R E S E A R C H
Open Access
Asymptotic behavior of third-order
differential equations with nonpositive
neutral coefficients and distributed deviating
arguments
Cuimei Jiang
1, Ying Jiang
1and Tongxing Li
2,3**Correspondence:
[email protected] 2LinDa Institute of Shandong
Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P.R. China 3School of Informatics, Linyi
University, Linyi, Shandong 276005, P.R. China
Full list of author information is available at the end of the article
Abstract
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.
MSC: 34K11
Keywords: asymptotic behavior; third-order neutral differential equation; nonpositive neutral coefficient; distributed deviating argument; oscillation
1 Introduction
Third-order differential equations have attracted noticeable interests due to their poten-tial applications in assorted fields, including physical sciences, technology, population dy-namics, 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 de-viating arguments; see the papers by Tianet al.[], Wang [], and Wang and Cai []. On the basis of these background details, the objective of this paper is to analyze the os-cillation and asymptotic properties of a class of third-order neutral differential equations
r(t)z(t)α+ d
c
q(t,ξ)fxσ(t,ξ)dξ= , (.)
wheret≥t> ,z(t) :=x(t) –
b
ap(t,μ)x[τ(t,μ)]dμ,α> is a quotient of odd positive integers,r(t)∈C([t,∞), (,∞)),
∞
t r
–/α(t)dt=∞,p(t,μ)∈C([t
,∞)×[a,b],R), ≤
b
a p(t,μ)dμ≤p< ,τ(t,μ)∈C([t,∞)×[a,b],R),τ(t,μ)≤t, lim inft→∞τ(t,μ) =∞ forμ∈[a,b],q(t,ξ)∈C([t,∞)×[c,d], [,∞)),q(t,ξ) is not identically zero for larget,
σ(t,ξ)∈C([t,∞)×[c,d],R) is a nondecreasing function forξ satisfyingσ(t,ξ)≤tand
lim inft→∞σ(t,ξ) =∞forξ∈[c,d],f(x)∈C(R,R), and there exists a positive constantk
such thatf(x)/xα≥kfor allx= .
We assume that solutions of (.) exist for anyt∈[t,∞). Our attention is restricted to
those solutions of (.) that are not identically zero for larget. 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
r(t)≥ (.)
and
≤– b
a
p(t,μ)dμ≤p< ,
asymptotic criteria for (.) have been reported in [, , ]. So far, there are few results dealing with the asymptotic properties of third-order differential equations with nonpos-itive coefficients; we refer the reader to [, , ]. In particular, Baculíková and Džurina [] and Zhanget al.[] established several Hille and Nehari type (see Agarwalet al.[]) criteria for the equation
r(t)x(t) –p(t)xτ(t)γ+q(t)xγσ(t)=
under the assumptions that ≤p(t)≤p< and (.) holds.
It should be noted that condition (.) is a restrictive condition in the study of asymp-totic behavior of third-order differential equations. To solve this problem without requir-ing (.), Liet al.[] obtained some oscillation criteria for a third-order neutral delay differential equation
r(t)x(t) +p(t)xτ(t)+q(t)xσ(t)=
by employing the Riccati substitution
w(t) :=ρ(t)r(t)z
(t)
z(t) ,
where ≤p(t)≤p< ,z(t) :=x(t) +p(t)x(τ(t)), andρ(t)∈C([t,∞), (,∞)). A natural
2 Several lemmas
Lemma . Assume that x(t)is an eventually positive solution of(.).Then there exists a t≥tsuch that,for t≥t,z(t)has the following four possible cases:
(i) z(t) > ,z(t) > ,z(t) > ,(r(t)(z(t))α)≤;
(ii) z(t) > ,z(t) < ,z(t) > ,(r(t)(z(t))α)≤;
(iii) z(t) < ,z(t) < ,z(t) > ,(r(t)(z(t))α)≤;
(iv) z(t) < ,z(t) < ,z(t) < ,(r(t)(z(t))α)≤.
Proof Letx(t) be an eventually positive solution of (.). Then there exists at≥tsuch
that, fort≥t,
x(t) > , xτ(t,μ)> , μ∈[a,b], and xσ(t,ξ)> , ξ∈[c,d].
It follows from (.) and the definition ofz(t) thatx(t)≥z(t) and
r(t)z(t)α= – d
c
q(t,ξ)fxσ(t,ξ)dξ≤.
Hence,r(t)(z(t))αis nonincreasing and of one sign, which implies thatz(t) is also of one
sign. Therefore, there exists at≥tsuch that, fort≥t,z(t) < orz(t) > . Case. The conditionz(t) < yields that there exists a constantM> such that
r(t)z(t)α≤–M< , that is,
z(t)≤–M
/α
r/α(t).
Integrating this inequality fromttot, we conclude that
z(t)≤z(t) –M/α
t
t
r–/α(s)ds.
Lettingt→ ∞, we have thatz(t)→–∞, and soz(t) < eventually. Note that the condi-tionsz(t) < andz(t) < imply thatz(t) < . Thus, we get case (iv).
Case . Assume that z(t) > . Thenz(t) is of one sign. Ifz(t) > , then z(t) > . If
z(t) < , thenz(t) > orz(t) < . Hence, we have three possible cases (i), (ii), and (iii) whenz(t) > . The proof is complete.
Lemma . Assume that x(t)is an eventually positive solution of(.)and the correspond-ing z(t)satisfies case(i)in Lemma..Then there exist two numbers t≥tand t>tsuch that,for t≥t,
z(t)≥ t
t s
tr–/
α(u)du ds
t
tr–/α(u)du
z(t)
Proof Letz(t) satisfy case (i) in Lemma .. Then
Hence, we deduce that
This completes the proof.
Lemma . Let x(t)be an eventually positive solution of(.)and assume that the corre-sponding z(t)satisfies case(ii)in Lemma..If
Proof It follows from property (ii) that there exists a finite constant l≥ such that limt→∞z(t) =l. We claim thatl= . Otherwise, assume thatl> . By the definition of
Integrating the latter inequality fromtto∞, we have
Integrating this inequality fromtto∞and then integrating the resulting inequality from
tto∞, we conclude that
z(t)≥lk/α
∞
t ∞
v
r(u) ∞
u d
c
q(s,ξ)dξds
/α
du dv,
which is a contradiction to (.). Hence,l= andlimt→∞z(t) = .
Next, we prove that x(t) is bounded. If not, then there exists a sequence{tm} such thatlimm→∞tm=∞andlimm→∞x(tm) =∞, wherex(tm) :=max{x(s);t≤s≤tm}. Since lim inft→∞τ(t,μ) =∞,τ(tm,μ) >tfor all sufficiently largem. Byτ(t,μ)≤t, we conclude
that
xτ(tm,μ)=maxx(s);t≤s≤τ(tm,μ)
≤maxx(s);t≤s≤tm
=x(tm),
and so
z(tm) =x(tm) – b
a
p(tm,μ)xτ(tm,μ)dμ≥x(tm) – b
a
p(tm,μ)x(tm)dμ
≥( –p)x(tm),
which yields limm→∞z(tm) =∞. This contradicts limt→∞z(t) = . Therefore, x(t) is bounded, and hence we may suppose thatlim supt→∞x(t) =a, where ≤a<∞. Then,
there exists a sequence {tk}such that limk→∞tk=∞andlimk→∞x(tk) =a. Assuming
now thata> and lettingε:=a( –p)/(p), we havex(τ(tk,μ)) <a+εeventually,
and thus
= lim
k→∞z(tk)≥klim→∞
x(tk) –p(a+ε)
=a( –p) > ,
which is a contradiction. Thus,a= andlimt→∞x(t) = . The proof is complete.
3 Main results In what follows, we let
ρ+(t) :=max,ρ(t), q∗(t) :=
d
c
q(t,ξ)dξ, and σ∗(t) :=σ(t,c),
where the meaning ofρ(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≥tand for some t>t>t,
lim sup t→∞
t
t
kρ(s)q∗(s)G(s) – (α+ )α+
r(s)(ρ+(s))α+
ρα(s) ds=∞, (.)
where
G(t) := σ∗(t)
t v
tr–/
α(u)du dv
t
tr–/α(u)du
α
, (.)
Proof Suppose to the contrary that (.) has a nonoscillatory solutionx(t). Without loss of generality, we may assume thatx(t) is eventually positive (since the proof of the case wherex(t) is eventually negative is similar). By Lemma ., we observe that, fort≥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. Fort≥t, define the Riccati transformationω(t) by
ω(t) :=ρ(t)r(t)(z
It follows from (.) and (i) that
Set
v:=ω(t), A:= α
(r(t)ρ(t))/α, and B:=
ρ+(t) ρ(t).
Using the inequality (see [])
Bv–Av+/α≤ α α
(α+ )α+ Bα+
Aα , A> , (.)
we have
ρ+(t) ρ(t)ω(t) –
αω+/α(t)
(r(t)ρ(t))/α ≤
(α+ )α+
r(t)(ρ+(t))α+
ρα(t) .
Substituting the latter inequality into (.), we conclude that
ω(t)≤–kρ(t)q∗(t)G(t) +
(α+ )α+
r(t)(ρ+(t))α+
ρα(t) .
Integrating this inequality fromt(t>t) tot, we arrive at
t
t
kρ(s)q∗(s)G(s) – (α+ )α+
r(s)(ρ+(s))α+
ρα(s) ds≤ω(t),
which contradicts (.).
Suppose that case (ii) is satisfied. By Lemma .,limt→∞x(t) = .
If case (iii) or case (iv) holds, then limt→∞z(t) = c < (possibly c = –∞) or
limt→∞z(t) = –∞, respectively. Proceeding similarly as in the proof of Lemma ., we
conclude thatx(t) andz(t) are bounded. Hence,cis finite, and case (iv) does not occur.
Similar analysis to that in Lemma . leads to the conclusion thatlimt→∞x(t) = . This
completes the proof.
Lettingρ(t) =tandρ(t) = , we can derive the following results from Theorem ..
Corollary . Let condition(.)hold.If for all sufficiently large t≥tand for some t> t>t,
lim sup t→∞
t
t
ksq∗(s)G(s) – (α+ )α+
r(s)
sα ds=∞,
where G(t)is as in(.),then the conclusion of Theorem.remains intact.
Corollary . Let condition(.)be satisfied.If for all sufficiently large t≥tand for some t>t>t,
∞
t
q∗(s)G(s)ds=∞,
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
ω(t) :=r(t)(z
(t))α
(z(t))α , (.)
¯
p:=lim inf t→∞ k
t
t
r–/α(s)ds
α ∞
t
q∗(s)G(s)ds,
¯
q:=lim inf t→∞
ktt(tsr–/α(u)du)α+q
∗(s)G(s)ds
t
tr–/α(u)du
,
¯
r:=lim inf t→∞
t
t
r–/α(s)ds
α
ω(t), and R¯:=lim sup t→∞
t
t
r–/α(s)ds
α
ω(t),
where G(t)is defined by(.),t≥tis sufficiently large,and t>t>t.
(I) Letp¯<∞,q¯<∞,and suppose that the correspondingz(t)satisfies case(i)in Lemma..Then
¯
p≤ ¯r–¯r+/α≤ α
α
(α+ )α+ and p¯+q¯≤. (.)
(II) Ifp¯=∞orq¯=∞,thenz(t)does not have property(i)in Lemma..
Proof Part(I). Assume thatx(t) is an eventually positive solution of (.) and the corre-spondingz(t) satisfies (i). By (.), we haveω(t) > and
ω(t) =(r(t)(z
(t))α)
(z(t))α –αr(t)
z(t)
z(t)
α+
.
As in the proof of Theorem ., we get (.) and (.), and so
ω(t)≤–kq∗(t)
z(σ∗(t)) z(t)
α
–αr(t)
ω(t)
r(t)
+/α
≤–kq∗(t)G(t) –αω +/α(t)
r/α(t) . (.)
On the other hand, we conclude that
r/α(t)z(t) z(t) ≤
t
tr–/α(s)ds due to the proof of Lemma .. Hence,
ω(t)≤
t
t
r–/α(s)ds
–α
which implies that ≤ ¯r≤ ¯R≤ andlimt→∞ω(t) = . Integrating (.) fromtto∞, we
Applications of (.) and the definitions ofr¯andp¯imply that
¯
r≥ ¯p+ (r¯–ε)+/α.
Sinceεis arbitrary, we conclude that
¯
r≥ ¯p+r¯+/α. (.)
Next, we prove that
Multiplying (.) by (ttr–/α(u)du)α+ and integrating the resulting inequality fromt
Integrating by parts, we deduce that
t
Using inequality (.) with
v:=ω(t), A:=αr–/α(t)
Thus, we arrive at
t
Taking thelim supof both sides of the latter inequality ast→ ∞, we have
¯
R≤ –q¯. (.)
It follows from (.) and (.) that
¯
Moreover, by inequality (.),
¯
r–¯r+/α≤ α α
(α+ )α+.
Therefore, the desired inequalities in (.) hold. This completes the proof of Part (I).
Part(II). Letx(t) be an eventually positive solution of (.). We show thatz(t) does not have property (i). Assume the contrary. Suppose first thatp¯=∞. Inequality (.) implies that
ω(t)
t
t
r–/α(s)ds α
≥
t
t
r–/α(s)ds α ∞
t
kq∗(s)G(s)ds.
Taking thelim infof both sides of the latter inequality ast→ ∞, we arrive at
≥ ¯r≥ ∞,
which is a contradiction. Assume now thatq¯=∞. An application of inequality (.) yields
≤ ¯R≤–∞,
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 ..
Theorem . Assume that condition(.)is satisfied.If for all sufficiently large t≥tand for some t>t>t,
lim inf t→∞
t
t
r–/α(s)ds α ∞
t
q∗(s)G(s)ds> α
α
k(α+ )α+ (.) or
¯
p+q¯> , (.)
where G(t)is defined by(.),p and¯ q are as in Lemma¯ .,then the conclusion of Theo-rem.remains intact.
4 Examples
The following examples illustrate applications of the main results in this paper.
Example . Fort≥, consider the third-order differential equation
x(t) –
π/
x(t–μ)dμ
+
–π
–π
x
t+ξ
dξ= . (.)
Letα= ,a= ,b=π/,c= –π,d= –π,k= ,r(t) = ,p(t,μ) = /,τ(t,μ) =t–μ,
q(t,ξ) = /, andσ(t,ξ) =t+ξ/. Note that
∞
t
r–/α(s)ds=
∞
ds=∞,
b
a
p(t,μ)dμ= π/
dμ=
σ∗(t) =σ(t, –π) =t–π
Hence, by Corollary ., every solutionx(t) of (.) is either oscillatory or converges to zero ast→ ∞. As a matter of fact,x(t) =sintis an oscillatory solution to (.).
Example . Fort≥ andq> , consider the third-order differential equation
Using Theorem ., every solutionx(t) of (.) is either oscillatory or converges to zero as
Remark . Observe that Theorems . and . cannot distinguish solutions of (.) with different behaviors. It is not easy to obtain sufficient conditions that ensure that all so-lutionsx(t) of (.) just satisfylimt→∞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?
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All three authors contributed equally to this work. They all read and approved the final version of the manuscript.
Author details
1Qingdao Technological University, Feixian, Shandong 273400, P.R. China.2LinDa Institute of Shandong Provincial Key
Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P.R. China.3School of Informatics, Linyi University, Linyi, Shandong 276005, P.R. China.
Acknowledgements
The authors express their sincere gratitude to the editors and two anonymous referees for careful reading of the manuscript and valuable suggestions that helped to improve the paper. This research is supported by NNSF of P.R. China (Grant Nos. 61503171, 61403061, and 11447005), CPSF (Grant No. 2015M582091), NSF of Shandong Province (Grant No. ZR2012FL06), DSRF of Linyi University (Grant No. LYDX2015BS001), and the AMEP of Linyi University, P.R. China.
Received: 12 November 2015 Accepted: 6 April 2016
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