Asymptotic behavior of third-order differential equations with nonpositive neutral coefficients and distributed deviating arguments

Using the Riccati transformation technique, we present several sufficient conditions that guarantee that all solutions to a third-order differential equation with nonpositive neutral coefficients and distributed deviating arguments are either oscillatory or converge to zero asymptotically. In particular, we establish Hille and Nehari type criteria. Two examples are given to demonstrate the practicability of the main results.


Introduction
Third-order differential equations have attracted noticeable interests due to their potential applications in assorted fields, including physical sciences, technology, population dynamics, and so on. Recently, the qualitative theory of third-order differential equations has become an interesting topic, and there have been some results on the oscillatory and asymptotic behavior of third-order equations; see, for example, the monographs [, ], the papers [-], and the references therein. In particular, it is a necessary and valuable issue, either theoretically or practically, to investigate differential equations with distributed deviating arguments; see the papers by Tian et al. [], Wang [], and Wang and Cai []. On the basis of these background details, the objective of this paper is to analyze the oscillation and asymptotic properties of a class of third-order neutral differential equations [, ∞)), q(t, ξ ) is not identically zero for large t, R), and there exists a positive constant k such that f (x)/x α ≥ k for all x = .
We assume that solutions of (.) exist for any t ∈ [t  , ∞). Our attention is restricted to those solutions of (.) that are not identically zero for large t. As usual, a solution of (.) is called oscillatory if it has arbitrarily large zeros on the interval [t  , ∞). Otherwise, it is termed nonoscillatory (i.e., it is either eventually positive or eventually negative).
It is known that analysis of neutral differential equations is more difficult in comparison with that of ordinary differential equations, although certain similarities in the behavior of solutions of these two classes of equations are observed; see, for example, [, , , -, , , -, -, -] and the references therein. Assuming that under the assumptions that  ≤ p(t) ≤ p  <  and (.) holds. It should be noted that condition (.) is a restrictive condition in the study of asymptotic behavior of third-order differential equations. To solve this problem without requiring (.), Li et al.
[] obtained some oscillation criteria for a third-order neutral delay differential equation by employing the Riccati substitution , the principal goal of this paper is to give an affirmative answer to this question. In Section , some lemmas are provided to prove the main results. In Section , some oscillation results for (.) are obtained by using the Riccati transformation technique, and these results also can be applied to the cases where r (t) ≤  or r (t) is oscillatory. In Section , two illustrative examples are included. All functional inequalities considered in the sequel are tacitly assumed to hold for all t large enough.

Several lemmas
Lemma . Assume that x(t) is an eventually positive solution of (.). Then there exists a t  ≥ t  such that, for t ≥ t  , z(t) has the following four possible cases: Proof Let x(t) be an eventually positive solution of (.). Then there exists a t  ≥ t  such that, for t ≥ t  , It follows from (.) and the definition of z(t) that x(t) ≥ z(t) and Hence, r(t)(z (t)) α is nonincreasing and of one sign, which implies that z (t) is also of one sign. Therefore, there exists a t Integrating this inequality from t  to t, we conclude that Letting t → ∞, we have that z (t) → -∞, and so z (t) <  eventually. Note that the conditions z (t) <  and z (t) <  imply that z(t) < . Thus, we get case (iv). Lemma . Assume that x(t) is an eventually positive solution of (.) and the corresponding z(t) satisfies case (i) in Lemma .. Then there exist two numbers t  ≥ t  and t  > t  such that, for t ≥ t  , Hence, we deduce that ds is nonincreasing eventually, and so This completes the proof.
Lemma . Let x(t) be an eventually positive solution of (.) and assume that the corre- Proof It follows from property (ii) that there exists a finite constant l ≥  such that lim t→∞ z(t) = l. We claim that l = . Otherwise, assume that l > . By the definition of z(t), x(t) ≥ z(t) > l. An application of (.) yields Integrating the latter inequality from t to ∞, we have which implies that Integrating this inequality from t to ∞ and then integrating the resulting inequality from t  to ∞, we conclude that which is a contradiction to (.). Hence, l =  and lim t→∞ z(t) = .
Next, we prove that x(t) is bounded. If not, then there exists a sequence {t m } such that lim m→∞ t m = ∞ and lim m→∞ x( which yields lim m→∞ z(t m ) = ∞. This contradicts lim t→∞ z(t) = . Therefore, x(t) is bounded, and hence we may suppose that lim sup t→∞ x(t) = a  , where  ≤ a  < ∞. Then, there exists a sequence {t k } such that lim k→∞ t k = ∞ and lim k→∞ x(t k ) = a  . Assuming now that a  >  and letting ε := a  (p  )/(p  ), we have x(τ (t k , μ)) < a  + ε eventually, and thus which is a contradiction. Thus, a  =  and lim t→∞ x(t) = . The proof is complete.

Main results
In what follows, we let where the meaning of ρ(t) will be explained later.

Theorem . Assume that condition (.) is satisfied. If there exists a function ρ(t) ∈
C  ([t  , ∞), (, ∞)) such that, for all sufficiently large t  ≥ t  and for some t  > t  > t  , then every solution x(t) of (.) is either oscillatory or converges to zero as t → ∞.
Proof Suppose to the contrary that (.) has a nonoscillatory solution x(t). Without loss of generality, we may assume that x(t) is eventually positive (since the proof of the case where x(t) is eventually negative is similar). By Lemma ., we observe that, for t ≥ t  ≥ t  , z(t) satisfies four possible cases (i), (ii), (iii), or (iv) (as those of Lemma .). We consider each of the four cases separately. Assume first that case (i) is satisfied. For t ≥ t  , define the Riccati transformation ω(t) by Then ω(t) >  for t ≥ t  . Differentiation of (.) yields It follows from (.) and (i) that Using (.) and (.) in (.), we deduce that Since σ * (t) ≤ t and z (t)/ t t  r -/α (s) ds is nonincreasing (see Lemma .), we have It follows now from Lemma . and (.) that where G(t) is defined by (.). Substituting (.) into (.), we get Set v := ω(t), A := α (r(t)ρ(t)) /α , and B := Using the inequality (see []) Substituting the latter inequality into (.), we conclude that Integrating this inequality from t  (t  > t  ) to t, we arrive at which contradicts (.). Suppose that case (ii) is satisfied. By Lemma ., lim t→∞ x(t) = . If case (iii) or case (iv) holds, then lim t→∞ z(t) = c  <  (possibly c  = -∞) or lim t→∞ z(t) = -∞, respectively. Proceeding similarly as in the proof of Lemma ., we conclude that x(t) and z(t) are bounded. Hence, c  is finite, and case (iv) does not occur. Similar analysis to that in Lemma . leads to the conclusion that lim t→∞ x(t) = . This completes the proof.
Letting ρ(t) = t and ρ(t) = , we can derive the following results from Theorem ..

Corollary . Let condition (.) hold. If for all sufficiently large t  ≥ t  and for some t
where G(t) is as in (.), then the conclusion of Theorem . remains intact.
Corollary . Let condition (.) be satisfied. If for all sufficiently large t  ≥ t  and for some where G(t) is defined by (.), then the conclusion of Theorem . remains intact.
In what follows, we establish Hille and Nehari type criteria for (.). To this end, we introduce the following lemma.
Lemma . Let x(t) be an eventually positive solution of (.). Define where G(t) is defined by (.), t  ≥ t  is sufficiently large, and t  > t  > t  .
(I) Letp < ∞,q < ∞, and suppose that the corresponding z(t) satisfies case (i) in Lemma .. Then Proof Part (I). Assume that x(t) is an eventually positive solution of (.) and the corresponding z(t) satisfies (i). By (.), we have ω(t) >  and As in the proof of Theorem ., we get (.) and (.), and so On the other hand, we conclude that due to the proof of Lemma .. Hence, Taking the lim sup of both sides of the latter inequality as t → ∞, we havē It follows from (.) and (.) that p ≤r -r +/α ≤r ≤R ≤  -q.
Moreover, by inequality (.), Therefore, the desired inequalities in (.) hold. This completes the proof of Part (I).
Part (II). Let x(t) be an eventually positive solution of (.). We show that z(t) does not have property (i). Assume the contrary. Suppose first thatp = ∞. Inequality (.) implies that Taking the lim inf of both sides of the latter inequality as t → ∞, we arrive at which is a contradiction. Assume now thatq = ∞. An application of inequality (.) yields which is also a contradiction. The proof of Part (II) is complete.
On the basis of Lemma ., we easily derive the following result with a proof similar to that of Theorem ..

Examples
The following examples illustrate applications of the main results in this paper.
Example . For t ≥ , consider the third-order differential equation Let α = , a = , b = π/, c = -π , d = -π , k = , r(t) = , p(t, μ) = /, τ (t, μ) = tμ, q(t, ξ ) = /, and σ (t, ξ ) = t + ξ /. Note that Remark . Observe that Theorems . and . cannot distinguish solutions of (.) with different behaviors. It is not easy to obtain sufficient conditions that ensure that all solutions x(t) of (.) just satisfy lim t→∞ x(t) =  and do not oscillate. Neither is it possible to utilize the technique exploited in this work for proving that all solutions of (.) are oscillatory. Therefore, two interesting problems for future research can be formulated as follows.
(P) Suggest a different method to establish asymptotic criteria that ensure that all solutions of (.) tend to zero asymptotically. (P) Is it possible to establish sufficient conditions that guarantee that all solutions of (.) are oscillatory?