Electronic Journal of Differential Equations, Vol. 2024 (2024), No. 55, pp. 1–13. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, https://ejde.math.unt.edu DOI: 10.58997/ejde.2024.55 EXISTENCE AND BOUNDS FOR KNESER-TYPE SOLUTIONS TO NONCANONICAL THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS GANESH PURUSHOTHAMAN, KANNAN SURESH, ETHIRAJU THANDAPANI, ERCAN TUNÇ Abstract. This article focuses on the existence and asymptotic behavior of Kneser-type solutions to third-order noncanonical differential equations with a delay or advanced argument in the neutral term( r2(t) ( r1(t)z ′(t) )′)′ + g(t)x(t) = 0, where z(t) = x(t) + p(t)x(τ(t)). This equation is transformed into a canonical equation, which reduces the number of classes of positive solutions from 4 to 2. This is done without assuming extra conditions, and greatly simplifies the process of obtaining conditions for the existence of Kneser-type solutions. Also we obtain lower and upper bounds for these solutions, and obtain their rate of convergence to zero. Two examples are provided to illustrate our main results, one with a delay neutral term, and one with an advanced neutral term. 1. Introduction This article concerns the asymptotic properties of Kneser-type solutions to the third-order neutral differential equation( r2(t) ( r1(t)z ′(t) )′)′ + g(t)x(t) = 0, t ≥ t0 > 0, (1.1) where z(t) = x(t)+p(t)x(τ(t)). During this study, we use the following assumptions: (H1) r1 ∈ C2([t0,∞), (0,∞)), r2 ∈ C1([t0,∞), (0,∞)), g ∈ C([t0,∞), [0,∞)), and p ∈ C3([t0,∞), [0,∞)) with 0 ≤ p(t) < 1; (H2) τ ∈ C3([t0,∞),R), either τ(t) ≤ t or τ(t) ≥ t, and limt→∞ τ(t) = ∞; (H3) equation (1.1) is in noncanonical form, i.e.,∫ ∞ t0 dt r1(t) <∞ and ∫ ∞ t0 dt r2(t) <∞. By a solution of (1.1), we mean a function x ∈ C([tx,∞),R) for some tx ≥ t0 such that z ∈ C1([tx,∞),R), r1z′ ∈ C1([tx,∞),R), r2(r1z′)′ ∈ C1([tx,∞),R), and x satisfies equation (1.1) on [tx,∞). We consider only those solutions of (1.1) that exist on some half-line [tx,∞) and such that sup{|x(t)| : T1 ≤ t <∞} > 0 for each 2020 Mathematics Subject Classification. 34C10, 34K11, 34K40. Key words and phrases. Kneser solution; neutral differential equation; noncanonical equation. ©2024. This work is licensed under a CC BY 4.0 license. Submitted July 25, 2024. Published September 19, 2024. 1 2 G. PURUSHOTHAMAN, K. SURESH, E. THANDAPANI, E. TUNÇ EJDE-2024/55 T1 ≥ tx; such solutions are said to be continuable. We tacitly assume that equation (1.1) possesses such solutions. A continuable solution x(t) of (1.1) is said to be oscillatory if it has infinitely many zeros; otherwise, it is called nonoscillatory. We say that (1.1) has property A if any solution x(t) of (1.1) is either oscillatory or limt→∞ x(t) = 0. Note that if y is a negative solution of (1.1), then x(t) = −y(t) is positive solution of the same equation; statements for positive solutions apply to non-oscillatory solutions. The qualitative theory of third-order differential equations has seen rapid devel- opment recently due to its numerous applications and the mathematical challenges it presents, as indicated by [11, 15, 16, 18]. In particular, the oscillation and asymptotic behavior of third-order functional differential equations have received significant attention, as evidenced by extensive literature; see the references in this article and the references therein. However, a literature review reveals limited find- ings on the existence and non-existence of Kneser-type solutions for third-order delay and neutral type differential equations. In [2], the authors studied the existence and estimates for Kneser-type solutions of (1.1) with an advanced neutral term (i.e., τ(t) ≥ t) and r1(t) = 1 and in canonical form, that is, ∫ ∞ t0 1 r2(t) dt = ∞. In [3], the authors generalized the results in [2] for the equation( r2(t) ( r1(t)(z ′(t))α )′)′ + g(t)xα(t) = 0 (1.2) under the condition ∫ ∞ t0 1 r 1/α 1 (t) dt = ∫ ∞ t0 1 r2(t) dt = ∞, (1.3) i.e., the authors considered the equation (1.2) in canonical form. In [7], the authors considered the equation( r2(t) ( r1(t)z ′(t) )′)′ + g(t)x(σ(t)) = 0, (1.4) under condition (1.3) and established conditions for the nonexistence of Kneser-type solutions of (1.4). The nonexistence of Kneser-type solutions was discussed in [21] for the equation( r2(t) ( z′′(t) )α)′ + g(t)xβ(σ(t)) = 0, under the condition ∫ ∞ t0 1 r 1/α 2 (t) dt = ∞. Recently in [23], the authors discussed the existence and estimates for the Kneser- type solutions for the semi-canonical equation with an advanced argument in the neutral term ( r2(t) ( r1(t)(z ′(t))α )′)′ + g(t)xα ( σ(t) ) = 0 (1.5) under the conditions∫ ∞ t0 1 r2(t) dt <∞ and ∫ ∞ t0 1 r 1/α 1 (t) dt = ∞, EJDE-2024/55 NONCANONICAL THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS 3 by transforming (1.5) into a canonical type equation. On the other hand, in [4], the authors discussed the existence and estimates for Kneser-type solutions of the third-order canonical type delay differential equation (i.e., σ(t) < t) x′′′(t) + g(t)x(σ(t)) = 0, and in [20] the authors studied the convergence to zero of Kneser solutions of general canonical type third-order delay differential equations. From the above discussion, it is clear that the existence and estimates for the Kneser-type solutions have been investigated for third-order delay differential equa- tions or advanced type neutral differential equations in canonical or semi-canonical form. To the best of the authors knowledge, no results exist on the existence of Kneser-type solutions for noncanonical type differential equations with a delay or advanced argument in the neutral term. This article aims to address this question by taking (1.1) in noncanonical form with a delay or advanced argument in the neutral term. To achieve this, we trans- form the noncanonical equation (1.1) into a canonical form. This reduces the number of classes of positive solutions to two instead of the usual four classes for noncanonical equations. Hence, this significantly simplifies finding conditions for the existence and finding estimates of Kneser-type solutions. Two examples are provided to illustrate our main results. 2. Preliminary results In this section, we introduce results necessary to demonstrate our main findings. In view of (H3), it is possible to define the following functions for t ≥ t0: R12(t) = ∫ ∞ t 1 r1(s) ∫ ∞ s 1 r2(w) dw ds, R21(t) = ∫ ∞ t 1 r2(s) ∫ ∞ s 1 r1(w) dw ds, q1(t) = r1(t)R 2 12(t) R21(t) , q2(t) = r2(t)R 2 21(t) R12(t) . Note that R12 and R21 are positive and decreasing. We begin with the following theorem whose proof can be found in [1, Theorem 2.1]. Theorem 2.1. The noncanonical equation (1.1) can be expressed in an equivalent canonical form as( q2(t) ( q1(t) ( z(t) R12(t) )′)′)′ +R21(t)g(t)x(t) = 0, (2.1) with ∫ ∞ t0 1 q1(t) dt = ∫ ∞ t0 1 q2(t) dt = ∞ . (2.2) Setting ψ(t) = z(t)/R12(t) in (2.1), we have the following 2 lemmas. Lemma 2.2. A function x is a solution of the noncanonical equation (1.1) if and only if x is solution of the canonical equation( q2(t) ( q1(t)ψ ′(t )′)′ +R21(t)g(t)x(t) = 0, t ≥ t0 . (2.3) Lemma 2.3. The noncanonical neutral differential equation (1.1) has an eventually positive solution if and only if the canonical equation (2.3) has an eventually positive solution. 4 G. PURUSHOTHAMAN, K. SURESH, E. THANDAPANI, E. TUNÇ EJDE-2024/55 It has been established, see for example [5, 19], that if x is an eventually positive solution of (1.1), then there exists t1 ≥ t0 such that for all t ≥ t1, the corresponding function z(t) = x(t) + p(t)x(τ(t)) belongs to one of the following four classes: S1 = {z : z > 0, L1z < 0, L2z < 0, L3z < 0}, S2 = {z : z > 0, L1z < 0, L2z > 0, L3z < 0}, S3 = {z : z > 0, L1z > 0, L2z > 0, L3z < 0}, S4 = {z : z > 0, L1z > 0, L2z < 0, L3z < 0}, where L1z = r1z ′, L2z = r2(r1z ′)′, L3z = ( r2(r1z ′)′ )′ . From the above classification, we see that (1.1) has two types of monotonically increasing solutions and two types of monotonically decreasing solutions. Lemma 2.3 simplifies the study of (1.1), since (2.3) has only of two types of solutions: one eventually decreasing and the other eventually increasing, as stated in the following lemma. This lemma follows from a generalization of the well-known Kiguradze lemma [13, Lemma 1.1] applied to (1.4). Lemma 2.4. Assume that x is an eventually positive solution of (2.3). Then the corresponding function ψ belongs to one of the following two classes: N0 = {ψ : ψ > 0, L1ψ < 0, L2ψ > 0, L3ψ < 0}, N2 = {ψ : ψ > 0, L1ψ > 0, L2ψ > 0, L3ψ < 0}, where L1ψ = q1ψ ′, L2ψ = q2(q1ψ ′)′, L3ψ = ( q2(q1ψ ′)′ )′ . Definition 2.5. A solution x is called Kneser solution, if its corresponding function ψ belongs to N0. 3. Main results In this section, first we derive conditions for making the class N2 empty, so the positive solutions of (2.3) belong N0. To simplify notation we define: Q1(t) = ∫ t t1 1 q1(s) ds, Q2(t) = ∫ t t1 1 q2(s) ds, Q12(t) = ∫ t t1 1 q1(s) ∫ s t1 1 q2(w) dw ds, ∆(t) = ∫ ∞ t 1 q2(s) ∫ ∞ s g(w)R21(w)R12(w) dw ds, ϕ(t) = exp (∫ t t1 ∆(s) q1(s) ds ) , where t ≥ t1 ≥ t0. Now we introduce 2 additional conditions on the coefficient p(t). There exits t2 ≥ t1 such that for all t ≥ t2, we have p(t) R12(τ(t)) R12(t) ϕ(t) ϕ(τ(t)) < 1 if τ(t) ≤ t, (3.1) p(t) R12(τ(t)) R12(t) Q12(τ(t)) Q12(t) < 1 if τ(t) ≥ t, (3.2) EJDE-2024/55 NONCANONICAL THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS 5 Since R12 and Q12 and ϕ are positive, the inequality 0 ≤ p(t) is preserved. Since ϕ is increasing, when τ(t) ≤ t, we have ϕ(t)/ϕ(τ(t)) ≥ 1. Therefore (3.1) implies p(t) R12(τ(t)) R12(t) < 1 . Assuming (3.1) and (3.2), we define G such that for t ≥ t2, 0 < G(t) = 1− p(t)R12(τ(t)) R12(t) if τ(t) ≤ t, 1− p(t)R12(τ(t)) R12(t) Q12(τ(t)) Q12(t) if τ(t) ≥ t, (3.3) Assuming (3.1), we define G1 such that for t ≥ t2, 0 < G1(t) = 1− p(t)R12(τ(t)) R12(t) if τ(t) ≥ t, 1− p(t)R12(τ(t)) R12(t) ϕ(t) ϕ(τ(t)) if τ(t) ≤ t. (3.4) Lemma 3.1. Let x be an eventually positive solution of (2.3) with the correspond- ing function ψ ∈ N2. Then (i) q1(t)ψ ′(t) Q2(t) is decreasing, and (ii) ψ(t) Q12(t) is decreasing. Proof. The proof of (i) is the same proof as in [1, Lemma 3.1]. Let x be an eventually positive solution of (2.3) with the corresponding function ψ ∈ N2. Since x(t) > 0, by (2.3), we have that q2(t)(q1(t)ψ ′(t))′ is decreasing, and q1(t)ψ ′(t) ≥ ∫ t t1 q2(s)(q1(s)ψ ′(s))′ q2(s) ds ≥ Q2(t)q2(t)(q1(t)ψ ′(t))′. Then (q1(t)ψ′(t) Q2(t) )′ = Q2(t)q2(t)(q1(t)ψ ′(t))′ − q1(t)ψ ′(t) Q2 2(t)q2(t) ≤ 0 . Hence, q1(t)ψ ′(t)/Q2(t) is decreasing, which proves (i). (ii) Using that q1(t)ψ ′(t)/Q2(t) is decreasing, we obtain ψ(t) ≥ ∫ t t1 q1(s)ψ ′(s) q1(s) Q2(s) Q2(s) ds ≥ Q12(t)q1(t)ψ ′(t) Q2(t) . (3.5) Therefore, ( ψ(t) Q12(t) )′ = q1(t)ψ ′(t)Q12(t)− ψ(t)Q2(t) q1(t)Q2 12(t) ≤ 0, which implies ψ(t)/Q12(t) is decreasing and so (ii) is proved. □ Lemma 3.2. Let (3.1) and (3.2) hold and let x be an eventually positive solution of (2.3) with the corresponding function ψ ∈ N2. Then G(t)ψ(t) ≤ x(t) R12(t) ≤ ψ(t), for all t ≥ t2 . (3.6) Proof. Let x(t) be an eventually positive solution of (2.3) with the corresponding function ψ(t) ∈ N2 for t ≥ t1 ≥ t0. Then, from the definition of ψ(t), it is easy to see that ψ(t) = z(t) R12(t) ≥ x(t) R12(t) . (3.7) 6 G. PURUSHOTHAMAN, K. SURESH, E. THANDAPANI, E. TUNÇ EJDE-2024/55 Moreover, x(t) = R12(t)ψ(t)− p(t)x(τ(t)) ≥ R12(t)ψ(t)− p(t)R12(τ(t))ψ(τ(t)) = R12(t) ( ψ(t)− p(t) R12(τ(t)) R12(t) ψ(τ(t)) ) . When τ(t) ≤ t, since ψ is increasing, ψ(τ(t)) ≤ ψ(t) which is used for obtain- ing G(t) in the above inequality. Also when τ(t) ≥ t, since ψ/Q12 is decreasing, ψ(τ(t))/Q12(τ(t)) ≤ ψ(t)/Q12(t), which is used for obtaining G(t) in the above inequality. In both cases we have x(t) ≥ R12(t)G(t)ψ(t). (3.8) Combining this inequality and (3.7) we obtain (3.6). □ Theorem 3.3. Let (3.1) and (3.2) hold and let x be an eventually positive solution of (1.1). If lim sup t→∞ ( 1 Q2(t) ∫ t t2 g(s)R21(s)R12(s)G(s)Q2(s)Q12(s) ds +Q12(t) ∫ ∞ t g(s)R21(s)R12(s)G(s) ds ) > 1 , (3.9) then the class N2 is empty. Proof. Let x be an eventually positive solution of (1.1). Since limt→∞ τ(t) = ∞, we can assume x(τ(t)) > 0. By Lemma 2.3, x is also a positive solution to (2.3). To obtain a contradiction assume that the corresponding function ψ(t) belongs to the class N2. Applying (3.6) to (2.3), one obtains( q2(t) ( q1(t)ψ ′(t) )′)′ + g(t)R21(t)R12(t)G(t)ψ(t) ≤ 0, t ≥ t2. Integrating the latter inequality from t to ∞, we obtain( q1(t)ψ ′(t) )′ ≥ 1 q2(t) ∫ ∞ t g(s)R21(s)R12(s)G(s)ψ(s) ds. Again integrating from t2 to t, we obtain q1(t)ψ ′(t) ≥ ∫ t t2 1 q2(s) (∫ ∞ s g(v)R21(v)R12(v)G(v)ψ(v) dv ) ds = ∫ t t2 1 q2(s) (∫ t s g(v)R21(v)R12(v)G(v)ψ(v) dv ) ds + ∫ t t2 1 q2(s) (∫ ∞ t g(v)R21(v)R12(v)G(v)ψ(v) dv ) ds ≥ ∫ t t2 g(s)R21(s)R12(s)G(s)Q2(s)ψ(s) ds +Q2(t) ∫ ∞ t g(s)R21(s)R12(s)G(s)ψ(s) ds. EJDE-2024/55 NONCANONICAL THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS 7 Using (3.5) in the latter inequality, we obtain Q2(t)ψ(t) Q12(t) ≥ ∫ t t2 g(s)R21(s)R12(s)G(s)Q2(s)ψ(s) ds +Q2(t) ∫ ∞ t g(s)R21(s)R12(s)G(s)ψ(s) ds. (3.10) Using that ψ(t) is increasing and ψ(t)/Q12(t) is decreasing in (3.10), we obtain 1 ≥ 1 Q2(t) ∫ t t2 g(s)R21(s)R12(s)G(s)Q2(s)Q12(s) ds +Q12(t) ∫ ∞ t g(s)R21(s)R12(s)G(s) ds. Computing the lim sup as t → ∞ on both sides of the inequality, we arrive at a contradiction to (3.9). The proof is complete. □ Next, we present lower and upper bounds for the positive solutions of (1.1). For simplicity, we define: F (t) = ϕ(t) q1(t) ∫ ∞ t 1 q2(v) ∫ ∞ v g(s)R21(s)R12(s)G1(s) ϕ(s) ds dv. Lemma 3.4. Let (3.1) hold and let x be an eventually positive solution of (2.3) with the corresponding function ψ ∈ N0. Then (i) ψ(t)ϕ(t) is increasing, and (ii) G1(t)ψ(t) ≤ x(t) R12(t) ≤ ψ(t). Proof. Since ψ ∈ N0, we have ψ(t) > 0, q1(t)ψ ′(t) < 0, q2(t)(q1(t)ψ ′(t))′ > 0, ( q2(t)(q1(t)ψ ′(t))′ )′ < 0, for all t ≥ t2. Since q2(t)(q1(t)ψ ′(t))′ is positive and decreasing, there exists a constant ℓ such that lim t→∞ q2(t) ( q1(t)ψ ′(t) )′ = ℓ ≥ 0. We claim that ℓ = 0. If not, then (q1(t)ψ ′(t))′ ≥ ℓ 2q2(t) > 0 and therefore q1(t)ψ ′(t) ≥ q1(t2)ψ ′(t2) + ℓ 2 ∫ t t2 1 q2(s) ds→ ∞ as t→ ∞, which contradicts that q1(t)ψ ′(t) < 0 for all t ≥ t2. Thus, lim t→∞ q2(t)(q1(t)ψ ′(t))′ = 0. Since q1(t)ψ ′(t) is negative and increasing, there exists a constant m such that lim t→∞ q1(t)ψ ′(t) = m ≤ 0. We claim that m = 0. If not, then ψ′(t) ≤ m q1(t) < 0 and we have ψ(t) ≤ ψ(t2) +m ∫ t t2 1 q1(s) ds→ −∞ as t→ ∞, 8 G. PURUSHOTHAMAN, K. SURESH, E. THANDAPANI, E. TUNÇ EJDE-2024/55 which contradicts that ψ(t) is positive. Therefore, limt→∞ q1(t)ψ ′(t) = 0. Now, an integration of (2.3) from t to ∞ yields( q1(t)ψ ′(t) )′ = 1 q2(t) ∫ ∞ t g(s)R21(s)x(s) ds ≤ 1 q2(t) ∫ ∞ t g(s)R21(s)R12(s)ψ(s) ds ≤ ψ(t) q2(t) ∫ ∞ t g(s)R21(s)R12(s) ds, (3.11) where we have used ψ(t) ≥ x(t) R12(t) and ψ(t) is decreasing. Again integrating (3.11), we obtain ψ′(t) ≥ − ψ(t) q1(t) ∫ ∞ t 1 q2(s) (∫ ∞ s g(v)R21(v)R12(v) dv ) ds = −ψ(t)∆(t) q1(t) . Hence, (ψ(t)ϕ(t))′ = ψ′(t)ϕ(t) + ψ(t)ϕ′(t) ≥ ψ(t) ( ϕ′(t)− ∆(t) q1(t) ϕ(t) ) . Since ϕ(t) is a solution of the differential equation ϕ′(t)− ∆(t) q1(t) ϕ(t) = 0, we conclude that ψ(t)ϕ(t) is increasing. From the definition of ψ(t), and 0 ≤ p(t), we see that ψ(t) = z(t) R12(t) = 1 R12(t) ( x(t) + p(t)x(τ(t)) ) . So, ψ(t) ≥ x(t) R12(t) and x(t) ≥ R12(t)ψ(t)− p(t)R12(τ(t))ψ(τ(t)) ≥ R12(t) ( ψ(t)− p(t) R12(τ(t)) R12(t) ψ(τ(t)) ) ≥ R12(t)G1(t)ψ(t). Hence, G1(t)ψ(t) ≤ x(t) R12(t) ≤ ψ(t), and the proof is complete. □ Theorem 3.5. Let conditions (3.1), (3.2) and (3.9) be satisfied. Then there exist positive constants α1 and α2 such that every positive solution x of (1.1) satisfies α1 G1(t) ϕ(t) ≤ x(t) R12(t) ≤ α2 exp ( − ∫ t t2 F (s) ds ) , for t ≥ t2 . (3.12) Proof. Assume that x is a positive solution of (1.1). Then, by Lemma 2.3, x is also a positive solution of (2.3). By Theorem 3.3, ψ(t) belongs to N0 for t ≥ t2. From (i) and (ii) of Lemma 3.4 we have x(t) R12(t) ≥ G1(t) ϕ(t) ϕ(t)ψ(t) ≥ G1(t) ϕ(t) ϕ(t2)ψ(t2). (3.13) On the other hand, integrating (2.3) from t to ∞ and taking into account Lemma 3.4(ii), we obtain (q1(t)ψ ′(t))′ = 1 q2(t) ∫ ∞ t g(s)R21(s)x(s) ds EJDE-2024/55 NONCANONICAL THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS 9 ≥ 1 q2(t) ∫ ∞ t g(s)R21(s)R12(s)G1(s)ψ(s) ds ≥ ψ(t)ϕ(t) q2(t) ∫ ∞ t g(s)R21(s)R12(s)G1(s) ϕ(s) ds. Once more integration yields −ψ′(t) ≥ ψ(t)ϕ(t) q1(t) ∫ ∞ t 1 q2(v) ∫ ∞ v g(s)R21(s)R12(s)G1(s) ϕ(s) ds dv, or equivalently ψ′(t) ψ(t) ≤ −F (t). Integrating the latter inequality from t2 to t yields x(t) R12(t) ≤ ψ(t) ≤ ψ(t2) exp ( − ∫ t t2 F (s) ds ) . (3.14) Combining (3.13) and (3.14) gives the desired result (3.12). □ 4. Examples In this section, we present two examples to illustrate the main results. Example 4.1. We consider the non-canonical differential equation with an ad- vanced neutral term,( t2 ( t2(x(t) + p0x(λt)) ′)′)′ + atx(t) = 0, t ≥ 1, (4.1) where a > 0, λ > 1, and 0 < p0 < 1/(2 − λ−1)2. Here r1(t) = r2(t) = t2, z(t) = x(t) + p0x(τ(t)), p(t) = p0, τ(t) = λt > t, g(t) = at, t0 = 1. Simple calculations show that R12(t) = R21(t) = 1 2t2 , q1(t) = q2(t) = 1 2 , Q1(t) = Q2(t) = (t− 1) 2 , Q12(t) = 2(t− 1)2. The transformation ψ(t) = z(t)/R12(t) yields the canonical equation ψ′′′(t) + 2a t x(t) = 0. First we check the conditions for applying Theorem 3.3. Condition (3.2) becomes p0 ( t− λ−1 t− 1 )2 < 1, ∀t ≥ t2 . Since λ > 1, for t > 1, it follows that t−λ−1 > 0 and t−λ−1 t−1 is decreasing. Therefore we select t2 = 2. Then (3.2) is implied by p0(2−λ−1)2 < 1, which follows from the choice of p0 in (4.1). Then for τ(t) ≥ t and t ≥ t2 = 2, we have G(t) = 1− p0 ( t− λ−1 t− 1 )2 ≥ 1− p0(2− λ−1)2 > 0 To check condition (3.9), we label the 2 integrals as I and II. Then I ≥ a ( 1− p0(2− λ−1)2 ) 2 1 (t− 1) ∫ t 2 ( 1− s−1)3 ds 10 G. PURUSHOTHAMAN, K. SURESH, E. THANDAPANI, E. TUNÇ EJDE-2024/55 and II ≥ a ( 1− p0(2− λ−1)2 ) 2 (t− 1)2 ∫ ∞ t s−3 ds Adding the above inequalities, and using L’Hopital’s Rule to compute the limits as t→ ∞, we have lim t→∞ I + II ≥ 3a 4 ( 1− p0(2− λ−1)2 ) which will be greater than 1 if we choose a > 4 3 ( 1− p0(2− λ−1)2 ) . Under this condition all positive solutions of (4.1) are Kneser-type solutions. To check the conditions for Theorem 3.5, we observe that G1(t) = 1 − p0λ −2. Since p0 < 1 and λ > 1, we have G1(t) > 0 for all t ≥ 1. Also we have ϕ(t) = ta/2, ∆(t) = ∫ ∞ t 2 ∫ ∞ s aw 1 2w2 1 2w2 dw ds = a 2 ∫ ∞ t ∫ ∞ s 1 w3 dw ds = a 4t , F (t) = 2ta/2 ∫ ∞ t 2 ∫ ∞ s aw 1 2w2 1 2w2 ( 1− p0λ −2 ) /wa/2 dw ds = 4a ( 1− p0λ −2 ) (4 + a)(2 + a) t−1, exp ( − ∫ t 1 F (s) ds ) = t−δ, where δ = 4a ( 1− p0λ −2 ) (4 + a)(2 + a) . By Theorem 3.5 there are positive constants α1 and α2 such that all positive solutions to (4.1) satisfy α1t −2− a 2 ≤ x(t) ≤ α2t −2−δ . As a particular case p0 = 1/18, λ = 2, and a = 2 yield the bounds α1t −3 ≤ x(t) ≤ α2t −503/216 . When p0 = 0, Theorem 3.5 yields an estimate for ordinary differential equation( t2 ( t2x′(t) )′)′ + atx(t) = 0, a > 0, t ≥ 1; namely α1t −2− a 2 ≤ x(t) ≤ α2t −2− 4a (4+a)(2+a) . Example 4.2. We consider the non-canonical differential equation with an delayed neutral term, ( t2 ( t2(x(t) + p0x(λt)) ′)′)′ + atx(t) = 0, t ≥ 1, (4.2) where a > 0, τ(t) = λt with λ < 1, 0 < p0 < λ2+ a 2 . The expression for g, q, q, r, R, ϕ,∆ are the same as in Example 4.1. First we check the conditions for applying Theorem 3.3. Condition (3.1) becomes p0 1 λ2 < 1, ∀t ≥ t2 , which is implied by the assumption 0 < p0 < λ2+ a 2 . Setting t2 = 1 we have G(t) = 1− p0λ −2 > 0, ∀t ≥ t2 = 1 . EJDE-2024/55 NONCANONICAL THIRD-ORDER NEUTRAL DIFFERENTIAL EQUATIONS 11 To check condition (3.9), we label the 2 integrals as I and II. Then I = a 2 ( 1− p0λ −2 ) 1 (t− 1) ∫ t 1 ( 1− s−1)3 ds and II = a 2 ( 1− p0λ −2 ) (t− 1)2 ∫ ∞ t s−3 ds . Adding the above equalities, and using L’Hopital’s Rule to compute the limits as t→ ∞, we have lim t→∞ I + II = 3a 4 ( 1− p0λ −2), which will be greater than 1 if we choose a > 4 3 ( 1− p0λ−2) . Under this condition all positive solutions of (4.2) are Kneser-type solutions. To check the conditions for Theorem 3.5, we compute the following expressions: G1(t) = 1− p0λ −2− a 2 , which is positive by the assumption 0 < p0 < λ2+ a 2 . Then we compute F (t) = 4a ( 1− p0λ −2− a 2 ) (4 + a)(2 + a) t−1, exp ( − ∫ t 1 F (s) ds ) = t−γ , where γ = 4a ( 1− p0λ −2− a 2 ) (4 + a)(2 + a) . By Theorem 3.5 there are positive constants α1 and α2 such that all positive solutions to (4.2) satisfy α1t −2− a 2 ≤ x(t) ≤ α2t −2−γ . As a particular case p0 = 1/16, λ = 1/2, and a = 2, we have the bounds α1t −3 ≤ x(t) ≤ α2t −13/6 . 5. Conclusion In this study, we derived conditions for the existence and estimates of Kneser- type solutions of the noncanonical third-order differential equations with a delay or advanced argument in the neutral term. This was achieved by transforming the noncanonical equation (1.1) into a canonical form without adding any new condi- tions. We obtained estimates for the Kneser-type solutions of (1.1), which are new contribution to the literature. These estimates for Kneser-type solutions are not easily obtained for the noncanonical equation (1.1) without such a transformation. Furthermore, the results from references [2, 3, 4, 7, 20, 21, 23] do not apply to equations (4.1) and (4.2) as they are noncanonical. It is interesting to study similar properties of (1.1) when the neutral term is of mixed type. 12 G. PURUSHOTHAMAN, K. SURESH, E. THANDAPANI, E. TUNÇ EJDE-2024/55 References [1] B. Bacuĺıková; Asymptotic properties of noncanonical third order differential equations, Math. 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Graef, E. Thandapani; New oscillation results for third-order half-linear neutral differential equations, Mathematics, 8 (2020), Art. 325, 1–9. Ganesh Purushothaman Department of Mathematics, St. Joseph’s College of Engineering, Chennai - 600119, India Email address: gpmanphd@gmail.com Kannan Suresh Department of Mathematics, St. Joseph’s College of Engineering, Chennai - 600119, India Email address: dhivasuresh@gmail.com Ethiraju Thandapani Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chen- nai - 600005, India Email address: ethandapani@yahoo.co.in Ercan Tunç Department of Mathematics, Faculty of Arts and Sciences, Tokat Gaziosmanpaşa Uni- versity, 60240, Tokat, Türkiye Email address: ercantunc72@yahoo.com 1. Introduction 2. Preliminary results 3. Main results 4. Examples 5. Conclusion References