R E S E A R C H
Open Access
Oscillation of second-order nonlinear neutral
dynamic equations with distributed deviating
arguments on time scales
Shao-Yan Zhang
1and Qi-Ru Wang
2**Correspondence:
[email protected] 2School of Mathematics and
Computational Science, Sun Yat-sen University, West Xingang Road 135, Guangzhou, 510275, P.R. China Full list of author information is available at the end of the article
Abstract
This paper concerns second-order nonlinear neutral dynamic equations with distributed deviating arguments on time scales of the form
(r(t)((y(t) +
p(t)y(τ(t))))
γ)
+b
a
f(t,y(δ(t,ξ)))ξ = 0,
where
γ
> 0 is a quotient of odd positive integers. By using the generalized Riccati technique and integral averaging techniques, we derive new oscillation criteria for the above equations, which generalize and improve some existing results in the literature.Keywords: neutral dynamic equations on time scales; distributed deviating arguments; oscillation; generalized Riccati technique
1 Introduction
In this paper, we consider second-order nonlinear neutral dynamic equations with dis-tributed deviating arguments of the following form:
r(t)y(t) +p(t)yτ(t)γ+
b
a
ft,yδ(t,ξ)ξ= (.)
on a time scaleTsatisfyinginfT=tandsupT=∞. Throughout this paper, we assume the following:
(H) γ > is a quotient of odd positive integers, <a<b,τ(t)∈Crd(T,T)such that
τ(t)≤tandlimt→∞τ(t) =∞,δ(t,ξ)∈Crd(T×[a,b],T)such that limt→∞δ(t,ξ) =∞;
(H) r(t)∈Crd(T, (,∞))such that
∞
t ( r(t))
γt=∞, andp(t)∈C
rd(T, [, )); (H) f :T×R→Ris a continuous function such thatuf(t,u) > for allu= , and
there exists a functionq(t)∈Crd(T, [,∞))such that|f(t,u)| ≥q(t)|u|γ.
Oscillation of some second-order nonlinear delay dynamic equations on time scales has been discussed; see [–] and the references therein. Recently, there has been much re-search activity concerning the oscillation of second-order nonlinear neutral delay dynamic
equation
r(t)y(t) +p(t)yτ(t)γ+ft,yδ(t)= , t∈T.
We refer the reader to [–].
In , Thandapani and Piramanantham [] discussed oscillation of the second-order nonlinear neutral delay dynamic equation with distributed deviating arguments
r(t)x(t) +p(t)x(t–τ)+
b
a
q(t,ξ)fxg(t,ξ)ξ= , t∈T,
whereg(t,ξ) is strictly increasing with respect totand decreasing with respect toξ, and
f ∈C(R,R) withuf(u) > foru= ,f(–u) = –f(u).
In , Candan [] discussed the oscillation of Eq. (.) forδ(t,ξ)≤tandδ(t,ξ) >t, respectively, whereγ > is a quotient of odd positive integers. In [],δ(t,ξ) is decreasing with respect toξ, <p(t) < is increasing andf∈C(T×R,R) withuf(t,u) > for allu= . There exists a positive functionq(t) defined onTsuch that|f(t,u)| ≥q(t)|u|β, whereβ> is a ratio of odd positive integers. In , Candan [] established other oscillation criteria of Eq. (.) forδ(t,ξ)≤t, whereγ ≥ is a quotient of odd positive integers,β (in []) is equal toγ,r(t) > , andδ(t,ξ) is decreasing with respect toξ.
The purpose of this paper is to establish new oscillation criteria of Eq. (.) forγ > , a quotient of odd positive integers, where functionsp(t) andr(t) may not be monotonic,
δ(t,ξ) may not be decreasing with respect toξ. Hence, our results will generalize and improve those in [, ] and others.
By a solution of Eq. (.), we mean a nontrivial real-valued function y(t) such that
y(t) +p(t)y(τ(t))∈Crd[τ∗(t),∞),r(t)((y(t) +p(t)y(τ(t))))γ∈Crd[τ∗(t),∞) and satisfies Eq. (.). Our attention is restricted to those solutions of Eq. (.) that satisfysup{|y(t)|:t≥ ty}> for anyty≥t. A solutiony(t) of Eq. (.) is said to be oscillatory if it is neither even-tually positive nor eveneven-tually negative. Otherwise, it is called nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
This paper is organized as follows. After this introduction, we introduce some basic lemmas in Section . In Section , we present the main results. In Section , we illustrate the versatility of our results by two examples.
2 Some preliminaries
In this section, we present several technical lemmas which will be used in the proofs of the main results. For convenience, we use the notation (x(σ(t)))γ= (xσ(t))γ and set
x(t) :=y(t) +p(t)yτ(t). (.)
Then Eq. (.) becomes
r(t)x(t)γ+
b
a
ft,yδ(t,ξ)ξ= .
Fort,T∈Twitht>T, we define
β(t,T) =
t
T
rγ(s)
s, and gξ(t,T) =
β
(δ(t,ξ),T)
β(t,T) , δ(t,ξ) <t,
Q(t,T) =q(t)
b
a
–pδ(t,ξ)γgξγ(t,T)ξ.
ForD={(t,s)∈T:t≥s≥}, we define
H= H(t,s)∈Crd D, [,∞):H(t,t) = ,H(t,s) > andHs(t,s)≥ fort>s≥,
C(t,s) =H
s (t,s)zσ(s) +H(t,s)z(s) forH(t,s)∈H,
wherez∈Crd(T, (,∞)) is to be given in Theorems . and ., andz
+(t) =max{z(t), }. First of all, we give the following lemma.
Lemma . Let conditions (H)-(H) hold. If y(t) is an eventually positive solution of Eq. (.), then there exists T ∈ T sufficiently large such that x(t) > , x(t) ≥, (r(t)(x(t))γ)≤,x(t)≥rγ(t)x(t)β(t,T),and x(δ(t,ξ))≥g
ξ(t,T)x(t)for t∈[T,∞)T.
Proof Sincey(t) is an eventually positive solution of Eq. (.), then by (H) there exists
T∈[t,∞)Tsuch that
δ(t,ξ) >T, y(t) > , yτ(t)> and yδ(t,ξ)> fort≥T.
From (.) and (H), we see thatx(t) is also positive and satisfiesx(t)≥y(t). Also by Eq. (.) and (H), we have thatx(t) satisfies
r(t)x(t)γ≤–
b
a
q(t)yγδ(t,ξ)ξ≤ fort≥T,
which implies thatr(t)(x(t))γ is decreasing on [T,∞)
T. So we can get
x(t) =x(T) +
t
T
(r(s)(x(s))γ)γ
rγ(s)
s
≥rγ(t)x(t)
t
T
rγ(s)
s:=rγ(t)x(t)β(t,T).
We claim thatr(t)(x(t))γ ≥ on [T,∞)
T. Assume not, there ist∈[T,∞)Tsuch that
r(t)(x(t))γ< . Sincer(t)(x(t))γ ≤r(t)(x(t))γ fort≥t, we have
x(t)≤rγ(t
)x(t)
/r(t)/γ.
Integrating the inequality above fromttot(≥T), by (H) we get
x(t)≤x(t) +r
γ(t
)x(t)
t
t
/r(s)/γs→–∞ (t→ ∞),
and this contradicts the fact thatx(t) > for allt≥T. Thus we haver(t)(x(t))γ ≥ on [T,∞)Tand sox(t)≥ on [T,∞)T.
Let t≥T be fixed such thatδ(t,ξ)≥T. We consider the two cases δ(t,ξ) <t and
3 Main results
In this section, we establish our main results.
Theorem . Letγ > .Assume that(H)-(H)hold.Furthermore,for sufficiently large T∈T,one of the following conditions is satisfied:
(a) eithert∞Q(s,T)s=∞,or
∞
t
Q(s,T)s<∞ and βγ(t,T)
∞
t
Q(s,T)s> for allt>T,
(b) there existsz∈C
rd(T, (,∞))such that
lim sup t→∞
t
T
z(s)Q(s,T) – z +(s)
βγ(s,T)
s=∞,
(c) there existsz∈C
rd(T, (,∞))such that
lim sup t→∞
t
T
z(s)Q(s,T) – (γ + )γ+
r(s)(z(s))γ+
zγ(s)
s=∞,
(d) there existz∈C
rd(T, (,∞))andH∈Hsuch that
lim sup t→∞
H(t,T)
t
T
H(t,s)z(s)Q(s,T) – C γ+(t,s)
Hγ(t,s)(γ + )γ+zγ(s)
s=∞.
Then every solution y(t)of Eq. (.)is oscillatory.
Proof Suppose to the contrary that Eq. (.) has a nonoscillatory solutiony(t). Without loss of generality, we may assume thaty(t) is eventually positive. Then, by (H)-(H), there existsT∈[t,∞)Tsuch that fort≥T,y(τ(t)) > ,y(δ(t,ξ)) > , and Lemma . holds.
The rest of the proof is divided into four parts corresponding to conditions (a)-(d), re-spectively.
Part I: Assume condition (a) holds.
Letφ(t) :=r(t)(x(t))γ. Thenφ(t)≥ andφ(t)≤ fort≥T, andlim
t→∞φ(t) =ζ≥. From (.), we have
φ(t) +xγ(t)q(t)
b
a
–pδ(t,ξ)γgξγ(t,T)ξ≤. (.)
Integrating both sides of (.) fromtto∞, we obtain
ζ–φ(t) +
∞
t
Q(s,T)xγ(s)s≤.
In view of x(t)≥, we have reached a contradiction if ∞
t Q(s,T)s=∞. If
∞
t Q(s,
T)s<∞, then
φ(t)≥
∞
t
Q(s,T)xγ(s)s≥xγ(t)
∞
t
By Lemma ., we obtain
βγ(t,T)
∞
t
Q(s,T)s≤,
which is a contradiction to condition (a). Therefore, every solutiony(t) of Eq. (.) is os-cillatory.
Part II: Assume condition (b) holds. Define
w(t) :=z(t)r(t)(x (t))γ
xγ(t) fort≥T. (.)
Thenw(t) > . From (.), we have
w(t) =r(t)x(t)γ
z(t)
xγ(t)
+r(t)x(t)γσ
z(t)
xγ(t)
≤–z(t)Q(t,T) +r(t)x(t)γσ
z(t)xγ(t) –z(t)(xγ(t))
xγ(t)(xσ(t))γ
≤–z(t)Q(t,T) +z
+(t)(r(t)(x(t))γ)σ (xσ(t))γ –
(r(t)(x(t))γ)σz(t)(xγ(t))
xγ(t)xγ(σ(t)) . (.)
Whenγ ≥, usingx(t) > and the Keller?s chain rule, we get
xγ(t)=γ
x(t) +hμ(t)x(t)γ–dh
x(t)
≥γx(t)
( –h)x(t) +hx(t)γ–dh=γxγ–(t)x(t). (.)
When <γ < , usingx(t) > and the Keller?s chain rule, we obtain
xγ(t)≥γx(t)
( –h)xσ(t) +hxσ(t)γ–dh=γxσ(t)γ–x(t). (.)
Noting thatr(t) > and from (.), (.), and Lemma ., we obtain
(r(t)(x(t))γ)σz(t)(xγ(t))
xγ(t)xγ(σ(t)) ≥.
Since (r(t)(x(t))γ)≤ andt≤σ(t), we have
rσ(t)(xσ(t)γ≤r(t)x(t)γ. (.)
Hence from (.) and Lemma . and noting thatx(t)≥, we have
w(t)≤–z(t)Q(t,T) + z +(t)
βγ(t,T).
Integrating the above inequality fromT totfort≥T, we get
t
T
z(s)Q(s,T) – z +(s)
βγ(s,T)
Taking limsup on both sides ast→ ∞, we obtain a contradiction to condition (b). There-fore, every solutiony(t) of Eq. (.) is oscillatory.
Part III: Assume condition (c) holds. Whenγ≥, from (.) and (.) we have
w(t)≤–z(t)Q(t,T) +z (t)
zσ(t)w σ
(t) –r(t)x(t)γσz(t)γx
γ–(t)x(t)
xγ(t)xγ(σ(t))
≤–z(t)Q(t,T) +z (t)
zσ(t)w σ
(t) –r(t)x(t)γσz(t)γx (t)
xγ+(σ(t)). (.)
From (.) we get
–r(t)x(t)γσz(t)γx (t)
xγ+(σ(t)) ≤–
rσ(t)
γ+
γ xσ
(t)γ+ z(t)γ
rγ(t)xγ+(σ(t))
= – z(t)γ
zγγ+(σ(t))rγ(t)
wγγ+σ(t).
Then
w(t)≤–z(t)Q(t,T) +z (t)
zσ(t)w σ
(t) – z(t)γ
zγ
+
γ (σ(t))rγ(t)
wγγ+σ(t). (.)
When <γ < , by (.) and (.) we have
w(t)≤–z(t)Q(t,T) +wσ(t)z (t)
zσ(t)–
r(t)x(t)γσz(t)γ(x
σ(t))γ–x(t)
xγ(t)(xσ(t))γ .
By (.) we have
–(r(t)(x(t))γ)σz(t)γ(xσ(t))γ–x(t)
xγ(t)(xσ(t))γ = –
(rσ(t))γγ+((x(t))σ)γ+z(t)γx(t)
xγ(t)xσ(t)(rσ(t))γ(x(t))σ
≤–(r
σ(t))γγ+((x(t))σ)γ+z(t)γx(t)
xγ(t)xσ(t)rγ(t)x(t)
≤– z(t)γ (zσ(t))γγ+rγ(t)
wσ(t)
γ+
γ .
It follows that
w(t)≤–z(t)Q(t,T) +wσ(t)z (t)
zσ(t)–
z(t)γ
(zσ(t))γγ+rγ(t)
wσ(t)
γ+
γ , (.)
which is the same as (.). Let
B=z (t)
zσ(t), A=
z(t)γ
(zσ(t))γγ+rγ(t)
Then by Lemma . and (.) we obtain that for allt≥T,
w(t)≤–z(t)Q(t,T) + (γ+ )γ+
r(t)(z(t))γ+
zγ(t) .
Integrating the above inequality fromT totfor≥T, we get
t
By taking limsup on both sides as t→ ∞, we obtain a contradiction to condition (c). Therefore, every solutiony(t) of Eq. (.) is oscillatory.
Part IV: Assume condition (d) holds.
From (.) and (.) we have that forH∈H∗andt≥T,
By integration by parts we obtain
By taking limsup on both sides ast→ ∞, we obtain a contradiction to condition (d). Therefore, every solutiony(t) of Eq. (.) is oscillatory.
The proof is complete.
The results in the next theorem hold only forγ≥.
Theorem . Letγ ≥.Assume that(H)-(H)hold.Furthermore,for sufficiently large T∈T,there exists z∈C
rd(T, (,∞))such that one of the following conditions is satisfied: (a)
lim sup t→∞
t
T
z(s)Q(s,T) – (z
(s))rγ(s)
γz(s)βγ–(s,T)
s=∞,
(b) there existsH∈Hsuch that
lim sup t→∞
H(t,T)
t
T
H(t,s)z(s)Q(s,T) – C
(t,s)rγ(s)
γz(s)H(t,s)βγ–(s,T)
s=∞.
Then every solution y(t)of Eq. (.)is oscillatory.
Proof Suppose to the contrary that Eq. (.) has a nonoscillatory solutiony(t). Without loss of generality, we may assume thaty(t) is eventually positive. Then, by (H)-(H) there existsT∈[t,∞)Tsuch that fort≥T,y(t) > ,y(δ(t,ξ)) > ,y(τ(t)) > , and Lemma .
holds.
The rest of the proof is divided into two parts corresponding to conditions (a) and (b), respectively.
Part I: Assume condition (a) holds.
Definew(t) as in (.). Byx(t)≥,σ(t)≥t, (.), and (.), we obtain
w(t)≤–z(t)Q(t,T) +wσ(t)z (t)
zσ(t)–
r(t)x(t)γσz(t)γx
γ–(t)x(t)
xγ(t)(xσ(t))γ
≤–z(t)Q(t,T) +wσ(t)z (t)
zσ(t)–
z(t)γxγ–(t)x(t) (zσ(t))(r(t)(x(t))γ)σ
wσ(t).
From (.) and Lemma ., we get
w(t)≤–z(t)Q(t,T) +wσ(t)z (t)
zσ(t)–
z(t)γ
(zσ(t))r(t)
xγ–(t) (x(t))γ–
wσ(t)
≤–z(t)Q(t,T) +wσ(t)z (t)
zσ(t)–
z(t)γ βγ–(t,T)
(zσ(t))rγ(t)
wσ(t). (.)
By completing the square forwσ(t) on the right-hand side, we have
w(t)≤–z(t)Q(t,T) + (z
(t))rγ(t)
γz(t)βγ–(t,T).
Integrating the above inequality fromT totfort≥T, we get
t
T
z(s)Q(s,T) – (z
(s))rγ(s)
γz(s)βγ–(s,T)
Taking limsup on both sides ast→ ∞, we obtain a contradiction to condition (a). There-fore, every solutiony(t) of Eq. (.) is oscillatory.
Part II: Assume condition (b) holds.
Based on (.), the proof is similar to those of Part IV of Theorem . and Part I of Theorem ., and hence is omitted.
The proof is complete.
4 Examples
In this section, we give two examples to illustrate our main results.
Example . Consider the equation
Hence condition (c) of Theorem . is satisfied.
By Theorem ., every solutiony(t) of Eq. (.) is oscillatory.
Example . Consider the equation
Hence (H)-(H) hold. Withz(t) = , we see that for sufficiently largeT∈T,
lim sup t→∞
t
T
Q(s,T)s≥lim sup t→∞ T
[ –A](b–a)
t
T
s
(s–T)(s–b–T) s
≥lim sup t→∞ T
[ –A](b–a)
t
T
s
–b
s – T
s
s=∞.
Hence condition (a) of Theorem . is satisfied.
By Theorem ., every solutiony(t) of Eq. (.) is oscillatory.
Competing interests
The authors declare that they have no competing interests.
Authors? contributions
All authors completed the paper together. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Guangdong University of Finance, Yingfu Road 527, Guangzhou, 510520, P.R. China.
2School of Mathematics and Computational Science, Sun Yat-sen University, West Xingang Road 135, Guangzhou,
510275, P.R. China.
Acknowledgements
This work was supported by the NNSF of P.R. China (No. 11271379), Foundation for Technology Innovation in Higher Education of Guangdong, P.R. China (No. 2013KJCX0136) and Foundation for Humanities and Social Science in Ministry of Education of P.R. China (No. 14YJC790141).
Received: 7 October 2014 Accepted: 19 December 2014 References
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