Oscillation of fourth-order delay differential equations
This article is concerned with oscillation of a certain class of fourth-order delay differential equations. Some new oscillation criteria are presented which include Hille and Nehari type. The results obtained improve some results obtained earlier in [Zhang C., Li T., Sun B., Thandapani E. On the os...
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irk-123456789-1771232021-02-11T01:28:44Z Oscillation of fourth-order delay differential equations Zhang, C. Li, T. Saker, S.H. This article is concerned with oscillation of a certain class of fourth-order delay differential equations. Some new oscillation criteria are presented which include Hille and Nehari type. The results obtained improve some results obtained earlier in [Zhang C., Li T., Sun B., Thandapani E. On the oscillation of higher-order half-linear delay differential equations // Appl. Math. Lett. — 2011. — 24. — P. 1618 – 1621]. Two examples are considered to illustrate the main results. Розглянуто коливання в деякому класi диференцiальних рiвнянь четвертого порядку з загаюванням. Знайдено новi критерiї коливання, якi включають в себе критерiї типу Хiлле та Нехарi. Отриманi результати покращують деякi результати з [Zhang C., Li T., Sun B., Thandapani E. On the oscillation of higher-order half-linear delay differential equations // Appl. Math. Lett. — 2011. — 24. — P. 1618 – 1621]. Розглянуто два приклади, якi iлюструють основнi результати. 2013 Article Oscillation of fourth-order delay differential equations / C. Zhang, T. Li, S.H. Saker // Нелінійні коливання. — 2013. — Т. 16, № 3. — С. 322-335. — Бібліогр.: 22 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177123 517.9 en Нелінійні коливання Інститут математики НАН України |
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This article is concerned with oscillation of a certain class of fourth-order delay differential equations. Some new oscillation criteria are presented which include Hille and Nehari type. The results obtained improve some results obtained earlier in [Zhang C., Li T., Sun B., Thandapani E. On the oscillation of higher-order half-linear delay differential equations // Appl. Math. Lett. — 2011. — 24. — P. 1618 – 1621]. Two examples are considered to illustrate the main results. |
| format |
Article |
| author |
Zhang, C. Li, T. Saker, S.H. |
| spellingShingle |
Zhang, C. Li, T. Saker, S.H. Oscillation of fourth-order delay differential equations Нелінійні коливання |
| author_facet |
Zhang, C. Li, T. Saker, S.H. |
| author_sort |
Zhang, C. |
| title |
Oscillation of fourth-order delay differential equations |
| title_short |
Oscillation of fourth-order delay differential equations |
| title_full |
Oscillation of fourth-order delay differential equations |
| title_fullStr |
Oscillation of fourth-order delay differential equations |
| title_full_unstemmed |
Oscillation of fourth-order delay differential equations |
| title_sort |
oscillation of fourth-order delay differential equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/177123 |
| citation_txt |
Oscillation of fourth-order delay differential equations / C. Zhang, T. Li, S.H. Saker // Нелінійні коливання. — 2013. — Т. 16, № 3. — С. 322-335. — Бібліогр.: 22 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT zhangc oscillationoffourthorderdelaydifferentialequations AT lit oscillationoffourthorderdelaydifferentialequations AT sakersh oscillationoffourthorderdelaydifferentialequations |
| first_indexed |
2025-07-15T15:08:39Z |
| last_indexed |
2025-07-15T15:08:39Z |
| _version_ |
1837726038143008768 |
| fulltext |
UDC 517.9
OSCILLATION OF FOURTH-ORDER DELAY
DIFFERENTIAL EQUATIONS*
КОЛИВАННЯ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ЧЕТВЕРТОГО ПОРЯДКУ З ЗАГАЮВАННЯМ
C. Zhang
School Control Sci. and Engineering, Shandong Univ.
Jinan, Shandong 250061, P. R. China
e-mail: zchui@sdu.edu.cn
T. Li
School Control Sci. and Engineering, Shandong Univ.
Jinan, Shandong 250061, P. R. China
e-mail: litongx2007@163.com
S. H. Saker
Mansoura Univ.
Mansoura 35516, Egypt
e-mail: shsaker@mans.edu.eg
This article is concerned with oscillation of a certain class of fourth-order delay differential equations.
Some new oscillation criteria are presented which include Hille and Nehari type. The results obtained
improve some results obtained earlier in [Zhang C., Li T., Sun B., Thandapani E. On the oscillation of
higher-order half-linear delay differential equations // Appl. Math. Lett. — 2011. — 24. — P. 1618 – 1621].
Two examples are considered to illustrate the main results.
Розглянуто коливання в деякому класi диференцiальних рiвнянь четвертого порядку з загаюван-
ням. Знайдено новi критерiї коливання, якi включають в себе критерiї типу Хiлле та Нехарi.
Отриманi результати покращують деякi результати з [Zhang C., Li T., Sun B., Thandapani E.
On the oscillation of higher-order half-linear delay differential equations // Appl. Math. Lett. — 2011. —
24. — P. 1618 – 1621]. Розглянуто два приклади, якi iлюструють основнi результати.
1. Introduction. In this paper, we are concerned with oscillation of the fourth-order quasilinear
delay differential equation
(
r(t)
(
x
′′′
(t)
)α)′
+ q(t)xα(τ(t)) = 0, for t ≥ t0. (1.1)
We will assume that the following assumptions hold:
(H1) α is a quotient of odd positive integers;
(H2) r ∈ C1[t0,∞), r
′
(t) ≥ 0, r(t) > 0, q, τ ∈ C[t0,∞), q(t) ≥ 0, τ(t) ≤ t, and
limt→∞ τ(t) = ∞.
∗ This research is supported by NNSF of P. R. China (Grant Nos. 61034007, 51277116, 51107069).
c© C. Zhang, T. Li, S. H. Saker, 2013
322 ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
OSCILLATION OF FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS 323
By a solution of (1.1), we mean a function x ∈ C3[Tx,∞), Tx ≥ t0, which has the property
r(x
′′′
)α ∈ C1[Tx,∞) and satisfies (1.1) on [Tx,∞). We consider only those solutions x of (1.1)
which satisfy sup{|x(t)| : t ≥ T} > 0 for all T ≥ Tx. We assume that (1.1) possesses such a
solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on [Tx,∞) and
otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are
oscillatory.
In recent decades, the oscillation of second-order and third-order differential equations
have been deeply studied in the literature, we refer the reader to the related books [1, 3 – 5, 13,
15, 21] and the papers [2, 6 – 12, 14, 16 – 20, 22]. In the following, we present some related results
that serve and motivate the contents of this paper. Agarwal et al. [2], Kamo and Usami [11,
12], and Kusano et al. [14] considered the oscillation of the fourth-order nonlinear differential
equation (
r(t)
(
x
′′
(t)
)α)′′
+ q(t)xβ(t) = 0.
Grace et al. [10] examined the oscillation behavior of the fourth-order nonlinear differential
equation (
r(t)
(
x
′
(t)
)α)′′′
+ q(t)f(x(g(t))) = 0.
Agarwal et al. [7] and Zhang et al. [22] studied the oscillatory properties of the higher-order
differential equation (
r(t)
(
x(n−1)(t)
)α)′
+ q(t)xβ(τ(t)) = 0, (1.2)
under the conditions
∞∫
t0
1
r1/α(t)
dt = ∞,
and
∞∫
t0
1
r1/α(t)
dt < ∞. (1.3)
Zhang et al. [22] obtained some results which ensure that every solution x of (1.2) is either
oscillatory or limt→∞ x(t) = 0 for the case where (1.3) holds. As a special case when n = 4,
they proved the following result: Let (H1), (H2), and (1.3) hold, and τ(t) < t. Further, assume
that for some constant λ0 ∈ (0, 1), the delay differential equation
y
′
(t) + q(t)
(
λ0τ
3(t)
6r1/α(τ(t))
)α
y(τ(t)) = 0 (1.4)
is oscillatory. If
lim sup
t→∞
t∫
t0
[
q(s)
(
λ1
2
τ2(s)
)α
δα(s)− αα+1
(α+ 1)α+1
1
δ(s)r1/α(s)
]
ds = ∞ (1.5)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
324 C. ZHANG, T. LI, S. H. SAKER
for some constant λ1 ∈ (0, 1), where δ(t) :=
∫ ∞
t
r−(1/α)(s) ds, then every solution of (1.1) is
either oscillatory or converges to zero as t → ∞.
Our aim in this paper is to employ the Riccati technique to establish some new conditions
for oscillation of all solutions of (1.1). The results not only differ from the results obtained
in [22], but also improve some of them. Some examples are considered to illustrate the main
results.
2. Main results. In this section, we will derive some new criteria for oscillation of (1.1). To
prove the main results we will need the following lemma.
Lemma 2.1 (see [3], Lemma 2.2.3). Let f ∈ Cn([t0,∞),R+). Assume that f (n)(t) is of fixed
sign and not identically zero on [t0,∞), and there exists a t1 ≥ t0 such that f (n−1)(t)f (n)(t) ≤ 0
for all t ≥ t1. If limt→∞ f(t) 6= 0, then for every k ∈ (0, 1), there exists tk ∈ [t1,∞) such that
f(t) ≥ k
(n− 1)!
tn−1|f (n−1)(t)|, for t ∈ [tk,∞).
Now, we are ready to state and prove the main results. For convenience, we denote
R(t) :=
∞∫
t
1
r
1
α (s)
ds, ρ
′
+(t) := max{0, ρ′(t)}, and θ
′
+(t) := max{0, θ′(t)}.
In the sequel, all occurring functional inequalities considered in this section are assumed to
hold eventually, that is, they are satisfied for all t large enough.
Theorem 2.1. Let (H1), (H2), and (1.3) hold. Assume that there exists a positive function
ρ ∈ C1[t0,∞) such that
∞∫
t0
[
q(s)
(
τ2(s)
s2
)α
ρ(s)− 2α
(α+ 1)α+1
r(s)(ρ
′
+(s))
α+1
(k1ρ(s)s2)α
]
ds = ∞, (2.1)
for some constant k1 ∈ (0, 1). Assume further that there exists a positive function θ ∈ C1[t0,∞)
such that
∞∫
t0
θ(s) ∞∫
s
1
r(ϑ)
∞∫
ϑ
q(ς)
(
τ2(ς)
ς2
)α
dς
1
α
dϑ−
(θ
′
+(s))
2
4θ(s)
ds = ∞. (2.2)
If
∞∫
t0
q(s)
∞∫
s
∞∫
u
R(v)dvdu
α
− αα+1
(α+ 1)α+1
∫∞
s R(v)dv∫∞
s
∫∞
u R(v)dvdu
ds = ∞, (2.3)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
OSCILLATION OF FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS 325
and
∞∫
t0
[
q(s)
(
k2
2
τ2(s)
)α
Rα(s)− αα+1
(α+ 1)α+1R(s)r1/α(s)
]
ds = ∞, (2.4)
for some constant k2 ∈ (0, 1), then every solution of (1.1) is oscillatory.
Proof. Assume that (1.1) has a nonoscillatory solution x. Without loss of generality we may
assume that x is eventually positive. It follows from (1.1) that there exist four possible cases for
t ≥ t1, where t1 ≥ t0 is large enough:
Case 1 : x(t) > 0, x
′
(t) > 0, x
′′
(t) > 0, x
′′′
(t) > 0, x(4)(t) ≤ 0, (r(x
′′′
)α)
′
(t) ≤ 0.
Case 2 : x(t) > 0, x
′
(t) > 0, x
′′
(t) < 0, x
′′′
(t) > 0, x(4)(t) ≤ 0, (r(x
′′′
)α)
′
(t) ≤ 0.
Case 3 : x(t) > 0, x
′
(t) < 0, x
′′
(t) > 0, x
′′′
(t) < 0, (r(x
′′′
)α)
′
(t) ≤ 0.
Case 4 : x(t) > 0, x
′
(t) > 0, x
′′
(t) > 0, x
′′′
(t) < 0, (r(x
′′′
)α)
′
(t) ≤ 0.
Assume that Case 1 holds. By Kiguradze Lemma [13], we have x(t) ≥ (t/2)x
′
(t), and so
x(τ(t))
x(t)
≥ τ2(t)
t2
. (2.5)
Define
ω(t) := ρ(t)
r(t)(x
′′′
)α(t)
xα(t)
, t ≥ t1. (2.6)
Then ω(t) > 0 for t ≥ t1, and
ω
′
(t) = ρ
′
(t)
r(t)(x
′′′
)α(t)
xα(t)
+ ρ(t)
(r(x
′′′
)α)
′
(t)
xα(t)
− αρ(t) x
α−1(t)x
′
(t)r(t)(x
′′′
)α(t)
x2α(t)
. (2.7)
From Lemma 2.1, we have
x
′
(t) ≥ k
2
t2x
′′′
(t) (2.8)
for every k ∈ (0, 1) and for all sufficiently large t. Hence, we obtain by (2.7) and (2.8) that
ω
′
(t) ≤ ρ
′
(t)
r(t)(x
′′′
)α(t)
xα(t)
+ ρ(t)
(r(x
′′′
)α)
′
(t)
xα(t)
− αk
2
t2ρ(t)
x
′′′
(t)r(t)(x
′′′
)α(t)
xα+1(t)
.
Hence by (1.1), we get
ω
′
(t) ≤ −q(t)
(
τ2(t)
t2
)α
ρ(t) +
ρ
′
+(t)
ρ(t)
ω(t)− αk
2
t2
(r(t)ρ(t))
1
α
ω
α+1
α (t). (2.9)
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
326 C. ZHANG, T. LI, S. H. SAKER
Set
A :=
αkt2
2(r(t)ρ(t))
1
α
, B :=
ρ
′
+(t)
ρ(t)
, y := ω(t).
Using the inequality
By −Ay
α+1
α ≤ αα
(α+ 1)α+1
Bα+1
Aα
, A,B > 0,
we have
ρ
′
+(t)
ρ(t)
ω(t)− αkt2
2(r(t)ρ(t))
1
α
ω
α+1
α (t) ≤ 2α
(α+ 1)α+1
r(t)(ρ
′
+(t))
α+1
(kρ(t)t2)α
.
Hence, we obtain
ω
′
(t) ≤ −q(t)
(
τ2(t)
t2
)α
ρ(t) +
2α
(α+ 1)α+1
r(t)(ρ
′
+(t))
α+1
(kρ(t)t2)α
,
which implies that
t∫
t1
[
q(s)
(
τ2(s)
s2
)α
ρ(s)− 2α
(α+ 1)α+1
r(s)(ρ
′
+(s))
α+1
(kρ(s)s2)α
]
ds ≤ ω(t1),
for every k ∈ (0, 1) and for all sufficiently large t. This is a contradiction to (2.1). Assume that
Case 2 holds. Integrating (1.1) from t to l, we have
r(l)(x
′′′
)α(l)− r(t)(x′′′)α(t) +
l∫
t
q(s)xα(τ(s)) ds = 0.
By virtue of x > 0, x
′
> 0, and x
′′
< 0, we get x(t) ≥ (t/2)x
′
(t), and so (2.5) holds. Then by
(2.5), we have
r(l)(x
′′′
)α(l)− r(t)(x′′′)α(t) +
l∫
t
q(s)
(
τ2(s)
s2
)α
xα(s) ds ≤ 0,
from which follows by x
′
> 0 that
r(l)(x
′′′
)α(l)− r(t)(x′′′)α(t) + xα(t)
l∫
t
q(s)
(
τ2(s)
s2
)α
ds ≤ 0.
Letting l → ∞, we have
−r(t)(x′′′)α(t) + xα(t)
∞∫
t
q(s)
(
τ2(s)
s2
)α
ds ≤ 0,
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
OSCILLATION OF FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS 327
i.e.,
−x′′′(t) + x(t)
1
r(t)
∞∫
t
q(s)
(
τ2(s)
s2
)α
ds
1
α
≤ 0.
Integrating again from t to∞, we get
x
′′
(t) + x(t)
∞∫
t
1
r(ϑ)
∞∫
ϑ
q(s)
(
τ2(s)
s2
)α
ds
1
α
dϑ ≤ 0. (2.10)
Define
ξ(t) := θ(t)
x
′
(t)
x(t)
, t ≥ t1.
Then ξ(t) > 0 for t ≥ t1, and
ξ
′
(t) = θ
′
(t)
x
′
(t)
x(t)
+ θ(t)
x
′′
(t)x(t)− (x
′
)2(t)
x2(t)
=
= θ(t)
x
′′
(t)
x(t)
+
θ
′
(t)
θ(t)
ξ(t)− ξ2(t)
θ(t)
.
Hence by (2.10), we get
ξ
′
(t) ≤ −θ(t)
∞∫
t
1
r(ϑ)
∞∫
ϑ
q(s)
(
τ2(s)
s2
)α
ds
1
α
dϑ+
θ
′
+(t)
θ(t)
ξ(t)− ξ2(t)
θ(t)
. (2.11)
Thus, we have
ξ
′
(t) ≤ −θ(t)
∞∫
t
1
r(ϑ)
∞∫
ϑ
q(s)
(
τ2(s)
s2
)α
ds
1
α
dϑ+
(θ
′
+(t))
2
4θ(t)
,
which yields
t∫
t1
θ(s) ∞∫
s
1
r(ϑ)
∞∫
ϑ
q(ς)
(
τ2(ς)
ς2
)α
dς
1
α
dϑ−
(θ
′
+(s))
2
4θ(s)
ds ≤ ξ(t1),
which contradicts (2.2). Assume that Case 3 holds. Recalling that r(x
′′′
)α is nonincreasing, we
have
r1/α(s)x
′′′
(s) ≤ r1/α(t)x
′′′
(t), s ≥ t ≥ t1.
ISSN 1562-3076. Нелiнiйнi коливання, 2013, т . 16, N◦ 3
328 C. ZHANG, T. LI, S. H. SAKER
Dividing the above inequality by r1/α(s) and integrating the resulting inequality from t to l, we
obtain
x
′′
(l) ≤ x
′′
(t) + r1/α(t)x
′′′
(t)
l∫
t
r−1/α(s) ds.
Letting l → ∞, we get
x
′′
(t) ≥ −r1/α(t)x′′′(t)R(t). (2.12)
Integrating (2.12) from t to∞, we have
−x′(t) ≥
∞∫
t
−r1/α(s)x′′′(s)R(s) ds ≥ −r1/α(t)x′′′(t)
∞∫
t
R(s) ds. (2.13)
Integrating (2.13) from t to∞, we get
x(t) ≥
∞∫
t
−r1/α(u)x′′′(u)
∞∫
u
R(s) ds du ≥ −r1/α(t)x′′′(t)
∞∫
t
∞∫
u
R(s) ds du. (2.14)
We define
ϕ(t) :=
r(t)(x
′′′
)α(t)
xα(t)
, t ≥ t1. (2.15)
Then ϕ(t) < 0, for t ≥ t1, and by (2.13), we have that
ϕ
′
(t) =
(r(x
′′′
)α)
′
(t)
xα(t)
− αr(t)(x
′′′
)α(t)x
′
(t)
xα+1(t)
≤
≤ −q(t)x
α(τ(t))
xα(t)
− αr
α+1
α (t)(x
′′′
)α+1(t)
xα+1(t)
∞∫
t
R(s) ds. (2.16)
Hence by (2.15) and (2.16), we obtain
ϕ
′
(t) ≤ −q(t)− αϕ
α+1
α (t)
∞∫
t
R(s) ds. (2.17)
From (2.14), we get
ϕ(t)
∞∫
t
∞∫
u
R(s) ds du
α
≥ −1. (2.18)
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OSCILLATION OF FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS 329
Multiplying (2.17) by
(∫ ∞
t
∫ ∞
u
R(s) ds du
)α
and integrating the resulting inequality from t1
to t, we have
∞∫
t
∞∫
u
R(s) ds du
α
ϕ(t)−
∞∫
t1
∞∫
u
R(s) ds du
α
ϕ(t1)+
+ α
t∫
t1
∞∫
s
R(v) dv
∞∫
s
∞∫
u
R(v) dv du
α−1
ϕ(s) ds+
+
t∫
t1
q(s)
∞∫
s
∞∫
u
R(v) dv du
α
ds+
+ α
t∫
t1
ϕ
α+1
α (s)
∞∫
s
∞∫
u
R(v) dv du
α ∞∫
s
R(v) dv ds ≤ 0.
Set
B :=
∞∫
s
R(v) dv
∞∫
s
∞∫
u
R(v) dv du
α−1
,
and
A :=
∞∫
s
∞∫
u
R(v) dv du
α ∞∫
s
R(v) dv, y := −ϕ(s).
Using the inequality
−By +Ay
α+1
α ≥ − αα
(α+ 1)α+1
Bα+1
Aα
, A,B > 0, (2.19)
we have
∞∫
s
R(v) dv
∞∫
s
∞∫
u
R(v) dv du
α−1
ϕ(s) + ϕ
α+1
α (s)
∞∫
s
∞∫
u
R(v) dv du
α ∞∫
s
R(v) dv ≥
≥ − αα
(α+ 1)α+1
∫∞
s R(v) dv∫∞
s
∫∞
u R(v) dv du
.
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330 C. ZHANG, T. LI, S. H. SAKER
Hence, we obtain by (2.18) that
t∫
t1
q(s)
∞∫
s
∞∫
u
R(v) dv du
α
− αα+1
(α+ 1)α+1
∫∞
s R(v) dv∫∞
s
∫∞
u R(v) dv du
ds ≤
≤
∞∫
t1
∞∫
u
R(s) ds du
α
ϕ(t1) + 1.
This is a contradiction to (2.3). Assume that Case 4 holds. In view of the proof of Case 3, we
have (2.12). On the other hand, by Lemma 2.1, we get
x(t) ≥ k
2
t2x
′′
(t) (2.20)
for every k ∈ (0, 1) and for all sufficiently large t. Now define
φ(t) :=
r(t)(x
′′′
)α(t)
(x′′)α(t)
, t ≥ t1. (2.21)
Then φ(t) < 0 for t ≥ t1, and by (2.20) and (2.21), we get that
φ
′
(t) = −q(t) xα(τ(t))
(x′′(τ(t)))α
(x
′′
(τ(t)))α
(x′′)α(t)
− α φ
α+1
α (t)
r
1
α (t)
≤ −q(t)
(
k
2
τ2(t)
)α
− αφ
α+1
α (t)
r
1
α (t)
. (2.22)
Multiplying the above inequality by Rα(t) and integrating the resulting inequality from t1 to t,
we have
Rα(t)φ(t)−Rα(t1)φ(t1) + α
t∫
t1
r−1/α(s)Rα−1(s)φ(s) ds ≤
≤ −
t∫
t1
q(s)
(
k
2
τ2(s)
)α
Rα(s) ds− α
t∫
t1
φ(α+1)/α(s)
r1/α(s)
Rα(s) ds.
Set B := r−1/α(s)Rα−1(s), A := Rα(s)/r1/α(s), and y := −φ(s). Using the inequality (2.19)
and (2.12), we have, for every k ∈ (0, 1) and for all sufficiently large t,
t∫
t1
[
q(s)
(
k
2
τ2(s)
)α
Rα(s)− αα+1
(α+ 1)α+1
1
R(s)r1/α(s)
]
ds ≤ Rα(t1)φ(t1) + 1.
This is a contradiction to (2.4).
Theorem 2.1 is proved.
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OSCILLATION OF FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS 331
It is well known (see [8]) that the differential equation
(a(t)(x
′
(t))α)
′
+ q(t)xα(t) = 0, (2.23)
where α > 0 is a ratio of odd positive integers, a, q ∈ C([t0,∞),R+) is nonoscillatory if and
only if there exist a number T ≥ t0 and a function v ∈ C1([T,∞),R) which satisfies the
inequality
v
′
(t) + αa−1/α(t)(v(t))(1+α)/α + q(t) ≤ 0, on [T,∞).
In the following, we compare the oscillatory behavior of equation (1.1) with second-order
half-linear equations of type (2.23). For the oscillation of equation (2.23), there are many
results; see e.g., [1, 3 – 5, 17, 18, 20, 21] which include Hille and Nehari type, Philos type, etc.
Theorem 2.2. Let (H1), (H2), and (1.3) hold. Assume that the equation(
r(t)
t2α
(x
′
(t))α
)′
+ q(t)
(
k1τ
2(t)
2t2
)α
xα(t) = 0 (2.24)
is oscillatory for some constant k1 ∈ (0, 1), the equation
x
′′
(t) + x(t)
∞∫
t
1
r(ϑ)
∞∫
ϑ
q(s)
(
τ2(s)
s2
)α
ds
1
α
dϑ = 0 (2.25)
is oscillatory, and the equation ∞∫
t
R(s) ds
−α (x′(t))α
′
+ q(t)xα(t) = 0 (2.26)
is oscillatory, and the equation(
r(t)(x
′
(t))α
)′
+ q(t)
(
k2
2
τ2(t)
)α
xα(t) = 0 (2.27)
is oscillatory for some constant k2 ∈ (0, 1). Then every solution of (1.1) is oscillatory.
Proof. Proceeding as in the proof of Theorem 2.1, we have (2.9), (2.11), (2.17), and (2.22).
Letting ρ(t) = 1 in (2.9), we have
ω
′
(t) +
αkt2
2(r(t))
1
α
ω
α+1
α (t) + q(t)
(
τ2(t)
t2
)α
≤ 0
for every constant k ∈ (0, 1). Then we can see that equation (2.24) is nonoscillatory for every
constant k1 ∈ (0, 1), which is a contradiction. Letting θ(t) = 1 in (2.11), we have
ξ
′
(t) + ξ2(t) +
∞∫
t
1
r(ϑ)
∞∫
ϑ
q(s)
(
τ2(s)
s2
)α
ds
1
α
dϑ ≤ 0.
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332 C. ZHANG, T. LI, S. H. SAKER
Then equation (2.25) is nonoscillatory, which is a contradiction. From (2.17), we have
ϕ
′
(t) + αϕ
α+1
α (t)
∞∫
t
R(s) ds+ q(t) ≤ 0.
Then we can find that equation (2.26) is nonoscillatory, which is a contradiction. From (2.22),
we have
φ
′
(t) + α
φ
α+1
α (t)
r
1
α (t)
+ q(t)
(
k
2
τ2(t)
)α
≤ 0
for every constant k ∈ (0, 1). Then we can see that equation (2.27) is nonoscillatory for every
constant k2 ∈ (0, 1), which is a contradiction.
Theorem 2.2 is proved.
It is well known (see [18]) that if
∞∫
t0
1
a(t)
dt = ∞, and lim inf
t→∞
t∫
t0
1
a(s)
ds
∞∫
t
q(s) ds >
1
4
,
then equation (2.23) with α = 1 is oscillatory. Also, it is well known (see [20], Theorem 3.3)
that if
∞∫
t0
1
a(t)
dt < ∞, and lim inf
t→∞
∞∫
t
1
a(s)
ds
−1 ∞∫
t
∞∫
s
1
a(v)
dv
2
q(s) ds >
1
4
,
then equation (2.23) with α = 1 is oscillatory.
Based on the above results and Theorem 2.2, we can easily obtain the following Hille and
Nehari type oscillation criteria for (1.1) when α = 1.
Theorem 2.3. Let α = 1, (H1), (H2), and (1.3) hold. Assume that
∞∫
t0
t2
r(t)
dt = ∞, and lim inf
t→∞
t∫
t0
s2
r(s)
ds
∞∫
t
q(s)
τ2(s)
s2
ds >
1
2k1
for some constant k1 ∈ (0, 1), and
lim inf
t→∞
t
∞∫
t
∞∫
η
1
r(ϑ)
∞∫
ϑ
q(s)
τ2(s)
s2
ds dϑ dη >
1
4
, (2.28)
and
∞∫
t0
∞∫
t
R(s) ds dt = ∞, and lim inf
t→∞
t∫
t0
∞∫
s
R(v) dv ds
∞∫
t
q(s) ds >
1
4
,
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OSCILLATION OF FOURTH-ORDER DELAY DIFFERENTIAL EQUATIONS 333
and
lim inf
t→∞
∞∫
t
1
r(s)
ds
−1 ∞∫
t
∞∫
s
1
r(v)
dv
2
q(s)τ2(s) ds >
1
2k2
(2.29)
for some constant k2 ∈ (0, 1). Then every solution of (1.1) with α = 1 is oscillatory.
Theorem 2.4. Let α = 1, (H1), (H2), (1.3), and (2.28) hold. Assume that
∞∫
t0
t2
r(t)
dt < ∞,
lim inf
t→∞
∞∫
t
s2
r(s)
ds
−1 ∞∫
t
∞∫
s
v2
r(v)
dv
2
q(s)
τ2(s)
s2
ds >
1
2k1
for some constant k1 ∈ (0, 1), and
∞∫
t0
∞∫
t
R(s) ds dt < ∞,
lim inf
t→∞
∞∫
t
∞∫
s
R(v) dv ds
−1 ∞∫
t
∞∫
s
∞∫
u
R(v) dv du
2
q(s) ds >
1
4
,
and (2.29) holds for some constant k2 ∈ (0, 1). Then every solution of (1.1) with α = 1 is
oscillatory.
3. Examples. In this section, we give two examples to illustrate the main results.
Example 3.1. Consider the differential equation(
t5x
′′′
(t)
)′
+ βtx(t) = 0, t ≥ 1. (3.1)
Here β > 0 is a constant. Let
α = 1, r(t) = t5, q(t) = βt, τ(t) = t.
Then, we have
R(t) =
1
4t4
,
∞∫
s
R(v) dv =
1
12s3
,
∞∫
s
∞∫
u
R(v) dv du =
1
24s2
.
Letting ρ(t) = θ(t) = 1, then we have that (2.1) and (2.2) are satisfied. By calculating, we
see that (2.3) and (2.4) hold when β > 12. Hence by Theorem 2.1, every solution of (3.1) is
oscillatory if β > 12. However, results of [22] cannot give this conclusion.
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334 C. ZHANG, T. LI, S. H. SAKER
Example 3.2. Consider the delay differential equation
(
etx
′′′
(t)
)′
+ 2
√
10 et+arcsin
√
10
10 x
(
t− arcsin
√
10
10
)
= 0, t ≥ 1. (3.2)
It is easy to see that every solution of (3.2) is oscillatory due to Theorem 2.1. One such soluti-
on is x(t) = et sin t. However, [22] (Corollary 2.1) implies that (3.2) may exist nonoscillatory
solutions x which satisfy limt→∞ x(t) = 0. Hence our results supplement and improve those
in [22].
4. Acknowledgements. 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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Received 17.12.11
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