On asymptotic properties of solutions to third-order delay differential equations

The purpose of the paper is to show that the canonical operator L3 given by L3(·) = ( r2 ( r1(·) )′)′ where the functions ri(t) ∈ C([t0, ∞), [0, ∞)) satisfy ∫ ∞ t0 ds ri(s) = ∞, (i = 1, 2), can be written in a certain strongly noncanonical form L3(·) ≡ b3 ( b2 ( b1 (b0(·)) )′)′ , such that the functions bi(t) ∈ C([t0, ∞), [0, ∞)) satisfy ∫ ∞ t0 ds bi(s) < ∞, (i = 1, 2). We study some relations between canonical and strongly noncanonical operators, showing the advantage of this reverse approach based on the use of a noncanonical representation of L3 in the study of oscillatory and asymptotic properties of third-order delay differential equations.

Under a solution of equation (E), we mean a nontrivial function y ∈ C 1 ([T y , ∞), R) with T y ≥ t 0 , which has the property L 1 y, L 2 y ∈ C 1 ([T y , ∞), R), and satisfies (E) on [T y , ∞).We only consider those solutions of (E) which exist on some half-line [T y , ∞) and satisfy the condition sup{|y(t)| : T ≤ t < ∞} > 0 for any T ≥ T y .
As is customary, a solution y of (E) is said to be oscillatory if it is neither eventually positive nor eventually negative.Otherwise, it is said to be nonoscillatory.The equation itself is termed oscillatory if all its solutions oscillate.From Trench theory [18], it is known that L 3 y can be always written in an equivalent canonical form and uniquely determined up to positive multiplicative constants with the product 1.The explicit forms of functions a i generally depend on the convergence or divergence of certain integrals and may be calculated using the proof of Lemmas 1 and 2 in [18].As a matter of fact, the investigation of asymptotic properties of canonical third-order differential equations, especially with regard to oscillation and nonoscillation, has became the subject of extensive research, see e.g.[1-9, 11, 12, 15, 17] and the references cited therein.The purpose of the paper is to show the reverse, i.e. that the canonical operator L 3 can be written in a certain strongly noncanonical form Consequently, we study some relations between canonical and strongly noncanonical operators and corresponding classes of nonoscillatory solutions of studied equations, showing the advantage and usefulness of this reverse approach based on the use of a noncanonical representation of L 3 in the study of oscillatory and asymptotic properties of solutions of third-order delay differential equations.

Noncanonical representation
Define the functions In the sequel, we will assume that L 3 is in canonical form, that is, The following result is a modification of the well known Kiguradze lemma [13, Lemma 1.1] based on (H 3 ).
Lemma 2.1.Assume (H 1 )−(H 3 ).The set of all nonoscillatory solutions y of (E) can be divided into the following two classes Proof.By some computations, we have Integrating the equality Therefore, 3) By virtue of (2.2), we see that Using the l'Hospital rule, we have Hence, Convergence of the second integral in (2.4) can be shown in the same way.The proof is complete.
Corollary 2.3.The equation (E) possesses a solution y if and only if the equation has a solution x = y/R 12 .
Similarly as before, one can define the operators where b i , i = 1, 2 are as in Theorem 2.2.Also, we set q(t) = q(t)R 21 (t)R 12 (τ(t)).
Then (E ) can be rewritten in the form Let us explore various asymptotic properties of (E ) which will be useful in the next.The following obvious result gives the structure of possible nonoscillatory solutions of (E ).Lemma 2.4.Assume (H 1 )−(H 3 ).The set of all nonoscillatory solutions x = y/R 12 of (E ) can be divided into the following four classes ) Proof.To show the nonexistence of solutions from classes N a and N b , we proceed as the in proof of cases ( 3) and ( 4), respectively, in [10, Theorem 1].
In the sequel, we consider the following auxiliary functions then every nonoscillatory solution x(t) ∈ N 0 of (E ) satisfies Proof.Let x(t) be a positive solution of (E ) such that x(t) ∈ N 0 eventually, say for t ≥ t 1 , where t 1 ∈ [t 0 , ∞) is large enough.Assume on the contrary that lim t→∞ x(t) = > 0. An integration of (E ) yields On the other of hand, since lim t→∞ π 1 (t) = 0, (2.7) implies that ∞ t 1 q(s) ds = ∞.In view of (2.8), this, however, contradicts the fact that L 2 x is decreasing and we conclude that x(t) → 0 as t → ∞.Now assume that lim t→∞ L 1 x(t) = − < 0. Then − L 1 x(t) ≥ eventually, and so Integrating (E ) from t 1 to ∞ and using (2.7) and (2.9) in the resulting inequality yield A contradiction and the proof is complete.
The next result is crucial in establishing important relations between solutions of (E) and those of the corresponding strongly noncanonical equation (E ). Hence, ≤ 0, which implies that L 1 x(t)/π 2 (t) is decreasing.Therefore, and we conclude that Now we assume that x(t) ∈ N * .By virtue of the fact that − L 1 x is increasing, we have .
In view of Lemma 2.5, the essential classes for (E ) are N 0 and N * .In the next main result, they will be shown, under weak assumptions, to be equivalent to classes N 0 and N 2 of (E), respectively.

Applications
In this section, we provide some oscillation criteria for (E) in two ways: using directly (E) and also using a strongly noncanonical corresponding equation (E ).Subsequently, we test the strength of these results on Euler type equations, showing the advantage of making use the strongly noncanonical equation (E ).
As usual, all functional inequalities considered in this paper are supposed to be satisfied for all t large enough.
then any nonoscillatory solution y of (E) belongs to the class N 0 .
Proof.Let y(t) be a nonoscillatory solution of (E).By Lemma 2.1, either y ∈ N 0 or y ∈ N 2 .
Assume on the contrary that y ∈ N 2 .Without loss of generality, we may take t 1 ≥ t 0 such that Next, we claim that (3.1) implies Assume not, i.e. lim t→∞ L 2 y(t) = > 0. Then L 2 y(t) ≥ eventually, say for t * ≥ t 1 and so y(t) ≥ R 12 (t).Using this inequality in (E) and integrating the resulting inequality from t * to t, we see that Since ∞ t 0 q(s)R 12 (τ(s))ds = ∞ is necessary for the validity of (3.1), condition (3.3) clearly contradicts the fact that L 2 y is decreasing.Thus, (3.2) holds.On the other hand, it follows from the monotonicity of L 2 y(t) that for t ≥ t 2 , where t 2 > t 1 is large enough.Dividing both sides of the latter inequality by r 1 (t) and integrating the resulting inequality from t 2 to t, we get Proof.By Theorem 2.8 and the proof of Theorem 3.3, it suffices to show that (E ) does not possess a solution x ∈ N 0 .Assume the contrary.Without loss of generality, we may take t 1 ≥ t 0 such that Proceeding the same as in the proof of case (2) of [9, Theorem 2], we arrive at contradiction with (3.7).The proof is complete.
Remark 3.7.In general, the nonexistence of solutions of (E) belonging to the class N 0 is due to a delay argument only.The idea of improving the criteria eliminating such solutions by rewriting the equation into a strongly noncanonical form which we present in this paper deserves to be further studied.