The purpose of this paper is to examine oscillatory properties of the third-order neutral delay differential equation [a(t)(b(t)(x(t)+p(t)x(σ(t)))′)′]′+q(t)x(τ(t))=0. Some oscillatory and asymptotic criteria are presented. These criteria improve and complement those results in the literature. Moreover, some examples are given to illustrate the main results.

1. Introduction

This paper is concerned with the oscillation and asymptotic behavior of the third-order neutral differential equation[a(t)(b(t)(x(t)+p(t)x(σ(t)))′)′]′+q(t)x(τ(t))=0.
We always assume that

We set z(t):=x(t)+p(t)x(σ(t)). By a solution of (E), we mean a nontrivial function x(t)∈C([Tx,∞)),Tx≥t0, which has the properties z(t)∈C1([Tx,∞)), b(t)z′(t)∈C1([Tx,∞)), a(t)(b(t)z′(t))′∈C1([Tx,∞)) and satisfies (E) on [Tx,∞). We consider only those solutions x(t) of (E) which satisfy sup{|x(t)|:t≥T}>0 for all T≥Tx. We assume that (E) possesses such a solution. A solution of (E) is called oscillatory if it has arbitrarily large zeros on [Tx,∞); otherwise, it is called nonoscillatory. Equation (E) is said to be almost oscillatory if all its solutions are oscillatory or convergent to zero asymptotically.

Recently, great attention has been devoted to the oscillation of differential equations; see, for example, the papers [1–30]. Hartman and Wintner [9], Hanan [10], and Erbe [8] studied a particular case of (E), namely, the third-order differential equation x′′′(t)+q(t)x(t)=0.
Equation (E) with p(t)=0 plays an important role in the study of the oscillation of third-order trinomial delay differential equation x′′′(t)+p(t)x′(t)+g(t)x(τ(t))=0,
see [6, 12, 27]. Baculíková and Džurina [21, 22], Candan and Dahiya [25], Grace et al. [28], and Saker and Džurina [30] examined the oscillation behavior of (E) with p(t)=0. It seems that there are few results on the oscillation of (E) with a neutral term. Baculíková and Džurina [23, 24] and Thandapani and Li [17] investigated the oscillation of (E) under the assumption b(t)=1,∫t0∞1a(t)dt=∞,a′(t)≥0.
Graef et al. [13] and Candan and Dahiya [26] considered the oscillation of [a(t)(b(t)(x(t)+p1x(t-σ))′)′]′+q(t)x(t-τ)=0,0≤p1<1.

In this paper, we shall further the investigation of the oscillations of (E) and E1. Three cases:∫t0∞1a(t)dt=∞,∫t0∞1b(t)dt=∞,∫t0∞1a(t)dt<∞,∫t0∞1b(t)dt=∞,∫t0∞1a(t)dt<∞,∫t0∞1b(t)dt<∞,
are studied.

In the following, all functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough. Without loss of generality, we can deal only with the positive solutions of (E).

2. Main Results

In this section, we will give the main results.

Theorem 2.1.

Assume that (1.4) holds, 0≤p(t)≤p1<1. If for some function ρ∈C1([t0,∞),(0,∞)), for all sufficiently large t1≥t0 and for t3>t2>t1, one has
limsupt→∞∫t3t(ρ(s)q(s)(1-p(τ(s)))∫t2τ(s)(∫t1v(1/a(u))du/b(v))dv∫t1s(1/a(u))du-a(s)(ρ′(s))24ρ(s))ds=∞,∫t0∞1b(v)∫v∞1a(u)∫u∞q(s)dsdudv=∞,
then (E) is almost oscillatory.

Proof.

Assume that x is a positive solution of (E). Based on the condition (1.4), there exist two possible cases:

z(t)>0,z′(t)<0,(b(t)z′(t))′>0,[a(t)(b(t)z′(t))′]′<0 for t≥t1, t1 is large enough.

Assume that case (1) holds. We define the function ω by
ω(t)=ρ(t)a(t)(b(t)z′(t))′b(t)z′(t),t≥t1.
Then, ω(t)>0 for t≥t1. Using z′(t)>0, we have
x(t)≥(1-p(t))z(t).
Since
b(t)z′(t)≥∫t1ta(s)(b(s)z′(s))′a(s)≥a(t)(b(t)z′(t))′∫t1t1a(s)ds,
we have that
(b(t)z′(t)∫t1t(1/a(s))ds)′≤0.
Thus, we get
z(t)=z(t2)+∫t2tb(s)z′(s)∫t1s(1/a(u))du∫t1s(1/a(u))dub(s)ds≥b(t)z′(t)∫t1t(1/a(u))du∫t2t∫t1s(1/a(u))dub(s)ds,
for t≥t2>t1. Differentiating (2.3), we obtain
ω(t)=ρ′(t)a(t)(b(t)z′(t))′b(t)z′(t)+ρ(t)(a(t)(b(t)z′(t))′)′b(t)z′(t)-ρ(t)a(t)(b(t)z′(t))′(b(t)z′(t))′(b(t)z′(t))2.
It follows from (E), (2.3), and (2.4) that
ω′(t)≤ρ′(t)ρ(t)ω(t)-ρ(t)q(t)(1-p(τ(t)))z(τ(t))b(t)z′(t)-ω2(t)ρ(t)a(t),
that is,
ω′(t)≤ρ′(t)ρ(t)ω(t)-ρ(t)q(t)(1-p(τ(t)))z(τ(t))b(τ(t))z′(τ(t))b(τ(t))z′(τ(t))b(t)z′(t)-ω2(t)ρ(t)a(t),
which follows from (2.6) and (2.7) that
ω′(t)≤ρ′(t)ρ(t)ω(t)-ρ(t)q(t)(1-p(τ(t)))∫t2τ(t)(∫t1s(1/a(u))du/b(s))ds∫t1τ(t)(1/a(u))du∫t1τ(t)(1/a(u))du∫t1t(1/a(u))du-ω2(t)ρ(t)a(t)=ρ′(t)ρ(t)ω(t)-ρ(t)q(t)(1-p(τ(t)))∫t2τ(t)(∫t1s(1/a(u))du/b(s))ds∫t1t(1/a(u))du-ω2(t)ρ(t)a(t).
Hence, we have
ω′(t)≤-ρ(t)q(t)(1-p(τ(t)))∫t2τ(t)(∫t1s(1/a(u))du/b(s))ds∫t1t(1/a(u))du+a(t)(ρ′(t))24ρ(t).
Integrating the last inequality from t3(>t2) to t, we get
∫t3t(ρ(s)q(s)(1-p(τ(s)))∫t2τ(s)(∫t1v(1/a(u))du/b(v))dv∫t1s(1/a(u))du-a(s)(ρ′(s))24ρ(s))ds≤ω(t3),
which contradicts (2.1).

Assume that case (2) holds. Using the similar proof of [23, Lemma 2], we can get limt→∞x(t)=0 due to condition (2.2). This completes the proof.

Theorem 2.2.

Assume that (1.5) holds, 0≤p(t)≤p1<1. Further, assume that for some function ρ∈C1([t0,∞),(0,∞)), for all sufficiently large t1≥t0 and for t3>t2>t1, one has (2.1) and (2.2). If
limsupt→∞∫t2t(δ(s)q(s)(1-p(τ(s)))∫t1τ(s)dvb(v)-14δ(s)a(s))ds=∞,
where
δ(t):=∫t∞1a(s)ds,
then (E) is almost oscillatory.

Proof.

Assume that x is a positive solution of (E). Based on the condition (1.5), there exist three possible cases (1), (2) (as those of Theorem 2.1), and

z(t)>0,z′(t)>0,(b(t)z′(t))′<0,[a(t)(b(t)z′(t))′]′<0, for t≥t1, t1 is large enough.

Assume that case (1) and case (2) hold, respectively. We can obtain the conclusion of Theorem 2.2 by applying the proof of Theorem 2.1.

Assume that case (3) holds. From [a(t)(b(t)z′(t))′]′<0, a(t)(b(t)z′(t))′ is decreasing. Thus, we get
a(s)(b(s)z′(s))′≤a(t)(b(t)z′(t))′,s≥t≥t1.
Dividing the above inequality by a(s) and integrating it from t to l, we obtain
b(l)z′(l)≤b(t)z′(t)+a(t)(b(t)z′(t))′∫tldsa(s).
Letting l→∞, we have
0≤b(t)z′(t)+a(t)(b(t)z′(t))′∫t∞dsa(s),
that is,
-∫t∞dsa(s)a(t)(b(t)z′(t))′b(t)z′(t)≤1.
Define function ϕ by
ϕ(t):=a(t)(b(t)z′(t))′b(t)z′(t),t≥t1.
Then, ϕ(t)<0 for t≥t1. Hence, by (2.19) and (2.20), we get
-δ(t)ϕ(t)≤1.
Differentiating (2.20), we obtain
ϕ′(t)=(a(t)(b(t)z′(t))′)′b(t)z′(t)-a(t)(b(t)z′(t))′(b(t)z′(t))′(b(t)z′(t))2.
Using z′(t)>0, we have (2.4). From (E) and (2.4), we have
ϕ′(t)≤-q(t)(1-p(τ(t)))z(τ(t))b(t)z′(t)-a(t)(b(t)z′(t))′(b(t)z′(t))′(b(t)z′(t))2.
In view of (3), we see that
z(t)≥b(t)∫t1tdsb(s)z′(t).
Hence,
(z(t)∫t1t(ds/b(s)))′≤0,
which implies that
z(τ(t))z(t)≥∫t1τ(t)(ds/b(s))∫t1t(ds/b(s)).
By (2.20) and (2.23), (2.24), and (2.26), we obtain
ϕ′(t)≤-q(t)(1-p(τ(t)))∫t1τ(t)dsb(s)-ϕ2(t)a(t).
Multiplying the last inequality by δ(t) and integrating it from t2(>t1) to t, we have
ϕ(t)δ(t)-ϕ(t2)δ(t2)+∫t2tδ(s)q(s)(1-p(τ(s)))∫t1τ(s)dvb(v)ds+∫t2tϕ2(s)δ(s)a(s)ds+∫t2tϕ(s)a(s)ds≤0,
which follows that
∫t2t(δ(s)q(s)(1-p(τ(s)))∫t1τ(s)dvb(v)-14δ(s)a(s))ds≤1+ϕ(t2)δ(t2)
due to (2.21), which contradicts (2.14). This completes the proof.

Theorem 2.3.

Assume that (1.6) holds, 0≤p(t)≤p1<1. Further, assume that for some function ρ∈C1([t0,∞),(0,∞)), for all sufficiently large t1≥t0 and for t3>t2>t1, one has (2.1), (2.2), and (2.14). If
∫t1∞1b(v)∫t1v1a(u)∫t1uη(s)q(s)ξ(τ(s))dsdudv=∞,
where
η(t):=1-p(τ(t))ξ(σ(τ(t)))ξ(τ(t))>0,ξ(t):=∫t∞1b(s)ds,
then (E) is almost oscillatory.

Proof.

Assume that x is a positive solution of (E). Based on the condition (1.6), there exist four possible cases (1), (2), (3) (as those of Theorem 2.2), and

Assume that case (1), case (2), and case (3) hold, respectively. We can obtain the conclusion of Theorem 2.3 by using the proof of Theorem 2.2.

Assume that case (4) holds. Since (b(t)z′(t))′<0, we get
z′(s)≤b(t)z′(t)b(s),s≥t,
which implies that
z(t)≥-ξ(t)b(t)z′(t)≥Lξ(t)
for some constant L>0. By (2.33), we obtain
(z(t)ξ(t))′≥0.
Using (2.34), we see that
x(t)=z(t)-p(t)x(σ(t))≥z(t)-p(t)z(σ(t))≥(1-p(t)ξ(σ(t))ξ(t))z(t).
From (E), (2.33), and (2.35), we have
[a(t)(b(t)z′(t))′]′+Lq(t)(1-p(τ(t))ξ(σ(τ(t)))ξ(τ(t)))ξ(τ(t))≤0.
Integrating the last inequality from t1 to t, we get
a(t)(b(t)z′(t))′+L∫t1tq(s)(1-p(τ(s))ξ(σ(τ(s)))ξ(τ(s)))ξ(τ(s))ds≤0.
Integrating again, we have
b(t)z′(t)+L∫t1t1a(u)∫t1uq(s)(1-p(τ(s))ξ(σ(τ(s)))ξ(τ(s)))ξ(τ(s))dsdu≤0.
Integrating again, we obtain
z(t1)≥L∫t1t1b(v)∫t1v1a(u)∫t1uq(s)(1-p(τ(s))ξ(σ(τ(s)))ξ(τ(s)))ξ(τ(s))dsdudv+z(t),
which contradicts (2.30). This completes the proof.

Theorem 2.4.

Assume that (1.6) holds, 0≤p(t)≤p1<1. Further, assume that for some function ρ∈C1([t0,∞),(0,∞)), for all sufficiently large t1≥t0 and for t3>t2>t1, one has (2.1), (2.2) and (2.14). If
∫t1∞1b(v)∫t1v1a(u)∫t1uq(s)dsdudv=∞,
then (E) is almost oscillatory.

Proof.

Assume that x is a positive solution of (E). Based on the condition (1.6), there exist four possible cases (1), (2), (3), and (4) (as those of Theorem 2.3).

Assume that case (1), case (2), and case (3) hold, respectively. We can obtain the conclusion of Theorem 2.4 by using the proof of Theorem 2.2.

Assume that case (4) holds. Then, limt→∞z(t)=l≥0(lisfinite). Assume that l>0. Then, from the proof of [23, Lemma 2], we see that there exists a constant k>0 such that
x(t)≥kl.
The rest of the proof is similar to that of Theorem 2.3 and hence is omitted.

3. Examples

In this section, we will present some examples to illustrate the main results.

Example 3.1.

Consider the third-order neutral delay differential equation
(t(x(t)+p1x(t2))′′)′+λt2x(t)=0,λ>0,t≥1,
where p1∈[0,1).

Let ρ(t)=t. It follows from Theorem 2.1 that every solution x of (3.1) is either oscillatory or limt→∞x(t)=0, if λ>1/(4k(1-p1)) for some k∈(1/4,1).

Note that (3.1) is almost oscillatory, if λ>2/(1-p1) due to [23, Corollary 3].

Example 3.2.

Consider the third-order neutral delay differential equation
(1t(t1/2(x(t)+12x(t-π))′)′)′+(t-1/22+38t-5/2)x(t-7π2)=0,t≥1.

Let ρ(t)=1. It follows from Theorem 2.1 that every solution x of (3.2) is almost oscillatory. One such solution is x(t)=sint.

Example 3.3.

Consider the third-order neutral delay differential equation
(t4/3(x(t)+p1x(t2))′′)′+λt5/3x(t)=0,λ>0,t≥1,
where p1∈[0,1).

Let ρ(t)=1. It follows from Theorem 2.2 that every solution x of (3.3) is either oscillatory or limt→∞x(t)=0, if λ>1/(36k(1-p1)) for some k∈(1/4,1).

Note that [22, Theorem 1] cannot be applied to (3.3) when p1=0.

Example 3.4.

Consider the third-order neutral delay differential equation
(t2(t2(x(t)+13x(t2))′)′)′+λt2x(t)=0,λ>0,t≥1.

Let ρ(t)=1. It follows from Theorem 2.3 that every solution x of (3.4) is either oscillatory or limt→∞x(t)=0, if λ>0.

4. RemarksRemark 4.1.

In [3], Agarwal et al. established a well-known result; see [4, Lemma 6.1]. Using [4, Lemma 6.1] and defining the function ω as in Theorem 2.1 with ρ(t)=1, we can replace condition (2.1) with
(a(t)y′(t))′+q(t)(1-p(τ(t)))∫t2τ(t)(∫t1s(1/a(u))du/b(s))ds∫t1t(1/a(u))duy(t)=0
that is oscillatory. Similarly, we can replace condition (2.14) by
(a(t)y′(t))′+q(t)(1-p(τ(t)))∫t1τ(t)dsb(s)y(t)=0
that is oscillatory.

Remark 4.2.

The results for (E) can be extended to the nonlinear differential equations.

Remark 4.3.

It is interesting to find a method to study (E) for the case when
∫t0∞1a(t)dt=∞,∫t0∞1b(t)dt<∞.

Remark 4.4.

It is interesting to find other methods to present some sufficient conditions which guarantee that every solution of (E) is oscillatory.

Acknowledgments

The authors would like to thank the referees and Professor P. J. Y. Wong for the comments which helped to improve the paper. This research is supported by NNSF of PR China (Grant nos. 61034007, 60874016, and 50977054). The second author would like to express his gratitude to Professors Ravi P. Agarwal and Martin Bohner for their selfless guidance.

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