R E S E A R C H
Open Access
Oscillation and asymptotic behavior of
third-order neutral differential equations with
distributed deviating arguments
Yazhou Tian
1,2*, Yuanli Cai
1, Youliang Fu
2and Tongxing Li
2*Correspondence:
1School of Electronic and
Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China
Full list of author information is available at the end of the article
Abstract
By employing a generalized Riccati transformation and integral averaging technique, two Philos-type criteria are obtained which ensure that every solution of a class of third-order neutral differential equations with distributed deviating arguments is either oscillatory or converges to zero. These results extend and improve related criteria reported in the literature. Two illustrative examples are provided.
MSC: 34K11
Keywords: oscillation; asymptotic behavior; third-order neutral differential equation; distributed deviating argument; generalized Riccati transformation
1 Introduction
Differential equations with distributed deviating arguments are often used for model-ing various problems arismodel-ing in the engineermodel-ing and natural sciences. Therefore, analy-sis of qualitative properties of solutions to such equations is crucial for applications; see Wang []. On the basis of these background details, we investigate the oscillation and asymptotic behavior of a third-order neutral differential equation with distributed deviat-ing arguments
r(t)
x(t) +
b a
p(t,ξ)xτ(t,ξ)dξ
α
+ d
c
q(t,ξ)fxσ(t,ξ)dξ= ,
t≥t, (.)
whereα≥ is the ratio of odd positive integers. Throughout, we suppose that the follow-ing assumptions hold.
(A) r(t)∈C([t,∞), (,∞)),r(t)≥, t∞ r–/α(s)ds=∞;
(A) p(t,ξ)∈C([t,∞)×[a,b], [,∞)),≤ abp(t,ξ)dξ≤P< ;
(A) τ(t,ξ)∈C([t,∞)×[a,b],R)is a nondecreasing function forξ satisfyingτ(t,ξ)≤t
andlim inft→∞τ(t,ξ) =∞forξ∈[a,b];
(A) q(t,ξ)∈C([t,∞)×[c,d], [,∞));
(A) σ(t,ξ)∈C([t,∞)×[c,d],R)is a nondecreasing function forξsatisfyingσ(t,ξ)≤t
andlim inft→∞σ(t,ξ) =∞forξ∈[c,d];
(A) f(x)∈C(R,R)and there exists a positive constantKsuch thatf(x)/xα≥Kfor allx= .
Define a new functionz(t) by
z(t) =x(t) + b
a
p(t,ξ)xτ(t,ξ)dξ.
By a solution of (.) we mean a nontrivial functionx(t)∈C([Tx,∞),R),Tx≥t, which
has the propertiesz(t)∈C([T
x,∞),R) andr(t)(z(t))α∈C([Tx,∞),R) forTx≥t. Our
attention is restricted to those solutions of (.) which satisfysup{|x(t)|:t≥T}> for any T≥Tx. A solutionx(t) of (.) is said to be oscillatory on [Tx,∞) if it is neither eventually
positive nor eventually negative. Otherwise it is called nonoscillatory. Equation (.) is called oscillatory if all its solutions are oscillatory.
Recently, there has been much research activity concerning the oscillation and asymp-totic properties of various classes of differential equations; see,e.g., [–] and the refer-ences cited therein. So far, there are few results dealing with the asymptotic behavior of third-order neutral differential equations with distributed deviating arguments, we refer the reader to [, ]. The third-order neutral differential equation
r(t)x(t) +p(t)xτ(t)α+q(t)fxσ(t)=
and its special cases have been studied by Baculíková and Džurina [, ], Candan [], Grace et al.[], Jiang and Li [], and Liet al.[]. Using Riccati transformation, Zhanget al.[] considered a class of third-order neutral differential equations
r(t)
x(t) +
b a
p(t,ξ)xτ(t,ξ)dξ
+ d
c
q(t,ξ)fxσ(t,ξ)dξ= , (.)
and they obtained several Philos-type (see []) criteria for (.), whereas Şenel and Utku [] studied (.).
In the study of oscillation of differential equations, there are two techniques which are used to reduce the higher-order equations to the first-order Riccati equations (or inequal-ities). One of them is the Riccati transformation technique which has been recently ex-tended to dynamic equations on time scales; see,e.g., Şenel and Utku []. The other one is termed the generalized Riccati transformation technique; we refer the reader to Li [], Liet al.[], Li and Saker [], and the related references cited therein. In particular, Li [] used the generalized Riccati substitution and established several oscillation criteria for a second-order ordinary differential equation
r(t)x(t)+q(t)x(t) = .
Furthermore, he proved that the equation
tx
(t)+
tx(t) =
In the special case whenα= , (.) reduces to (.). Now the following question arises. Could we obtain new Philos-type oscillation criteria for (.) by using a generalized Riccati transformation which differs from that of []? Motivated by Li [], Liet al.[], and Li and Saker [], our purpose in this paper is to give a positive answer to this question. In Section , four lemmas are given to prove the main results. In Section , we establish two Philos-type theorems for (.). In Section , two examples and some conclusions are pre-sented to illustrate the main results. As customary, all functional inequalities considered in this paper are supposed to hold for alltlarge enough.
2 Some lemmas
Lemma . Suppose that conditions(A)-(A)are satisfied and let x(t)be a positive
solu-tion of(.).Then z(t)has only one of the following two properties: (I) z(t) > ,z(t) > ,z(t) > ,z(t)≤;
(II) z(t) > ,z(t) < ,z(t) > ,z(t)≤, for t≥t,where t≥tis sufficiently large.
Proof Assume thatx(t) is a positive solution of (.). Then there exists at≥tsuch that,
fort≥t,
x(t) > , xτ(t,ξ)> , ξ∈[a,b], and xσ(t,ξ)> , ξ∈[c,d].
From (.) and the definition ofz(t), we havez(t) > and
r(t)z(t)α= – d
c
q(t,ξ)fxσ(t,ξ)dξ≤.
Thusr(t)(z(t))αis nonincreasing and of one sign. Therefore,z(t) is also of one sign and
so we have two possibilities:z(t) < orz(t) > fort≥t≥t. We assert thatz(t) >
fort≥t. Otherwise, there exists a constantM> such that, fort≥t,
z(t)≤–Mα
rα(t)
< .
Integrating this inequality fromttot, we obtain
z(t)≤z(t) –M
α
t t
rα(s)
ds.
Lettingt→ ∞and using (A), we getlimt→∞z(t) = –∞. Thusz(t) < eventually. But
conditions z(t) < andz(t) < imply thatz(t) < , which contradicts our assumption z(t) > . Hence,z(t) > fort≥t. Furthermore, we have, fort≥t,
r(t)z(t)α=r(t)z(t)α+αr(t)z(t)α–z(t)≤.
This yieldsz(t)≤ fort≥tdue to condition (A). Therefore,z(t) has only one of the
Lemma . Let x(t)be a positive solution of(.)and assume that corresponding z(t)has the property(II).If
∞
Proof Letx(t) be a positive solution of (.). Sincez(t) has the property (II), there exists a finite constantl≥ such thatlimt→∞z(t) =l≥. We prove thatl= . Assume now that
Noting thatz(t) has the property (II) and using (A), we have
Integrating inequality (.) fromtto∞, we have
–z(t)≥l
Integrating the latter inequality fromtto∞, we obtain
which contradicts (.). Hencel= andlimt→∞z(t) = . Then it follows from ≤x(t)≤
z(t) thatlimt→∞x(t) = . The proof is complete.
Lemma . ([], Lemma ) Assume that u(t) > ,u(t) > ,and u(t) < for t≥t.If σ(t)∈C([t,∞), [,∞)),σ(t)≤t,andlimt→∞σ(t) =∞,then,for everyβ∈(, ),there
exists a Tβ≥tsuch that,for t≥Tβ,
uσ(t)≥βσ(t)
t u(t).
Lemma . Assume that u(t) > ,u(t) > ,u(t) > ,and u(t)≤for t≥t.Then,for
everyγ∈(, ),there exists a Tγ ≥tsuch that,for t≥Tγ,
u(t)≥ γtu
(t).
Proof The proof is similar to that of Baculíková and Džurina ([], Lemma ), and hence it
is omitted.
3 Main results
Let
D=(t,s)∈R:t≥s≥t
and D=
(t,s)∈R:t>s≥t
.
The function H(t,s)∈C(D,R) is said to belong to the classX(denoted byH∈X) if it satisfies
(i) H(t,t) = ,t≥t,H(t,s) > ,(t,s)∈D;
(ii) ∂H(t,s)/∂s≤, there existρ(t)∈C([t
,∞), (,∞)),b(t)∈C([t,∞), [,∞)), and
h(t,s)∈C(D,R)satisfying
–∂H(t,s)
∂s =H(t,s)
ρ(s)
ρ(s) + (α+ )b
α(s)
+h(t,s).
Theorem . Assume that conditions(A)-(A)and(.)are satisfied. If there exists a
function H∈X such that,for someβ∈(, )andγ∈(, ),
lim sup
t→∞
H(t,t)
t t
H(t,s)ψ(s) – (α+ )α+
ρ(s)r(s)|h(t,s)|α+
Hα(t,s)
ds=∞, (.)
whereσ(t) =σ(t,c)and
ψ(t) =K( –P)αρ(t)
βγ
σ(t) t
α d c
q(t,ξ)dξ
+ρ(t)r(t)b+α(t) –ρ(t)r(t)b(t), (.)
then every solution x(t)of(.)is either oscillatory or satisfieslimt→∞x(t) = .
Suppose first thatz(t) has the property (I). Then we obtain
formationω(t) by
ω(t) =ρ(t)
By virtue of (.), we conclude that
From Lemma ., for everyβ∈(, ), there exists aTβ≥Tγ such that, fort≥Tβ,
Using the inequality (see [])
we obtain t
T
H(t,s)ψ(s)ds≤H(t,T)ω(T) + t
T
(α+ )α+
ρ(s)r(s)|h(t,s)|α+
Hα(t,s) ds.
Hence
H(t,T)
t T
H(t,s)ψ(s) – (α+ )α+
ρ(s)r(s)|h(t,s)|α+
Hα(t,s)
ds≤ω(T) (.)
for all sufficiently larget, which contradicts (.).
Assume now thatz(t) has the property (II). By Lemma ., we havelimt→∞x(t) = . The
proof is complete.
It may happen that assumption (.) in Theorem . fails to hold. Consequently, Theo-rem . cannot be applied. The following theoTheo-rem provides a new oscillation criterion for (.).
Theorem . Let conditions(A)-(A)and(.)be satisfied.Assume that there exists a
function H∈X such that
<inf
s≥t
lim inf
t→∞
H(t,s) H(t,t)
≤ ∞ (.)
and
lim sup
t→∞
H(t,t)
t t
ρ(s)r(s)|h(t,s)|α+
Hα(t,s) ds<∞ (.)
hold.If there exists a functionϕ(t)∈C([t,∞),R)such that,for all T≥t,
lim sup
t→∞
t t
ρ–α(s)r–α(s)ϕ+(s) α+
α ds=∞ (.)
and
lim sup
t→∞
H(t,T)
t T
H(t,s)ψ(s) – (α+ )α+
ρ(s)r(s)|h(t,s)|α+
Hα(t,s)
ds≥ϕ(T), (.)
whereψ(t)is defined by(.)andϕ+(t) =max{ϕ(t), },then the conclusion of Theorem.
remains intact.
Proof Assuming thatz(t) has the property (I) and proceeding as in the proof of Theo-rem ., we have (.) and (.) for allt>T. Hence, by virtue of (.),
lim sup
t→∞
H(t,T)
t T
H(t,s)ψ(s) – (α+ )α+
ρ(s)r(s)|h(t,s)|α+
Hα(t,s)
ds≤ω(T) (.)
for allt>T. Thus, by (.) and (.), we have
and
From (.), we obtain
∞
To complete the proof, we show that (.) does not hold. Let
u(t) = α
Now by (.), there exists a positive constantξsatisfying
By (.), there exists aT≥Tsuch thatH(t,T)/H(t,t)≥ξfor allt≥T. Thusu(t)≥ξ
for allt≥T. Sinceξis an arbitrary constant,
lim
t→∞u(t) =∞. (.)
Consider now a sequence{ti}∞i=in (T,∞) withlimi→∞ti=∞such that
lim
i→∞
u(ti) –v(ti)
=lim inf
t→∞
u(t) –v(t)<∞.
By virtue of (.), there exist a natural numberNand a constantL> such that, for all
i>N,
u(ti) –v(ti)≤L. (.)
It follows from (.) that
lim
i→∞u(ti) =∞. (.)
Combining (.) and (.), we conclude that
lim
i→∞v(ti) =∞ (.)
and, forilarge enough,
v(ti)
u(ti)
>
. (.)
From (.) and (.), we obtain
lim
i→∞
vα+(t
i)
uα(t
i)
=∞. (.)
On the other hand, by Hölder’s inequality, we have
v(ti)≤
α
H(ti,T)
ti
T
H(ti,s)ρ–
α(s)r–α(s)ωαα+(s)ds
α α+
×
ααH(t
i,T)
ti
T
ρ(s)r(s)|h(ti,s)|α+
Hα(t
i,s)
ds
α+ .
Therefore, for allilarge enough,
vα+(t
i)
uα(t
i) ≤
ααξ
H(ti,t)
ti
t
ρ(s)r(s)|h(ti,s)|α+
Hα(t
i,s)
ds. (.)
From (.) and (.), we deduce that
lim
i→∞
H(ti,t)
ti
t
ρ(s)r(s)|h(ti,s)|α+
Hα(t
i,s)
so
which contradicts (.). Therefore, (.) cannot hold. By virtue of (.), we get
∞
T
ρ–α(s)r–α(s)ϕ+(s) α+
α ds<∞,
which contradicts (.).
Suppose thatz(t) has the property (II). By Lemma ., we obtainlimt→∞x(t) = . This
completes the proof.
4 Examples and conclusions
Example . Fort≥, consider a third-order differential equation
It is not difficult to verify that
∞
Therefore, the conditions (A)-(A) and (.) are satisfied. Furthermore, we chooseK= ,
ifq> /(βγ) for someβ∈(, ) andγ ∈(, ). Hence, by Theorem ., every solution
x(t) of (.) is either oscillatory or converges to zero ast→ ∞in the case where q>
,,/(,)≈. (by lettingβ=γ = /). Observe that the results reported in [] cannot be applied to (.) sinceα= .
Example . Fort≥, consider a third-order differential equation
It is easy to verify that
∞
Therefore, the conditions (A)-(A) and (.) are satisfied. Furthermore, we chooseK= ,
P= /,ρ(t) =t,b(t) = /t, andH(t,s) = (t–s). Thenh(t,s) = (t–s)( – ts–),
Remark . With an appropriate choice of the functionH, one can derive from Theorems . and . a number of oscillation criteria for (.). For example, consider a Kamenev-type functionH(t,s) byH(t,s) = (t–s)n–, (t,s)∈D, wheren> is an integer. The remainder of the details are left to the reader.
Remark . Note that Theorems . and . ensure that every solution x(t) to (.) is either oscillatory or satisfieslimt→∞x(t) = and, unfortunately, these results cannot
dis-tinguish solutions with different behaviors. Since the sign of the derivativez(t) is not fixed, it is not easy to establish sufficient conditions which guarantee that all solutions to (.) are just oscillatory and do not satisfylimt→∞x(t) = . Neither is it possible to use the
tech-nique exploited in this paper for proving that all solutions of (.) satisfylimt→∞x(t) = .
Hence, these two interesting problems are left for future research.
Remark . It would be interesting to find a different method to investigate (.) when <α< . It would also be of interest to find another method to study (.) in the case where t∞ r–/α(s)ds<∞.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All four authors contributed equally to this work. They all read and approved the final version of the manuscript.
Author details
1School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P.R. China.2Qingdao
Technological University, Feixian, Shandong 273400, P.R. China.
Acknowledgements
The authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This research was supported by NNSF of P.R. China (Grant Nos. 61174217, 61374074, and 61473133) and NSF of Shandong Province (Grant No. JQ201119).
Received: 23 February 2015 Accepted: 11 August 2015 References
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