ (η) ( ℓ j∑ i=1 ai (η) ( b3i (η) η3 )p−1 + εν (1+(p−1))/p−1 1 η2 − 2ν1p− 1 2κ 1 p−1 (η)R(p−1)+1(η) ) , σ (η) := µ′ + (η) µ (η) + ((p− 1) + 1) ν 1/p−1 1 εη2 2κ 1 (p−1) (η)R(η) , σ∗ (η) := ς ′+ (η) ς (η) + 2ν2 R(η) , and ϱ∗ (η) := ς (η) ∫ ∞ η ( ℓ κ (v) ∫ ∞ v j∑ i=1 ai (s) bp−1 i (s) sp−1 ds )1/p−1 dv + ν22 − ν2κ −1 p−1 (η) R2(η)  , where µ, ς ∈ C1 ([η0,∞) , (0,∞)) and ν1, ν2 are constants. Remark 1. The generalized Riccati substitutions are defined by us. ζ (η) := µ (η) ( κ (η) (z′′′)p−1 (η) zp−1 (η) + ν1 Rp−1(η) ) , (9) and w (η) := ς (η) ( z′ (η) z (η) + ν2 R(η) ) . (10) Lemma 1. [2] Assume that V > 0, U be constant, and v is the ratio of two odd values. Then P (ν+1)/ν − (P − a)(ν+1)/ν ≤ 1 ν a1/ν [(1 + ν)P − a] , Pa ≥ 0, ν ≥ 1 and Uz − V z(ν+1)/ν ≤ νν (ν + 1)ν+1 Uν+1 V ν . A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 5 of 11 Lemma 2. [23] Suppose that g ∈ Cu ([η0,∞) , (0,∞)) , g(u) is of a fixed sign on [η0,∞) , g(u) not identically zero and there exists a η1 ≥ η0 such that g(u−1) (η) g(u) (η) ≤ 0, for all η ≥ η1. If we have limη→∞ g (η) ̸= 0, then there exists ην ≥ η1 such that g (η) ≥ ν (u− 1)! ηu−1 ∣∣∣g(u−1) (η) ∣∣∣ , for all ν ∈ (0, 1) and η ≥ ην . Lemma 3. [10] If κ(j) > 0 and κ(u+1) < 0, then u! ηu κ (η)− (u− 1)! ηu−1 d dη κ (η) ≥ 0, for all j = 0, 1, ..., u. 3. Oscillation criteria We will define various oscillation criterion for equation (1) in this part. Lemma 4. Let z be a positive solution of (1) in the end, and for all r = 1, 2, 3, z(r) (η) > 0. In the case where µ ∈ C1 ([η0,∞) , (0,∞)) , and ζ ∈ C1[η,∞) defined as (9), then ζ ′ (η) ≤ −ϱ (η) + σ (η) ζ (η)− εη2(p− 1) 2 (κ (η)µ (η))1/p−1 (ζ (η)) p p−1 , (11) for all η > η1. Proof. Let z > 0. From Lemma 2, wefind z′ (η) ≥ ε 2 η2z′′′ (η) , ε ∈ (0, 1). (12) By (9), we find ζ (η) > 0 for η ≥ η1, and ζ ′ (η) = µ′ (η) ( κ (η) (z′′′)p−1 (η) zp−1 (η) + ν1 Rp−1(η) ) + µ (η) ( κ (z′′′)p−1 )′ (η) zp−1 (η) −(p− 1)µ (η) z(p−1)−1 (η) z′ (η)κ (η) (z′′′)p−1 (η) z2p−1 (η) + (p− 1)ν1µ (η) κ 1 p−1 (η)Rp(η) . Using (12) and (9), we obtain ζ ′ (η) ≤ µ′ + (η) µ (η) ζ (η) + µ (η) ( κ (η) (z′′′ (η))p−1 )′ zp−1 (η) A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 6 of 11 −(p− 1)µ (η) ε 2 η2 κ (η) (z′′′ (η))p zp (η) + (p− 1)ν1µ (η) κ 1 p−1 (η)Rp(η) ≤ µ′ (η) µ (η) ζ (η) + µ (η) ( κ (η) (z′′′ (η))p−1 )′ zp−1 (η) −(p− 1)µ (η) ε 2 η2κ (η) ( ζ (η) µ (η)κ (η) − ν1 κ (η)Rp−1(η) ) p p−1 + (p− 1) ν1µ (η) κ 1 p−1 (η)Rp(η) .(13) Using Lemma 1 with P = ζ (η) / (µ (η)κ (η)) , a = ν1/ ( κ (η)R(p−1)(η) ) and ν = (p− 1), we get( ζ (η) κ (η)µ (η) − ν1 κ (η)R(p−1)(η) ) p p−1 ≥ ( ζ (η) µ (η)κ (η) ) p p−1 − ν 1/(p−1) 1 (p− 1)κ 1 (p−1) (η)R(η) ( ((p− 1) + 1) ζ (η) µ (η)κ (η) − ν1 κ (η)R(p−1)(η) ) .(14) From Lemma 3, we have that z (η) ≥ η 3z ′ (η) and hence, z (bi (η)) z (η) ≥ b3i (η) η3 . (15) From (1), (13) and (14), we obtain ζ ′ (η) ≤ µ′ + (η) µ (η) ζ (η)− ℓµ (η) j∑ i=1 ai (η) [ b3i (η) η3 ]p−1 − (p− 1)µ (η) ε 2 η2κ (η) ( ζ (η) µ (η)κ (η) ) p (p−1) − (p− 1)µ (η) ε 2 η2κ (η) ( −ν 1/(p−1) 1 (p− 1)κ 1 (p−1) (η)R(η) ( pζ (η) µ (η)κ (η) − ν1 κ (η)R(p−1)(η) )) + (p− 1) ν1µ (η) κ 1 (p−1) (η)Rp(η) . This implies that ζ ′ (η) ≤ ( µ′ + (η) µ (η) + pν 1/(p−1) 1 εη2 2κ 1 (p−1) (η)R(η) ) ζ (η)− εη2 (p− 1) 2κ1/(p−1) (η)µ1/(p−1) (η) ζ p p−1 (η) −µ (η) ( ℓ j∑ i=1 ai (η) ( b3i (η) η3 )(p−1) + εν p/(p−1) 1 η2 − 2ν1 (p− 1) 2κ 1 (p−1) (η)Rp(η) ) . Thus, ζ ′ (η) ≤ −ϱ (η) + σ (η) ζ (η)− (p− 1) εη2 2 (κ (η)µ (η))1/(p−1) ζ p (p−1) (η) . The proof is complete. A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 7 of 11 Lemma 5. Let z be a positive solution of (1) in the end and Case (2) hold, then w′ (η) ≤ −ϱ∗ (η) + σ∗ (η)w (η)− 1 ς (η) w2 (η) , (16) where ς ∈ C1 ([η0,∞) , (0,∞)) . Proof. Let z ultimately be a positive solution of (1) and Case (2) hold. Lemma 3 gives us the result that z (η) ≥ ηz′ (η). This inequality can be integrated from bi (η) to η to obtain z (bi (η)) ≥ bi (η) η z (η) . From (2), we so have f (z (bi (η))) ≥ ℓ b (p−1) i (η) η(p−1) z(p−1) (η) . (17) Integrating (1) from η to κ and using z′ (η) > 0, we obtain κ (κ) ( z′′′ (κ) )(p−1) − κ (η) ( z′′′ (η) )(p−1) = − ∫ κ η j∑ i=1 ai (s) f (z (bi (s))) ds ≤ −ℓz(p−1) (η) ∫ κ η j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds. Letting κ → ∞ , we find κ (η) ( z′′′ (η) )(p−1) ≥ ℓz(p−1) (η) ∫ ∞ η j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds and so z′′′ (η) ≥ z (η) ( ℓ κ (η) ∫ ∞ η j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds )1/(p−1) . Once more integrating from η to ∞, we obtain z′′ (η) ≤ −z (η) ∫ ∞ η ( ℓ κ (v) ∫ ∞ v j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds )1/(p−1) dv. (18) By differentiating w (η), we find w′ (η) = ς ′ (η) ς (η) w (η) + ς (η) z′′ (η) z (η) − ς (η) ( w (η) ς (η) − ν2 R(η) )2 + ς (η) ν2 κ1/(p−1) (η)R2(η) . (19) Using Lemma 1 with P = w (η) /ς (η) , a = ν2/R(η) and ν = 1, we get( w (η) ς (η) − ν2 R(η) )2 ≥ ( w (η) ς (η) )2 − ν2 R(η) ( 2w (η) ς (η) − ν2 R(η) ) . (20) A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 8 of 11 By (1), (19) and (20), we get w′ (η) ≤ ς ′ (η) ς (η) w (η)− ς (η) ∫ ∞ η ( ℓ κ (v) ∫ ∞ v j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds )1/(p−1) dv −ς (η) (( w (η) ς (η) )2 − ν2 R(η) ( 2w (η) ς (η) − ν2 R(η) )) + ν2ς (η) κ 1 (p−1) (η)R2(η) . This implies that w′ (η) ≤ ( ς ′+ (η) ς (η) + 2ν2 R(η) ) w (η)− 1 ς (η) w2 (η) −ς (η) ∫ ∞ η ( ℓ κ (v) ∫ ∞ v j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds )1/(p−1) dv + ν22 − ν2κ −1 (p−1) (η) R2(η)  . Thus, w′ (η) ≤ −ϱ∗ (η) + σ∗ (η)w (η)− 1 ς (η) w2 (η) . The proof is finished. Lemma 6. Let z be a positive solution of (1) in the end. If∫ ∞ η0 ( ϱ (s)− ( 2 εs2 )(p−1) κ (s)µ (s) (σ (s))p pp ) ds = ∞, (21) where µ ∈ C ([η0,∞)) and ε ∈ (0, 1), then z doesn’t fulfill Case (1). Proof. Let z be a positive solution of (1) in the end. By Lemma 4, we see (11) holds and from Lemma 1 with U = σ (η) , V = (p− 1) εη2/ ( 2 (κ (η)µ (η))1/(p−1) ) and η = ζ, we get ζ ′ (η) ≤ −ϱ (η) + ( 2 εη2 )(p−1) κ (η)µ (η) (σ (η))(p−1)+1 pp . (22) Once more integrating from η1 to η, we find∫ η η1 ( ϱ (s)− ( 2 εs2 )(p−1) κ (s)µ (s) (σ (s))p pp ) ds ≤ ζ (η1) , which contradicts (21). So, The proof is finished. A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 9 of 11 Lemma 7. Let z be an eventually positive solution of (1) and Case (2) hold. If∫ ∞ η0 ( ϱ∗ (s)− 1 4 ς (s) (σ∗ (s))2 ) ds = ∞, where ς ∈ C ([η0,∞)) (23) then z doesn’t fulfill Case (2). Proof. Let z be a positive solution of (1) in the end. (16) holds according to Lemma 5. Lemma 1 is used with U = σ∗ (η) , V = 1/ς (η) , p = 2 and η = w, We obtainThe formula is ζ ′ (η) ≤ −ϱ∗ (η) + 1 4 ς (η) (σ∗ (η))2 . (24) Once more integrating from η1 to η, we find∫ η η1 ( ϱ∗ (s)− 1 4 ς (s) (σ∗ (s))2 ) ds ≤ ζ (η1) . This runs counter to (23). The proof is finished. Theorem 3. Assume that (21) and (23) hold. Then (1) is oscillatory. The oscillation requirements that result from applying µ (η) = η3 and ς (η) = η to Theorem 3 are as follows: Corollary 1. Let (3) hold. Assume that lim sup η→∞ ∫ η η1 ( β (s)− ( 2 εs2 )(p−1) κ (s)µ (s) (β (s))p pp ) ds = ∞, (25) for some ε ∈ (0, 1) . If lim sup η→∞ ∫ η η1 ( β1 (s)− 1 4 ς (s) (β1 (s)) 2 ) ds = ∞, (26) where β (η) : = η3 ( ℓ j∑ i=1 ai (η) ( b3i (η) η3 )(p−1) + εν p/(p−1) 1 η2 − 2ν1 (p− 1) 2κ 1 (p−1) (η)Rp(η) ) β (η) : = 3 η + ((p− 1) + 1) ν 1/(p−1) 1 εη2 2κ 1 (p−1) (η)R(η) , β1 (η) := 1 η + 2ν2 R(η) and β1 (η) := η ∫ ∞ η ( ℓ κ (v) ∫ ∞ v j∑ i=1 ai (s) b (p−1) i (s) s(p−1) ds )1/(p−1) dv + ν22 − ν2κ −1 (p−1) (η) R2(η)  , then (1) is oscillatory. A. Almutairi / Eur. J. Pure Appl. Math, 18 (1) (2025), 5591 10 of 11 Example 1. Let equation z(4) (η) + c0 η4 z ( 1 2 η ) = 0, η ≥ 1, (27) where p = 2,κ (η) = 1, c0 > 0, a (η) = c0/η 4 and b (η) = η/2. Hence, we have R (η0) = ∞, β (s) = c0 8s . If we set ℓ = ν1 = 1, then condition (25) becomes lim sup η→∞ ∫ η η1 ( β (s)− ( 2 εs2 )(p−1) κ (s)µ (s) (β (s))p pp ) ds = lim sup η→∞ ∫ η η1 ( c0 8s − 9 2s ) ds = ∞ if c0 > 36. 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