On Sharp Oscillation Criteria for General Third-Order Delay Differential Equations

: In this paper, effective oscillation criteria for third-order delay differential equations of the form, (cid:16) r 2 ( r 1 y (cid:48) ) (cid:48) (cid:17) (cid:48) ( t ) + q ( t ) y ( τ ( t )) = 0 ensuring that any nonoscillatory solution tends to zero asymptotically, are established. The results become sharp when applied to a Euler-type delay differential equation and, to the best of our knowledge, improve all existing results from the literature. Examples are provided to illustrate the importance of the main results.


Introduction
In this article, we consider linear third-order delay differential equations of the form r 2 r 1 y (t) + q(t)y(τ(t)) = 0, t ≥ t 0 , where r 1 , r 2 , q, τ ∈ C(I, R), I = [t 0 , ∞) ⊂ R, t 0 > 0 is a fixed constant such that r 1 > 0, r 2 > 0, q ≥ 0 does not vanish eventually, τ(t) ≤ t, and lim t→∞ τ(t) = ∞. For any solution y of (1), we denote the ith quasi-derivative of y as L i y, that is, L 0 y = y, L 1 y = r 1 y , L 2 y = r 2 r 1 y , L 3 y = r 2 r 1 y on I and assume that By a solution of Equation (1), we mean a nontrivial function y with the property L i y ∈ C 1 ([T y , ∞), R) for i = 0, 1, 2 and a certain T y ≥ t 0 , which satisfies (1) on [T y , ∞). Our attention is restricted to proper solutions of (1), which exist on some half-line [T y , ∞) and satisfy the condition sup{|x(s)| : t ≤ s < ∞} > 0 for any t ≥ T y .
The oscillatory nature of the solutions is understood in the usual way, that is, a proper solution is termed oscillatory or nonoscillatory according to whether it does or does not have infinitely many zeros.
The oscillation theory of third-order differential equations with variable coefficients has been attracting considerable attention over the last decades, which is evidenced by a large number of published studies in the area, most of which have been collected and presented in the monographs [4,5].
In particular, various criteria for property A of (1) have been presented in the literature, see [3,[6][7][8][9][10][11][12][13][14][15][16][17] and the references cited therein. The methodology in these articles has been mainly based on the use of the so-called Riccati technique or suitable comparison principles with lower-order delay differential inequalities. In [3], the authors point out that the proofs essentially use the estimates relating a solution y ∈ N 2 of (1) with its first and second quasi-derivatives and "despite the differences in the proofs of the cited works, the resulting criteria have in common that their strength depends on the sharpness of these estimates". Here, it is worth noting that in order to test the strength of the oscillation criteria derived by different methods, Euler-type differential equations are mostly used.
For our comparison purposes, let us consider a particular case of (1)-the third-order Euler differential equation with proportional delay of the form where τ ∈ (0, 1], q 0 > 0, α < 1, and γ < 1. It is easy to verify by a direct substitution that (3) has a nonoscillatory solution y = t µ belonging to the class N 2 , when µ ∈ (1 − α, 2 − α − γ) is a root of the characteristic equation For a special case of (3) with α = γ = 0 and τ = 1, i.e., for the linear third-order Euler differential equation condition (5) for the existence of a solution from the class N 2 reduces to which is sharp in the sense that if We stress that there is no result so far in the literature on the property A of (1), which would be sharp for (3). The main purpose of the paper is to positively answer this open problem. Following the direction initiated in [3], we present new asymptotic properties of solutions belonging to the class N 2 . Our approach differs from that applied in [3] and allows us to relax the assumption of the monotonicity of the delay function τ(t), which is generally required in previous works. As a consequence, we establish efficient criteria for detecting property A for Equation (1), which are unimprovable in the sense that they give a necessary and sufficient condition for the delay Euler Equation (3) to have property A. Our motivation comes from the recent papers [18][19][20], where a similar technique leads to obtaining sharp oscillation results for second-order half-linear differential equations with deviating arguments. Such an idea was successfully adopted for the third-order Equation (1) with r 1 = r 2 = 1 in a recent work [21]. However, it turns out that the general functions r i require a carefully modified the approach.
The organization of the paper is as follows. In Section 2, we introduce the basic notations and assumptions. In Section 3, we state the main results of the paper. In particular, we present a single condition criteria for property A of (1) in case when the functions r 1 and r 2 are of the same type (see Definition 1 and condition (15) below). In Section 4, we illustrate the importance of the main results by means of a couple of examples.
As the limit inferior triple λ * , β * , and k * is defined on an extended range of real R ∪ {∞}, in our proofs, we will rather make use of real constants λ ≤ λ * , β ≤ β * and k ≤ k * defined by (C) λ , (C) β , and (C) k , respectively, for the particular cases that can occur depending on the delay function τ.
It is useful to note that there are two situations when the impact of the delay would not influence the value of β n in the sequence (11): λ * = 1 or k i = 1, i = 0, 1, . . . , n. Below, we point out that the second one cannot occur in a particular case, when coefficients r 1 and r 2 are of the same type, e.g., either r i = e a i t or r i = t a i and likewise. With this aim, we use a concept of asymptotically similar functions. Definition 1. We say that the functions f and g are asymptotically similar ( f ∼ g) if there exists a positive constant such that As a special case of (1), we will consider the case when Lemma 1. Assume (15). Then, for any c ∈ (0, 1), eventually.

Lemma 2. Let
Proof. Using l'Hôspital's rule, it is easily seen that Taking into account the fact that R 2 is increasing and (16) holds, we have, for any c ∈ (0, 1), The proof is complete.
Proof. It is simple to compute the limit (18) when r 1 = r 2 ; hence, we omit the details.
For the sake of convenience, we assume here that all functional inequalities hold eventually, that is, they are satisfied for all t that are large enough. As usual and without loss of generality, we can assume from now on that nonoscillatory solutions of (1) are eventually positive.

Nonexistence of Solutions from the Class N 2
In this section, we give a series of lemmas about the asymptotic properties of solutions belonging to the class N 2 , which will play a crucial role in proving our main oscillation results stated in Section 3.3.
(ii) Again, using the fact that L 2 y is positive and decreasing, it follows that In view of (i), there is a t 2 > t 1 , such that Thus, L 1 y(t) > L 2 y(t)R 2 (t), t ≥ t 2 and consequently, which proves (ii).
(iii) In view of the fact that L 1 y/R 2 is a decreasing function tending to zero, we have which proves (iii). The proof is complete.
The next lemma provides some additional properties of solutions from the class N 2 .
By virtue of (iii), we have for t ≥ t 2 for some t 2 ≥ t 1 . Integrating from t 2 to t and using the fact that L 1 y/R 2 is decreasing and tends to zero asymptotically (see (i) and (ii)), there exists t 3 ≥ t 2 such that Then, and It follows directly from (23) and the fact that L 1 y is increasing that β < 1. Using this in (22) and taking (19) into account, we find that there is t 4 ≥ t 3 such that and The conclusion of this is in the following. (b 0 ) Clearly, (24) also implies that L 1 y/R 1−β * 2 → 0 as t → ∞, since otherwise which is a contradiction.
(c 0 ) Using the fact that by (a 0 ) and (b 0 ), L 1 y/R 1−β * 2 is a decreasing function tending to zero, we have Therefore, The proof is complete.
Proof. This follows directly from (a 0 ) and the fact that L 2 y is positive.
Since λ can be arbitrarily large, we can set λ > (1/kβ) k/(k−1) , which contradicts the positivity of L 2 y. The proof is complete. Proof. Using the fact that k can be arbitrarily large, the proof follows the lines of Corollary 3, and so we omit it.

Lemma 5.
Assume β * > 0 and let y be an eventually positive solution of (1) belonging to N 2 . Then, for any n ∈ N 0 , β n and k n defined by (11) and (12), respectively, and for a t that is sufficiently large: (a n ) (1 − β n )L 1 y > R 2 L 2 y and L 1 y/R 1−β n 2 decrease; (c n ) y > ε n k n (R 12 /R 2 )L 1 y and y/R 1/(ε n k n ) 12 is decreasing for any ε n ∈ (0, 1).
Proof. Let y ∈ N 2 with y(τ(t)) > 0 satisfy the conclusion of Lemma 3 for t ≥ t 1 ≥ t 0 and choose fixed but arbitrarily large β ≤ β * and k ≤ k * , satisfying (9) and (10), respectively, for t ≥ t 1 . We will proceed by induction on n. For n = 0, the conclusion follows from Lemma 4 with ε 0 = k/k * . Next, assume that (a n )-(c n ) hold for n ≥ 1 for t ≥ t n ≥ t 1 . We need to show that they each hold for n + 1.
Choose µ such that where n satisfies (14). Then, and there exist two constants c 1 ∈ (0, 1) and c 2 > 0 such that In view of the definition (20) of z, we see that and Using the above monotonicity in (27), we find that there exists t n ≥ t n that is sufficiently large such that Then, and from which the conclusion follows. (b n+1 ) Clearly, (30) also implies that L 1 y/R which is a contradiction.
In view of the above corollary and (13), the sequence {β n } defined by (11) is increasing and bounded from the above, i.e., there exists a limit lim n→∞ β n = β f ∈ (0, 1) satisfying the equation where Then, the following crucial result on the nonexistence of N 2 -type solutions is immediate.

Convergence to Zero of Kneser Solutions
In this section, we state some results ensuring that any Kneser solution converges to zero asymptotically. We start by pointing out the useful fact that is necessary for the existence of an unbounded nonoscillatory solution. For the reader's convenience, we state its one-line proof.
Proof. Assume, on the contrary, that y is a nonvanishing, nonoscillatory, positive solution of (1), i.e., y(t) ≥ ξ > 0 for t ≥ t 1 . Then, the integration of (1) from t 2 to t yields which contradicts the positivity of L 2 y.
Integrating (40) from t 2 to t, we obtain −L 1 y(t) ≤ −L 1 y(t 2 ) − t t 2 ξ r 2 (u) ∞ u q(s)ds du → −∞ as t → ∞, which contradicts the positivity of −L 1 y. which is depicted in Figure 1-see the orange line. We can also observe from Figure 1 (see the green line) that if q 0 < max{c(µ) : µ < 0} 2.944, then (5) has a couple of Kneser solutions tending to zero asymptotically. The remaining open problem stated below in Remark 2 is to prove a general criterion for the nonexistence of Kneser solutions of (1), which would reduce to q 0 > max{c(µ) : µ < 0} when applied to the Euler equation (5).
Next, we consider the situation when r 1 and r 2 are not of same type.
Clearly, a positive β * implies that the integral (37) is divergent, i.e., Finally, we illustrate the case with non-proportional delay argument.