On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations

By using comparison principles, we analyze the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Due to less restrictive assumptions on the coefficients of the equation and on the deviating argument τ , our criteria improve a number of related results reported in the literature. ©


Introduction
Higher-order functional differential equations have numerous applications in engineering and natural sciences, see Hale [1].For instance, one can describe the behavior of solutions to third-order partial differential equations using information about the asymptotics of solutions to associated delay differential equations; see Agarwal et al. [2] for more details.In this paper, we are concerned with the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations where t ∈ I := [t 0 , ∞) ⊂ (0, ∞), γ is a ratio of odd positive integers, z(t) := x(t) + p 0 x(t − δ 0 ), p 0 ≥ 0, p 0 ̸ = 1, and δ 0 are constants, δ 0 ≥ 0 (delayed argument) or δ 0 ≤ 0 (advanced argument), a, q, τ ∈ C(I, R), a(t) > 0, q(t) ≥ 0, q(t) is not identical to zero for large t, and lim t→∞ τ (t) = ∞.Let t * := min{t 0 − δ 0 , min t∈I τ (t)}.By a solution of Eq. ( 1) we understand a function x ∈ C([t * , ∞), R) such that a(z ′′ ) γ ∈ C 1 (I, R) and x satisfies (1) on I.We consider only solutions of Eq. (1) which satisfy sup{|x(t)| : t ≥ T } > 0 for all T ≥ t 0 and tacitly assume that (1) possesses such solutions.A solution x (t) of ( 1) is said to be oscillatory if it has arbitrarily large zeros on [t x , ∞) for some t x ∈ I depending on the solution; otherwise, it is called nonoscillatory.
Usually, a second-order differential equation is called oscillatory if all its solutions oscillate.This is not the case for third-order equations whose solutions often exhibit different asymptotic behavior.Thus a thirdorder differential equation is called oscillatory if it has at least one oscillatory solution, see Erbe [3] and Hanan [4].Furthermore, the presence of functional argument in the equation may significantly affect the nature of solutions.For example, a third-order linear differential equation ) .However, one can completely eliminate all nonoscillatory solutions introducing the delayed argument and considering a third-order linear delay differential equation By the result due to Ladas et al. [5,Theorem 1], all solutions to the latter equation are oscillatory since the associated characteristic equation λ 3 + e −πλ = 0 has no real roots.We note that such drastic changes in the asymptotic behavior of solutions are not specific for third-order equations and can be observed also for first-order differential equations.Taking into account that under our assumptions differential equation ( 1) can be both delayed and advanced and that we are concerned in this paper only with the asymptotic behavior of solutions, we tacitly assume that solutions to the equation under study exist and can be continued to infinity.
Candan [13] analyzed behavior of solutions to (1) assuming that (3) holds and Finally, Li and Rogovchenko [15] investigated Eq. ( 1) under conditions (6) and 0 ≤ p 0 < ∞.The objective of this paper is to analyze the asymptotic nature of solutions to Eq. ( 1) in the case where condition ( 2) is satisfied but without assumptions (3)-( 6).In the sequel, all functional inequalities are supposed to hold for all t large enough.Without loss of generality, we deal only with eventually positive solutions of (1) since, under our assumption on γ, if x(t) is a solution of Eq. ( 1), so is −x(t).

Auxiliary lemmas
The following lemmas will be used to establish our main results.

Lemma 1. Assume that condition (2) is satisfied and let x(t) be an eventually positive solution of Eq. (1).
Then there exists a sufficiently large t 1 ≥ t 0 such that, for all t ≥ t 1 , either Proof .Thanks to condition (2) employed also by Baculíková and Džurina [10, Lemma 1], the proof follows the same lines as in the cited paper since assumptions (3)-( 5 Lemma 3. Let x(t) be an eventually positive solution of Eq. ( 1) and assume that z(t) satisfies (8).If Proof .By virtue of inequalities z(t) > 0 and z ′ (t) < 0, there exists a constant z 0 ≥ 0 such that lim t→∞ z(t) = z 0 .We claim that z 0 = 0. Otherwise, using Lemma 2, we conclude that lim t→∞ x(t) = z 0 / (1 + p 0 ) > 0. Then there should exist a t 2 ≥ t 0 such that, for all t ≥ t 2 , It follows from ( 1) and ( 11) that Integrating this inequality from t to ∞, we conclude that Integrating ( 12) from t to ∞, we have One more integration from t 2 to ∞ yields which contradicts condition (9).Therefore, lim t→∞ z(t) = 0, and the desired property (10) follows now from the inequality 0 < x(t) ≤ z(t).■

Main results
Theorem 4. Let conditions (2) and (9) be satisfied and assume that Suppose that there exists a function η ∈ C(I, R) such that η(t) ≤ τ (t), η(t) < t, and lim t→∞ η(t) = ∞.If the first-order delay differential equation is oscillatory for all large t 1 ≥ t 0 and for some t 2 ≥ t 1 , then every solution x(t) of Eq. ( 1) is either oscillatory or satisfies (10).
Proof .Condition (21) ensures that, by virtue of the result in Ladde et al. [9, Theorem 2.1.1],Eq. ( 14) is oscillatory.An application of Theorem 4 completes the proof.■ The next result relates oscillation of (1) in the case when to that of an associated first-order delay differential equation.

Examples and discussion
The following examples illustrate theoretical results presented in the previous section.In both examples, t ≥ 1 and p 0 ̸ = 1 is a nonnegative real number.
Remark 10.An important feature that distinguishes our results from many related theorems reported in the literature is that we do not impose specific restrictions on the deviating argument τ , that is, τ may be delayed, advanced, or change back and forth from advanced to delayed.On the other hand, we would like to point out that, contrary to Baculíková and Džurina [10,11], Candan [13], Džurina et al. [14], Li and Rogovchenko [15], Li et al. [16,17], and Yang and Xu [19], in our results we do not need restrictive conditions (3)-( 6), which is an improvement compared to the results in the cited papers.
Remark 11.Theorems 4 and 6 and Corollaries 5 and 7 ensure that every solution x(t) of Eq. ( 1) is either oscillatory or tends to zero as t → ∞.Since the sign of the derivative z ′ (t) changes, it is hard to derive sufficient conditions which ensure that all solutions of Eq. ( 1) are just oscillatory and do not satisfy (10).
Neither is it possible to utilize the method exploited in this paper for proving that all solutions of Eq. ( 1) only have the property (10).These two interesting problems remain open for now.