R E S E A R C H
Open Access
Oscillation criteria for third-order neutral
dynamic equations with continuously
distributed delay
Mehmet Tamer ¸Senel
1*and Nadide Utku
2*Correspondence: [email protected] 1Department of Mathematics, Faculty of Sciences, Erciyes University, Kayseri, 38039, Turkey Full list of author information is available at the end of the article
Abstract
It is the purpose of this paper to give oscillation criteria for the third-order neutral dynamic equations with continuously distributed delay,
r(t)
x(t) +
b
a
p(t,
η
)xτ
(t,η
)η
γ
+
d
c
q(t,
ξ
)f(
xφ
(t,ξ
))ξ
= 0,on a time scaleT, where
γ
is the quotient of odd positive integers. By using a generalized Riccati transformation and an integral averaging technique, we establish some new sufficient conditions which ensure that every solution of this equation oscillates or converges to zero.Keywords: oscillation; time scales; third-order neutral dynamic equation; asymptotic behavior
1 Introduction
We are concerned with the oscillatory behavior of third-order neutral dynamic equations with continuously distributed delay,
r(t) x(t) +
b
a
p(t,η)xτ(t,η)η
γ
+
d
c
q(t,ξ)fxφ(t,ξ)ξ= , ()
on an arbitrary time scaleT, whereγ is a quotient of odd positive integers. Throughout this paper, we will assume the following hypotheses:
(H) randqare positive rd-continuous functions onTand
∞
t
r(t)
γ
t=∞; ()
(H) p(t,η)∈Crd([t,∞)×[a,b],R),≤p(t)≡
b
a p(t,η)η≤P< ;
(H) τ(t,η)∈Crd([t,∞)×[a,b],T)is not a decreasing function forηand such that
τ(t,η)≤t and lim
t→∞ηmin∈[a,b]τ(t,η) =∞;
(H) φ(t,ξ)∈Crd([t,∞)×[c,d],T)is not decreasing function forξand such that
φ(t,ξ)≤t and lim
t→∞ξmin∈[c,d]φ(t,ξ) =∞;
(H) the functionf∈Crd(T,R)is assumed to satisfyuf(u) > and there exists a positive rd-continuous functionδ(t)onTsuch thatfu(γu)≥δ, foru= . Define the function by
z(t) =x(t) +
b
a
p(t,η)xτ(t,η)η. ()
Furthermore, () is like the following:
r(t)z(t)γ+
d
c
q(t,ξ)fxφ(t,ξ)ξ= . ()
A solutionx(t) of () is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is non-oscillatory.
Much recent attention has been given to dynamic equations on time scales, or mea-sure chains, and we refer the reader to the landmark paper of Hilger [] for a comprehen-sive treatment of the subject. Since then, several authors have expounded various aspects of this new theory; see the survey paper by Agarwalet al.[]. A book on the subject of time scales by Bohner and Peterson [] also summarizes and organizes much of the time scale calculus. In the recent years, there has been increasing interest in obtaining suffi-cient conditions for the oscillation and non-oscillation of solutions of various equations on time scales; we refer the reader to the papers [–]. Candan [] considered oscilla-tion of second-order neutral dynamic equaoscilla-tions with distributed deviating arguments of the form
r(t)y(t) +p(t)yτ(t)γ+
d
c
ft,yθ(t,ξ)ξ= ,
whereγ > is a ratio of odd positive integers withr(t) andp(t) real-valued rd-continuous positive functions defined on T. He established some new oscillation criteria and gave sufficient conditions to ensure that all solutions of nonlinear neutral dynamic equation are oscillatory on a time scaleT.
To the best of our knowledge, there is very little known about the oscillatory behavior of third-order dynamic equations. Erbeet al.[] are concerned with the oscillatory behavior of solutions of the third-order linear dynamic equation
x(t) +p(t)x(t) = ,
on an arbitrary time scaleT, wherep(t) is a positive real-valued rd-continuous function defined onT. Liet al.[] considered third-order nonlinear delay dynamic equation
x+p(t)xγτ(t)= ,
Erbeet al.[, ] established some sufficient conditions which guarantee that every solution of the third-order nonlinear dynamic equation
c(t)a(t)x(t)+q(t)fx(t)= ,
and the third-order dynamic equation
c(t)a(t)x(t)γ+ft,x(t)=
oscillate or converge to zero. Liet al.[] considered the third-order delay dynamic equa-tions
a(t)r(t)x(t)γ+ft,xτ(t)= ,
on a time scaleT, whereγ > is quotient of odd positive integers,aandrare positive rd-continuous functions onT, and the so-called delay functionτ :T→Tsatisfiesτ(t)≤t, andτ(t)→ ∞ast→ ∞,f(x)∈Crd(T×R,R) is assumed to satisfyuf(t,u) > , foru= , and there exists a functionponTsuch thatf(utγ,u)≥p(t) > , foru= .
Saker [] considered the third-order nonlinear functional dynamic equations
p(t)r(t)x(t)γ+q(t)fxτ(t)= ,
on a time scaleT, whereγ > is quotient of odd positive integers. Recently Hanet al.[] and Graceet al.[] considered the third-order neutral delay dynamic equation
r(t)x(t) –a(t)xτ(t)+p(t)xγδ(t)= ,
on a time scaleT.
In this paper, we consider third-order neutral dynamic equation with continuously dis-tributed delay on time scales which is not in literature. We obtain some conclusions which contribute to oscillation theory of third-order neutral dynamic equations.
2 Several lemmas
Before stating our main results, we begin with the following lemmas which play an impor-tant role in the proof of the main results. Throughout this paper, we let
d+(t) :=max
,d(t), d–(t) :=max
, –d(t),
and
β(t) :=b(t), <γ ≤, β(t) :=bγ(t), γ > ,
b(t) = t
σ(t), R(t,t∗) :=
t
t∗
r(s)
γ s,
In order to prove our main results, we will use the formula
zγ(t)=γ
hzσ+ ( –h)zγ–z(t)dh,
wherez(t) is delta differentiable and eventually positive or eventually negative, which is a simple consequence of Keller’s chain rule (see Bohner and Peterson []).
Lemma . Let x(t)be a positive solution of(),z(t)is defined as in().Then z(t)has only
one of the following two properties: (I) z(t) > ,z(t) > ,z(t) > ,
(II) z(t) > ,z(t) < ,z(t) > ,
with t≥t,tsufficiently large.
Proof Letx(t) be a positive solution of () on [t,∞), so thatz(t) >x(t) > , and
r(t)z(t)γ= –
d
c
q(t,ξ)fxφ(t,ξ)ξ< .
Thenr(t)([z(t)])γis a decreasing function and therefore eventually of one sign, soz(t)
is either eventually positive or eventually negative ont≥t≥t. We assert thatz(t) >
ont≥t≥t. Otherwise, assume thatz(t) < , then there exists a constantM> , such
that
r(t)z(t)γ ≤–M< .
By integrating the last inequality fromttot, we obtain
z(t)≤z(t
) –M
γ
t
t
r(s)
γ s.
Lett→ ∞. Then from (H), we have (z(t))→–∞, and therefore eventuallyz(t) < .
Since z(t) < and z(t) < , we have z(t) < , which contradicts our assumption
z(t) > . Therefore,z(t) has only one of the two properties (I) and (II).
This completes the proof.
Lemma . Let x(t)be an eventually positive solution of(),correspondingly z(t)has the property(II).Assume that()and
∞
t
∞
v r(u)
∞
u
q(s)s
γ
uv=∞ ()
hold.Thenlimt→∞x(t) = .
I+>z(t) >Ifor all> . Choosing<I(–PP) and using () and (H), we obtain
x(t) =z(t) –
b
a
p(t,η)xτ(t,η)η
>I–
b
a
p(t,η)xτ(t,η)η
≥I–p(t)zτ(t,a)
≥I–P(I+) >Kz(t), ()
whereK=I–PI+(+)> . Using (H) and (), we find from () that
r(t)z(t)γ = –
d
c
q(t,ξ)fxφ(t,ξ)ξ
≤–
d
c
q(t,ξ)xφ(t,ξ)γδξ
≤–Kγδ d
c
q(t,ξ)zφ(t,ξ)γξ.
Note thatz(t) has property (II) and (H), and we have
r(t)z(t)γ≤–Kγ·δ·zφ(t,d)γ
d
c
q(t,ξ)ξ= –q(t)
zφ(t)
γ
, ()
whereq(t) =Kγδ
d
c q(t,ξ)ξ,φ(t) =φ(t,d). Integrating inequality () fromtto∞, we obtain
r(t)z(t)γ ≥
∞
t
q(s)
zφ(s)
γ
s.
Using (z(φ(s)))γ ≥Iγ, we obtain
z(t)≥ I
rγ ∞
t
q(s)
γ
(s). ()
Integrating inequality () fromtto∞, we have
–z(t)≥I
∞
t r(u)
∞
u
q(s)(s)
γ u.
Integrating the last inequality fromtto∞, we obtain
z(t)≥I
∞
t
∞
v r(u)
∞
u
q(s)(s)
γ uv.
Because () and the last inequality contradict (), we haveI= . Since ≤x(t)≤z(t),
Lemma . Assume that x(t)is a positive solution of(),z(t)is defined as in()such that z(t) > ,z(t) > ,on [t
∗,∞)T,t∗≥.Then
z(t)≥R(t,t∗)rγ(t)z(t). ()
Proof Sincer(t)(z(t))γ is strictly decreasing on [t
∗,∞)T, we get fort∈[t∗,∞)T
z(t) >z(t) –z(t∗)
=
t
t∗
(r(s)(z(t))γ)γ
rγ(s)
s
≥r(t)z(t)γ
γ
t
t∗
r(s)
γ s.
Using the definition ofR(t,t∗), we obtain
z(t) >R(t,t
∗)r
γ(t)z(t) on [t∗,∞)
T.
Lemma . Assume that x(t)is a positive solution of(),correspondingly z(t)has the prop-erty(I).Such that z(t) > ,z(t) > ,on[t
∗,∞)T,t∗≥t.Furthermore,
t
t
q(s)φ
γ
(s)s=∞. ()
Then there exists a T∈[t∗,∞)T,sufficiently large,so that
z(t) >tz(t),
z(t)/t is strictly decreasing,t∈[T,∞)T.
Proof LetU(t) =z(t) –tz(t). HenceU(t) = –σ(t)z(t) < . We claim there exists a
t∈ [t∗,∞)Tsuch thatU(t) > ,z(φ(t,ξ)) > on [t,∞)T. Assume not. ThenU(t) < on
[t,∞)T. Therefore,
z(t) t
=tz
(t) –z(t)
tσ(t) = – U(t)
tσ(t)> , t∈[t,∞)T,
which implies that z(t)/t is strictly increasing on [t,∞)T. Pick t ∈ [t,∞)T so that
φ(t,ξ)≥φ(t,ξ), fort≥t. Then
z(φ(t,ξ)) φ(t,ξ) ≥
z(φ(t,ξ))
φ(t,ξ)
=d> ,
so thatz(φ(t,ξ)) >dφ(t,ξ), fort≥t. By (), (), and (H), we obtain
x(t) =z(t) –
b
a
p(t,η)xτ(t,η)η
≥z(t) –
b
a
≥z(t) –zτ(t,b)
Now by integrating both sides of last equation fromttot, we have
r(t)z(t)γ –r(t)
In this section we give some new oscillation criteria for ().
Theorem . Assume that(), (),and()hold.Furthermore,assume that there exists
a positive functionρ∈C
Proof Assume () has a non-oscillatory solutionx(t) on [t,∞)T. We may assume without
loss of generality thatx(t) > ,t≥t;x(τ(t,η)) > , (t,η)∈[t,∞)×[a,b] andx(φ(t,ξ)) > ,
(t,ξ)∈[t,∞)×[c,d] for allt∈[t,∞)T.z(t) is defined as in (). We suppose thatz(t) > .
We shall consider only this case, since the proof whenz(t) is eventually negative is similar. Therefore Lemma . and Lemma ., we get
Define the functionw(t) by the Riccati substitution
w(t) =ρ(t)r(t)([z(t)] Using the Keller chain rule (see []), we have
–γρσ(t)R(t,t∗)r
In the second caseγ> . Applying the Keller chain rule, we have
Then using the inequality []
λABλ––Aλ≤(λ– )Bλ, ()
From this last inequality and (), we find
Integrating both sides fromT tot, we get
t
T
ρσ(s)q(s)
φ(s)
σ(s)
γ
– ((ρ
(s))
+)γ+
(γ+ )γ+(β(s)ρσ(s)R(s,t
∗))γ
s≤w(T) –w(t)≤w(T),
which contradicts assumption (). This completes the proof of Theorem ..
Remark . From Theorem ., we can obtain different conditions for oscillation of ()
with different choices ofρ(t).
Remark . The conclusion of Theorem . remains intact if assumption () is replaced
by the two conditions
lim sup
t→∞
t
T
ρσ(s)q(s)
φ(s)
σ(s)
γ
s=∞,
lim sup
t→∞
t
T
((ρ(s))
+)γ+
(γ+ )γ+(β(s)ρσ(s)ψ(s,t∗))γs<∞.
For example, letρ(t) =t. Now Theorem . yields the following results.
Corollary . Assume that(H)-(H), (),and()hold.If
lim sup
t→∞
t
T
σ(s)q(s)
φ(s)
σ(s)
γ
–
(γ + )γ+(β(s)σ(s)R(s,t∗))γ
s=∞ ()
holds,then every solution()is either oscillatory orlimt→∞x(t) = .
For example, letρ(t) = . Now Theorem . yields the following results.
Corollary . Assume that(H)-(H), (),and()hold.If
lim sup
t→∞
t
T
q(s)
φ(s)
σ(s)
γ
s=∞, ()
then every solution()is either oscillatory orlimt→∞x(t) = .
Theorem . Assume that(), (),and()hold.Furthermore,suppose that there exist
functions H,h∈Crd(D,R),whereD≡(t,s) :t≥s≥tsuch that
H(t,t) = , t≥,
H(t,s) > , t>s≥t,
and H has a nonpositive continuous-partial derivative Hs(t,s)with respect to the
sec-ond variable and satisfies
Hsσ(t),s+Hσ(t),σ(s)ρ
(s)
ρ(s) = – h(t,s)
ρ(s) H
σ(t),σ(s) γ
and for all sufficiently large T∈[t,∞)T,there is a T>Tsuch that
whereρis a positive-differentiable function and
K(t,s) =Hσ(t),σ(s)ρσ(s)q(s)
Then every solution of()is either oscillatory or tends to zero.
Proof Suppose thatx(t) is a non-oscillatory solution of () andz(t) is defined as in ().
It then follows from () that
It then follows from () that
Therefore, as in Theorem ., by letting
Aλ:=Hσ(t),σ(s)γρσ(t)β(t)R(t,T)
Then using the inequality []
and this implies that
for all largeT, which contradicts (). This completes the proof of Theorem ..
Remark . The conclusion of Theorem . remains intact if assumption () is replaced
by the two conditions
lim sup
similar to the proofs of Theorem ., we can obtain different results. We leave the details to the reader.
Example . Consider the following third-order neutral dynamic equationt∈[t,∞)T:
It is clear that condition (), (), and () hold. Therefore, by Theorem ., picking ρ(t) =t, we have
Hence, by Theorem . every solution of () is oscillatory or tends to zero ifβ> .
Example . Consider the following third-order neutral dynamic equationt∈[t,∞)T:
It is clear that condition (), (), and () hold. Therefore, by Theorem ., picking ρ(t) = , we have
lim sup
t→∞
t
T
q(s)
φ(s)
σ(s)
s=lim sup
t→∞
t
T β
ss=∞.
Hence, by Theorem . every solution of () is oscillatory or tends to zero ifβ> .
Competing interests
The authors declare that they have no competing interests. Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details
1Department of Mathematics, Faculty of Sciences, Erciyes University, Kayseri, 38039, Turkey.2Institute of Sciences, Erciyes University, Kayseri, 38039, Turkey.
Received: 17 January 2014 Accepted: 19 July 2014 Published:05 Aug 2014
References
1. Hilger, S: Analysis on measure chains a unified approach to continuous and discrete calculus. Results Math.18, 18-56 (1990)
2. Agarwal, RP, Bohner, M, O’Regan, D, Peterson, A: Dynamic equations on time scales: a survey. J. Comput. Appl. Math.
141, 1-26 (2002)
3. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)
4. Agarwal, RP, O’ Regan, D, Saker, SH: Oscillation criteria for second-order nonlinear neutral delay dynamic equations. J. Math. Anal. Appl.300(1), 203-217 (2004)
5. ¸Sahiner, Y: Oscillation of second-order neutral delay and mixed-type dynamic equations on time scales. Adv. Differ. Equ.2006, Article ID 65626 (2006)
6. Wu, H-W, Zhuang, R-K, Mathsen, RM: Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations. Appl. Math. Comput.178(2), 321-331 (2006)
7. Saker, SH: Oscillation of second-order nonlinear neutral delay dynamic equations on time scales. J. Comput. Appl. Math.187(2), 123-141 (2006)
8. Zhang, SY, Wang, QR: Oscillation of second-order nonlinear neutral dynamic equations on time scales. Appl. Math. Comput.216(10), 2837-2848 (2010)
9. ¸Senel, MT: Kamenev-type oscillation criteria for the second-order nonlinear dynamic equations with damping on time scales. Abstr. Appl. Anal.2012, Article ID 253107 (2012)
10. ¸Senel, MT: Oscillation theorems for dynamic equation on time scales. Bull. Math. Anal. Appl.3, 101-105 (2011) 11. Li, T, Agarwal, RP, Bohner, M: Some oscillation results for second-order neutral dynamic equations. Hacet. J. Math. Stat.
41, 715-721 (2012)
12. Erbe, L, Hassan, TS, Peterson, A: Oscillation of third order nonlinear functional dynamic equations on time scales. Differ. Equ. Dyn. Syst.18, 199-227 (2010)
13. ¸Senel, MT: Behavior of solutions of a third-order dynamic equation on time scales. J. Inequal. Appl.2013, Article ID 47 (2013)
14. Zhang, Q, Gao, L, Yu, Y: Oscillation criteria for third order neutral differential equations with continuously distributed delay. Appl. Math. Lett.10, 10-16 (2012)
15. Hardy, GH, Littlewood, JE, Pólya, G: Inequalities. Reprint of the 1952 edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1988)
16. Graef, J, Saker, SH: Oscillation of third-order nonlinear neutral functional dynamic equations. Dyn. Syst. Appl.21, 583-606 (2012)
17. Saker, SH: On oscillation of a certain class of third-order nonlinear functional dynamic equations on time scales. Bull. Math. Soc. Sci. Math. Roum.54, 365-389 (2011)
18. Saker, SH: Oscillation Theory of Dynamic Equations on Time Scales. Lambert Academic Publishing, Colne (2010) 19. Agarwal, RP, O’Regan, D, Saker, SH: Philos-type oscillation criteria of second-order half-linear dynamic equations on
time scales. Rocky Mt. J. Math.37, 1085-1104 (2007)
20. Candan, T: Oscillation of second order nonlinear neutral dynamic equations on time scales with distributed deviating arguments. Comput. Math. Appl.62, 4118-4125 (2011)
21. Erbe, L, Peterson, A, Saker, SH: Hille and Nehari type criteria for third-order dynamic equations. J. Math. Anal. Appl.
329, 112-131 (2007)
22. Li, T, Han, Z, Zhang, C, Sun, Y: Oscillation criteria for third-order nonlinear delay dynamic equations on time scales. Bull. Math. Anal. Appl.3, 52-60 (2011)
23. Erbe, L, Peterson, A, Saker, SH: Asymptotic behavior of solutions of a third-order nonlinear dynamic equation on time scales. J. Comput. Appl. Math.181, 92-102 (2005)
25. Li, T, Han, Z, Sun, S, Zhao, Y: Oscillation results for third order nonlinear delay dynamic equations on time scales. Bull. Malays. Math. Soc.34, 639-648 (2011)
26. Saker, SH: Oscillation of third-order functional dynamic equations on time scales. Sci. China Math.54, 2597-2614 (2011)
27. Han, Z, Li, T, Sun, S, Zhang, C: Oscillation behavior of third order neutral Emden-Fowler delay dynamic equations on time scales. Adv. Differ. Equ.2010, Article ID 586312 (2010)
28. Grace, SR, Graef, JR, El-Beltagy, MA: On the oscillation of third order delay dynamic equations on time scales. Appl. Math. Comput.63, 775-782 (2012)
10.1186/1687-1847-2014-220
