Oscillation criteria for certain second order superlinear differential equations
In this paper, a class of second-order superlinear equations is studied. New oscillations criteria are established by using a general class of the parameter functions in the averaging techniques. We extend and improve the oscillation criteria of several authors. One of our results is based on the in...
Збережено в:
| Дата: | 2014 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут математики НАН України
2014
|
| Назва видання: | Нелінійні коливання |
| Онлайн доступ: | http://dspace.nbuv.gov.ua/handle/123456789/177106 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Oscillation criteria for certain second order superlinear differential equations / A. Tiryaki, R. Oinarov // Нелінійні коливання. — 2014. — Т. 17, № 4. — С. 533-545 — Бібліогр.: 21 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-177106 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-1771062021-02-11T01:28:14Z Oscillation criteria for certain second order superlinear differential equations Tiryaki, A. Oinarov, R. In this paper, a class of second-order superlinear equations is studied. New oscillations criteria are established by using a general class of the parameter functions in the averaging techniques. We extend and improve the oscillation criteria of several authors. One of our results is based on the information of the whole half-line and the other is based on the information on a sequence subintervals of the whole half-line. Вивчено клас суперлiнiйних рiвнянь другого порядку. Отримано новi критерiї осциляцiй за допомогою застосування загального класу параметричних функцiй у процедурi усереднення. Поширено та покращено критерiї осциляцiй, якi були отриманi деякими авторами. Один з отриманих результатiв базується на даних на всiй пiвпрямiй, iнший — на даних на послiдовностi пiдiнтервалiв всiєї пiвпрямої. 2014 Article Oscillation criteria for certain second order superlinear differential equations / A. Tiryaki, R. Oinarov // Нелінійні коливання. — 2014. — Т. 17, № 4. — С. 533-545 — Бібліогр.: 21 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177106 517.9 en Нелінійні коливання Інститут математики НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
In this paper, a class of second-order superlinear equations is studied. New oscillations criteria are established by using a general class of the parameter functions in the averaging techniques. We extend and improve the oscillation criteria of several authors. One of our results is based on the information of the whole half-line and the other is based on the information on a sequence subintervals of the whole half-line. |
| format |
Article |
| author |
Tiryaki, A. Oinarov, R. |
| spellingShingle |
Tiryaki, A. Oinarov, R. Oscillation criteria for certain second order superlinear differential equations Нелінійні коливання |
| author_facet |
Tiryaki, A. Oinarov, R. |
| author_sort |
Tiryaki, A. |
| title |
Oscillation criteria for certain second order superlinear differential equations |
| title_short |
Oscillation criteria for certain second order superlinear differential equations |
| title_full |
Oscillation criteria for certain second order superlinear differential equations |
| title_fullStr |
Oscillation criteria for certain second order superlinear differential equations |
| title_full_unstemmed |
Oscillation criteria for certain second order superlinear differential equations |
| title_sort |
oscillation criteria for certain second order superlinear differential equations |
| publisher |
Інститут математики НАН України |
| publishDate |
2014 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/177106 |
| citation_txt |
Oscillation criteria for certain second order superlinear differential equations / A. Tiryaki, R. Oinarov // Нелінійні коливання. — 2014. — Т. 17, № 4. — С. 533-545 — Бібліогр.: 21 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT tiryakia oscillationcriteriaforcertainsecondordersuperlineardifferentialequations AT oinarovr oscillationcriteriaforcertainsecondordersuperlineardifferentialequations |
| first_indexed |
2025-07-15T15:04:15Z |
| last_indexed |
2025-07-15T15:04:15Z |
| _version_ |
1837725758307434496 |
| fulltext |
UDC 517.9
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER
SUPERLINEAR DIFFERENTIAL EQUATIONS
КРИТЕРIЇ ОСЦИЛЯЦIЙ ДЛЯ ДЕЯКИХ СУПЕРЛIНIЙНИХ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ДРУГОГО ПОРЯДКУ
A. Tiryaki
Faculty of Arts and Sci., Izmir Univ.
35350 Uckuyular, Izmir, Turkey
e-mail: aydin.tiryaki@izmir.edu.tr
R. Oinarov
L. N. Gumilyov Eurasian Nat. Univ.
Astana, Kazakhstan
In this paper, a class of second-order superlinear equations is studied. New oscillations criteria are establi-
shed by using a general class of the parameter functions in the averaging techniques. We extend and
improve the oscillation criteria of several authors. One of our results is based on the information of the
whole half-line and the other is based on the information on a sequence subintervals of the whole half-line.
Вивчено клас суперлiнiйних рiвнянь другого порядку. Отримано новi критерiї осциляцiй за до-
помогою застосування загального класу параметричних функцiй у процедурi усереднення. По-
ширено та покращено критерiї осциляцiй, якi були отриманi деякими авторами. Один з отри-
маних результатiв базується на даних на всiй пiвпрямiй, iнший — на даних на послiдовностi
пiдiнтервалiв всiєї пiвпрямої.
1. Introduction. This paper is concerned with the second-order nonlinear differential equation
of superlinear type
(r(t)|y′|p−2y′)′ + p(t)|y′|p−2y′ + q(t)f(y) = 0, (1.1)
where r ∈ C(I,R+), p ∈ C(I,R), q ∈ C(I,R), f ∈ C1(R,R) such that yf(y) > 0 and
f ′(y) ≥ 0 for y 6= 0, I = [t0,∞], R+ = (0,∞), R = (−∞,∞) and p > 1 is a real number.
This equation can be considered as a generalization of the second-order equation with
damping
(r(t)y′)′ + p(t)y′ + q(t)f(y) = 0 (1.2)
which have been the subject of intensive studies in the recent years.
By a solution of (1.1), we mean a function y : [Ty,∞) → R, Ty ≥ t0, such that y and
r(t)|y′|p−2y′ are continuously differentiable and satisfy (1.1) for t ≥ Ty. A solution is said to be
global if it exists on the whole interval. On the other hand a solution, y of (1.1) which exists on
some interval [Ty,∞), Ty ≥ t0, is called proper if sup{|y(t)| : t ≥ T} 6= 0 for all T ≥ Ty.
c© A. Tiryaki, R. Oinarov, 2014
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4 533
534 A. TIRYAKI, R. OINAROV
The existence of proper (or global) solutions for the nonlinear second-order differential
equations was investigated in [4, 5]. In [4] Kiguradze and Chanturia established sufficient condi-
tions for all global solutions to be proper. So we shall suppose that (1.1) has proper solutions
and our attention will be restricted to these solutions only.
The oscillation of (1.1) is considered in the usual sense; that is, a solution y of (1.1) is said to
be oscillatory if it has arbitrary large zeros on [Ty,∞); otherwise it is said to be nonoscillatory.
Equation (1.1) is said to be oscillatory if all solutions are oscillatory.
Equation (1.1) is said to be superlinear if
0 <
∞∫
ε
du
f
1
p−1 (u)
,
−ε∫
−∞
du
(−f(u))
1
p−1
< ∞ (1.3)
for all ε > 0, and it is said to be sublinear if
0 <
ε∫
0+
du
f
1
p−1 (u)
,
0−∫
−ε
du
(−f(u))
1
p−1
< ∞ (1.4)
for all ε > 0.
Here we are interested in the oscillation of solutions of (1.1) when (1.3) is satisfied, including
the so-called Emden – Fowler equation
y′′ + q(t)|y|γsgn y = 0, (1.5)
where γ is a positive real number.
Since many physical problems are modeled by second-order nonlinear differential equati-
ons, the oscillatory and nonoscillatory behavior of solutions of such differential equations have
been considerably investigated by many authors [1 – 21] and references therein. Probably, the
most considered equation is Eq. (1.5), which attracted the attention for the first time around
the turn of the century with earlier theories concerning gaseous dynamics in astrophysics. This
equation also appears in the study of fluid mechanics, relativistic mechanics and nuclear physics.
Recently, Philos et al. [11, 12], Meng [9], Li and Yan [6], Yu [21], Lu and Meng [7], Ma-
nojlovic [8], and Tiryaki [16] have studied the oscillatory behavior of superlinear differential
equations. Some of these results involve the Philos type averaging technique. More recently
Qing-Hai Hau and Fang Lu [13] obtained some new oscillation criteria for the superlinear
equation (1.2) including (1.5), by allowing more general means along the lines given in [2, 3].
Our purpose here is to develop oscillation theory for (1.1), without any restriction on the
signs of p(t), q(t) and ρ′(t) including the well-known Emden – Fowler and half-linear equations,
where ρ′ is the differentiation of the test function used in the main section. We obtain some new
oscillation criteria extending and improving some earlier results. We believe that our approach
is simpler and more general than the recent results for (1.1) with p(t) ≡ 0 in [17]. We should
note that, in most of the oscillation results for (1.1) and its special cases (e.g.p(t) ≡ 0), the basic
condition on f is given by
f ′(y)
|f(y)|
p−2
p−1
≥ k > 0 (1.6)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER . . . 535
for y 6= 0 and k is constant.
For instance for p = 2, when we take f(y) = |y|γsgn y, γ > 1 or when we choose f as
f(y) = |y|γ [λ+ sin(ln(1 + |y|))]sgn y
condition (1.6) is not satisfied, hence most of the known methods in the literature for example
[1, 14, 15, 17] cannot be applied. To the best of our knowledge there is no oscillation result on
(1.1), except for the special case p = 2 [13].
We will present theorems in the main results which will be applicable for functions f like
the ones given above.
2. Preliminaries. In order to discuss our main results, we introduce the general mean given
in [2, 3, 17, 20] and we shall present some properties, which will be used in the proof of our
results.
Let D1 = {(t, s) : t0 ≤ s ≤ t} denote a subset of R2 and let D2 = {(t, s) : t0 ≤ s < t}.
Consider a kernel function k(t, s), which is defined, continuous, and sufficiently smooth on D1,
so that the following conditions are satisfied:
(K1) k(t, t) = 0 and k(t, s) > 0 for (t, s) ∈ D2,
(K2)
∂k
∂s
(t, s) ≤ 0 and λ(t, s) :=
−∂k
∂s (t, s)
(k(t, s))1/q
for (t, s) ∈ D2, where
1
p
+
1
q
= 1.
Let ρ ∈ C1(I) and ρ(t) > 0 on I. We take the integral operator Aρτ , which is defined in [20]
for the first time, in terms of k(t, s) and ρ(s) and as
Aρτ (h; t) =
t∫
τ
k(t, s)h(s)ρ(s)ds, t ≥ τ ≥ t0, (2.1)
where h ∈ C(I). It is easily seen thatAρτ is linear and positive, and in fact satisfies the following:
Aρτ (α1h1 + α2h2; t) = α1A
ρ
τ (h1; t) + α2A
ρ
τ (h2; t), Aρτ (h; t) ≥ 0, (2.2)
whenever h ≥ 0
Aρτ (h′; t) = −k(t, τ)h(τ)ρ(τ)−Aρτ
([
−λk−
1
p +
ρ′
ρ
]
h; t
)
≥
≥ −k(t, τ)h(τ)ρ(τ)−Aρτ
(∣∣∣∣−λk− 1
p +
ρ′
ρ
∣∣∣∣ |h|; t) .
Here h1, h2, h ∈ C(I) and α1, α2 are real numbers.
Let D3 = {H ∈ C1([a, b]) : H(t) 6= 0 for t ∈ I1 = [a, b] ⊂ I and H(a) = H(b) = 0}. We
take the integral operator Aba in terms of H ∈ D3 and ρ(t) as
Aba(h; t) =
b∫
a
H2(p−1)(t)h(t)ρ(t)dt, a ≤ t ≤ b, (2.3)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
536 A. TIRYAKI, R. OINAROV
where ρ and h are defined as before. As the operator Aρτ , Aba is also linear, positive and also
satisfies the following:
Aba(h
′, t) = −Aba
([
2(p− 1)
H ′
H
+
ρ′
ρ
]
h; t
)
≥ −Aba
(∣∣∣∣2(p− 1)
H ′
H
+
ρ′
ρ
∣∣∣∣ |h|; t) . (2.4)
Note that the first operator Aρτ is defined on the entire half-line I = [t0,∞). The second
operator Aba is defined on the subinterval I1 ⊂ D chosen according to our propose.
3. Main results. In this section we establish some oscillation criteria for (1.1) and its special
cases in the superlinear case. Additional assumptions on the function f will be imposed.
Suppose that
∞∫
ε
(
f ′(u)
f2(u)
) 1
p
du < ∞,
−∞∫
−ε
(
f ′(u)
f2(u)
) 1
p
du < ∞ (3.1)
for all ε > 0, and
min
{
inf
x>0
f ′(x)
f
p−2
p−1 (x)
G(x), inf
x<0
f ′(x)
(−f(x))
p−2
p−1
G(x)
}
> 0, (3.2)
where
G(x) =
∫ ∞
x
(
f ′(u)
f2(u)
1
p
)
du, x > 0,
∫ −∞
x
(
f ′(u)
f2(u)
) 1
p
du, x < 0.
We define the following functions that will be used in the proof of our results. Suppose that
there exists a function φ ∈ C1(I,R+) such that
ξ(t) := r(t)φ′(t)− p(t)φ(t),
η(t) :=
1
r(t)φ(t)
,
and
ν(t, T ) := η
1
p−1 (t)
t∫
T
η(s)
1
p−1ds
−1 , T ≥ t0.
Let us state the main results.
Theorem 3.1. Let conditions (1.3), (3.1), (3.2) and p > 1 and p 6= 2 hold. Assume that k(t, s)
satisfies conditions (K1) and (K2) and Aρτ is defined by (2.1). If there exist k(t, s), φ ∈ C(I,R+),
and ρ ∈ C1(I,R+) for any constant C > 0, such that
ξ(t) ≥ 0, ξ′(t) < 0
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER . . . 537
for t ≥ t0,
∞∫
T
(η(s))
1
p−1 ds = ∞, (3.3)
lim inf
t→∞
t∫
T
φ(s)q(s)ds > −∞, (3.4)
and
lim sup
t→∞
1
k(t, t0)
Aρt0
[
qφ−
(
p− 1
p
)p 1
p− 1
(
1
C
)(p−1)(∣∣∣∣λk− 1
p +
ρ′
ρ
∣∣∣∣+ ξη
)p
ν−(p−1); t
]
=∞,
(3.5)
then any solution y(t) of Eq. (1.1) such that y′(t) is bounded is oscillatory.
Proof. Suppose (1.1) possesses a nonoscillatory solution y(t) on [T,∞), T ≥ t0. Without
loss of generality, we assume that y(t) > 0 for every t ≥ T. We observe that the substitution
x = −y transforms (1.1) into the equation
(r(t)|x′|p−2x′)′ + p(t)|x′|p−2x′ + q(t)f∗(x) = 0,
where f∗(x) = −f(−y), y ∈ R. Since the function f∗(x) is subject to the same conditions as
on f(y), we can restrict our discussion to the case where the solution y(t) is positive on [T,∞).
Further assume that |y′(t)| ≤ L for some L > 0. Let w(t) be defined by
w(t) = φ(t)
r(t)|y′(t)|p−2y′(t)
f(y(t))
, t ≥ T, (3.6)
we have
w′(t) = −q(t)φ(t) + ξ(t)
|y′(t)|p−2y′(t)
f(y(t))
− f ′(y(t))|w(t)|q
(f(y(t)))
p−2
p−1 (φ(t)q(t))
1
p−1
. (3.7)
Let us consider the boundedness of the following term:
t∫
t1
ξ(s)
|y′(s)|p−2y′(s)
f(y(s))
ds, t ≥ t1 ≥ T. (3.8)
Applying the integral mean value theorem and using the boundedness of y′(t), there exists
t∗ ∈ [t1, t] for every t ≥ t1 such that
t∫
t1
ξ(s)
|y′(s)|p−2y′(s)
f(y(s))
ds ≤ ξ(t1)L
p−2
y(t)∫
y(t1)
du
f(u)
≤ ξ(t1)L
p−2
∞∫
y(t1)
du
f(u)
< K1, (3.9)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
538 A. TIRYAKI, R. OINAROV
where K1 > 0 is a constant. We consider the following three cases for the behavior of y′(t).
Case 1. Suppose that y′(t) > 0 for t ≥ t1 ≥ T then w(t) > 0 for t ≥ t1. Integrating both
sides of (3.7) from t1 to t, we have
w(t) =w(t1)−
t∫
t1
q(s)φ(s)ds+
t∫
t1
ξ(s)
|y′(s)|p−2y′(s)
f(y(s))
ds−
t∫
t1
f ′(y(s))|w(s)|qds
[[f(y(s))]p−2(φ(s)r(s))]
1
p−1
. (3.10)
Hence, for t ≥ t1, we obtain
w(t) ≤ N −
t∫
t1
q(s)φ(s)ds−
t∫
t1
f ′(y(s))|w(s)|qds
[(fp−2(y(s)))(φ(s)r(s))]
1
p−1
, (3.11)
where N = K1 + w(t0). Using (3.4) we see that
∞∫
t1
f ′y(s))|w(s)|qds
[(fp−2(y(s)))(φ(s)r(s))]
1
p−1
< ∞. (3.12)
There exists a constant M > 0 such that
∞∫
t1
f ′(y(s))|w(s)|qds
[(fp−2(y(s)))(φ(s)r(s))]
1
p−1
≤ M, t ≥ t1. (3.13)
By using Hölder’s inequality and (3.13), we get∣∣∣∣∣∣
t∫
t1
y′(s)
(
f ′(y(s))
f2(y(s))
) 1
p
ds
∣∣∣∣∣∣ =
∣∣∣∣∣∣
t∫
t1
(φ(s)r(s))
1
p (φ(s)r(s))
− 1
p y′(s)
(
f ′(y(s))
f2(y(s))
) 1
p
ds
∣∣∣∣∣∣ ≤
≤
∣∣∣∣∣∣∣
t∫
t1
r(s)φ(s)|y′(s)|pf ′(y(s))ds
f2(y(s))
1
p
t∫
t1
ds
(r(s)φ(s))
1
p−1
1
q
∣∣∣∣∣∣∣ ≤
≤ M
1
p
t∫
t1
η
1
p−1 (s)
1
q
. (3.14)
Applying the condition (3.2), we see that
(
f ′(y(t))
f
p−2
p−1 (y(t))
) 1
q
∞∫
y(t)
(
f ′(u)
f2(u)
) 1
p
du ≥ N1, t ≥ t1, (3.15)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER . . . 539
where N1 is a positive constant. Let
N2 =
∞∫
y(t1)
(
f ′(u)
f2(u)
) 1
p−1
du > 0.
Then, applying (3.15), we have
f ′(y(t))
f
p−2
p−1 (y(t))
≥ N q
1
N2−
y(t)∫
y(t1)
(
f ′(u)
f2(u)
) 1
p
du
−q
=N q
1
N2 −
t∫
t1
y′(s)
(
f ′(y(s))
f2(y(s))
) 1
p
ds
−q ≥
≥ N q
1
N2 +
∣∣∣∣∣∣
t∫
t1
y′(s)
(
f ′(y(s))
f2(y(s))
) 1
p
ds
∣∣∣∣∣∣
−q .
A use of (3.14) in the above inequality leads to
f ′(y(t))
f
p−2
p−1 (y(t))
≥ N q
1[
N2 +M
1
p
(∫ t
t1
η
1
p−1 (s)ds
) 1
q
]q .
Hence, there exists a constant C > 0 and t2 > t1 such that
f ′(y(t))
f
p−2
p−1 (y(t))
≥ C[∫ t
t1
η
1
p−1 (s)ds
] (3.16)
for all t ≥ t1, where C is a positive constant which depends on N1, N2, M and p.
Now from (3.16), Eq. (3.7) gives
w′(t) ≤ −q(t)φ(t) + ξ(t)η(t)|w(t)| − Cν(t, t1)|w(t)|q. (3.17)
Applying the operator Aρτ to (3.17) and using (2.2), we obtain
Aρτ (qφ; t) ≤ k(t, τ)w(τ)ρ(τ) +Aρτ
((∣∣∣∣λk− 1
p +
ρ′
ρ
∣∣∣∣+ ξη
)
|w| − Cv(·, t1)|w|q; t
)
. (3.18)
By using the inequality
Du− Euq ≤
(
p− 1
p
)p 1
p− 1
DpE−(p−1), D ≥ 0, E ≥ 0, u ≥ 0, (3.19)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
540 A. TIRYAKI, R. OINAROV
which can be easily obtained by using the extremum of one variable function, we get
Aρτ (qφ, t) ≤ k(t, τ)w(τ)ρ(τ) +Aρτ
((
p− 1
p
)p 1
p− 1
(∣∣∣∣λk−1/p +
ρ′
ρ
∣∣∣∣+ ξη
)p
×
× (Cν(·, t1))−(p−1) ; t
)
. (3.20)
If we set τ = t0 and divide (3.20) through by k(t, t0), then we have
1
k(t, t0)
Aρt0
(
qφ−
(
p− 1
p
)p C−(p−1)
p− 1
(∣∣∣∣λk−1/p +
ρ′
ρ
∣∣∣∣+ ξη
)p
ν(·, t1)−(p−1); t
)
≤ ρ(t0)w(t0).
(3.21)
Taking lim sup in (3.21) as t → ∞, condition (3.5) gives the desired contradiction in (3.21).
Thus, the existence of a nonoscillatory solution y(t) is ruled out, so Eq. (1.1) is oscillatory.
Case 2. Suppose that y′(t) is oscillatory. Then, there exists a sequence {tm}m=1,2,... such that
limm→∞ tm = ∞ and y′(tm) = 0, m = 1, 2, . . . . Choose m such that tm ≥ t0. Without loss of
generality we assume that y′(t) > 0 for t ∈ (tm, tm+1). Further, in view of (3.4),
tm+1∫
tm
f ′(y(s))|w(s)|q
[fp−2(y(s))(r(s)φ(s))]1/p−1
ds ≤ N −
tm+1∫
tm
q(s)φ(s)ds. (3.22)
There is an infinite number of m’s such that y′(t) > 0 for t ∈ (tm, tm+1). Summing all these
inequalities (3.22) and using (3.4), we have
∞∫
tm1
f ′(y(s))|w(s)|q
[fp−2(y(s))(r(s)φ(s))]1/p−1
ds < ∞.
The rest of the proof is as in Case 1.
Case 3. Suppose that y′(t) < 0 for t ≥ t1 ≥ t0. Meanwhile if the inequality (3.12) holds,
then we can have a similar contradiction as in Case 1. If the integration in (3.12) is divergent,
we obtain the following inequality from (3.4) and (3.10):
N3 +
t∫
t1
f ′(y(s))|w(s)|q
f
p−2
p−1 (y(s))
η(s)
1
p−1 ds ≤ −w(t), t ≥ t1, (3.23)
where N3 is a constant. By choosing t2 ≥ t1, we can get
N4 = N3 +
t2∫
t1
η
1
p−1 (s)
f ′(y(s))|w(s)|q
f
p−2
p−1 (y(s))
ds > 1. (3.24)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER . . . 541
From (3.23) and (3.10), we obtain
w(t) < 0, t ≥ t2.
Then using (3.23), we find
η
1
p−1 (t)|w(t)|qf ′(y(t))
f
p−2
p−1 (y(t))
N3 +
∫ t
t1
η
1
p−1 (s)|w(s)|qf ′(y(s))
f
p−2
p−1 (y(s))
ds
≥ −y
′(t)f ′(y(t))
f(y(t))
, t ≥ t2.
Integrating the above inequality, we get
ln
N3 +
t∫
t1
η
1
p−1 (s)|w(s)|qf ′(y(s))
f
p−2
p−1 (y(s))
ds
≥ ln
f(y(t2))
f(y(t))
.
Hence
N3 +
t∫
t1
η
1
p−1 (s)|w(s)|qf ′(y(s))
f
p−2
p−1 (y(s))
ds ≥ f(y(t2))
f(y(t))
, t ≥ t2. (3.25)
Applying (3.23) and (3.24), we have
y(t) ≤ y(t2)− f
1
p−1 (y(t2))
t∫
t2
η
1
p−1 (s)ds.
In this inequality, the right-hand side tends to −∞ as t → ∞ by (3.3). However, the left-
hand side is positive, which is a contradiction.
Theorem 3.1 is proved.
For p = 2, we do not require the boundedness condition for y′(t) and we have the following
result.
Theorem 3.2. Let conditions (1.3), (3.1) and (3.2) hold with p = 2. Assume that k(t, s) sati-
sfies conditions (K1) and (K2) and Aρτ is defined by (2.1). If there exist k(t, s), φ ∈ C(I,R+) and
ρ ∈ C1(I,R+) for any constant C > 0, such that
ξ(t) ≥ 0, ξ′(t) < 0
for t ≥ t0,
∞∫
T
η(s)ds = ∞, (3.26)
lim inf
t→∞
t∫
T
φ(s)q(s)ds > −∞, (3.27)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
542 A. TIRYAKI, R. OINAROV
and
lim sup
t→∞
1
k(t, t0)
Aρt0
[
qφ− 1
4C
(∣∣∣∣λk− 1
2 +
ρ′
ρ
∣∣∣∣+ ξη
)2
ν−1; t
]
= ∞, (3.28)
then any solution y(t) of Eq. (1.2) is oscillatory.
Proof. Let p = 2. As in [13], by using conditions (1.3) and (3.6) we have
t∫
t1
ξ(s)
y′(s)
f(y(s))
ds ≤ ξ(t1)
∞∫
y(t1)
du
f(u)
= K2, (3.29)
where K2 > 0 is a constant. Replacing K1 with K2 in the proof of Theorem 3.1, the rest of the
proof stays the same.
A close look at the proof of Theorem 3.1 reveals that condition (3.5) may be replaced by
the conditions
lim sup
t→∞
1
k(t, t0)
Aρt0(qφ) = ∞ (3.30)
and
lim sup
t→∞
1
k(t, t0)
Aρt0
((∣∣∣∣λk− 1
p +
ρ′
ρ
∣∣∣∣+ ξη
)p
ν−(p−1); t
)
< ∞. (3.31)
This leads to the following result.
Corollary 3.1. Let the conditions of Theorem 3.1 be satisfied except that condition (3.5) is
replaced by (3.30) and (3.31). Then any solution y(t) of Eq. (1.1) such that y′(t) is bounded is
oscillatory.
By making the same replacement in Theorem 3.2, we get similar results.
Note that the above oscillation criteria as well as most of the known results in the literature
require information of (1.1) on the entire half-line I.Now, motivated by [2, 3, 13], we present the
following oscillation criteria for equation (1.1) which depend on the behavior of the coefficients
only on a sequence of subintervals I1 of I.
Theorem 3.3. Let conditions (1.3), (3.1), (3.2) and p > 1 and p 6= 2 hold. Assume that for
any T ≥ t0, there exist a, b satisfying T ≤ a < b such that D3 and Aba be defined as before. If
there exist H ∈ D3, φ ∈ C(I,R+) and ρ ∈ C1(I,R+) for any constant C such that (3.1), (3.3)
and (3.4) hold and
Aba(qφ, t) >
(
p− 1
p
)p 1
p− 1
(
1
C
)p−1
Aba
((∣∣∣∣2(p− 1)
H ′
H
+
ρ′
ρ
∣∣∣∣+ ξη
)p
ν−(p−1); t
)
, (3.32)
then any solution y(t) of Eq. (1.1) such that y′(t) is bounded is oscillatory.
Proof. Again let y(t) be nonoscillatory solution of (1.1), say y(t) > 0 for t ≥ t0. Further
assume that |y′(t)| ≤ L for some L > 0. We shall discuss three cases for the behavior of y′(t),
namely y′(t) > 0, y′(t) is oscillatory and y′(t) < 0.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER . . . 543
Case 1. y′(t) > 0 for t ≥ t1 for some t1 ≥ t0. We proceed as in the proof of Theorem 3.1,
by the assumption, we can choose a, b ≥ t1 and a < b, i.e., for a given T ≥ t1, there exist
a, b satisfying T ≤ a < b. Applying the operator Aba to (3.17) using (2.4) and the fact that
H(a) = H(b) = 0, we obtain
Aba(qφ; t) ≤ Aba
((∣∣∣∣2(p− 1)
H ′
H
+
ρ′
ρ
∣∣∣∣+ ξη
)
|w| − Cν|w|q; t
)
. (3.33)
By using the inequality (3.19),
Aba(qφ; t) ≤ Aba
(
p− 1
p
)p 1
p− 1
((∣∣∣∣2(p− 1)
H ′
H
+
ρ′
ρ
)
+ ξη
)p
(Cν)−(p−1); t
)
which contradicts the assumption (3.32).
Theorem 3.3 is proved.
Proof of the other cases is similar to the cases given in the proof of the Theorem 3.1. Thus,
the existence of the nonoscillatory solution y(t) is ruled out, so Eq. (1.1) is oscillatory.
For the sake of completeness, we should note that the following result is given as Theorem 2.1
in [13]:
Theorem 3.4. Let conditions (1.3), (3.1) and (3.2) hold with p = 2. Assume that for any
T ≥ t0, there exist a, b satisfying T ≤ a < b such that D3 and Aba be defined as before. If there
exist H ∈ D3, φ ∈ C(I, R+) and ρ ∈ C1(I,R+) for any constant C such that (3.2), (3.26) and
(3.27) hold and
Aba(qφ, t) >
1
4C
Aba
((∣∣∣∣2H ′H +
ρ′
ρ
∣∣∣∣+ ξη
)2
ν−1; t
)
, (3.34)
then any solution y(t) of Eq. (1.2) is oscillatory.
Remarks. 3.1. In the above results, the conditions that the integral
∫
dt
r(t)
is either conver-
gent or divergent, and the dampeing coefficient p(t) is a “small” function are not necessary.
Therefore, the restraint for r(t) and p(t) is relaxed. Also there is no sign conditions on p(t) and
q(t).
3.2. When p = 2, it is not necessary to assume that y′(t) is bounded. For p > 1 and p 6= 2,
we require a boundedness condition for the derivative of the solution. However the conditions
on f(y) are relaxed. Removing the boundedness condition still remains as an open problem
and will be interesting.
3.3. Since the conditions on f(y) are relaxed, Theorem 3.1 improves some results in [17]
related to the special case of Eq. (1.1) when p(t) ≡ 0.
3.4. Note that Theorem 3 given in [16], which also uses the averaging technique, contain
different sufficient conditions than Theorem 3.1 with p = 2 and p(t) ≡ 0.
4. An example. Let us consider the following second-order superlinear equation with dam-
ping
(tλ−1|y′|p−2y′)′ − tλ−2|y′|p−2y′ +Ktλ|y|γsgn y = 0, t ≥ 1. (4.1)
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
544 A. TIRYAKI, R. OINAROV
Let the constants γ, λ and K satisfy the following conditions depending on the value of p:
For 1 < p < 2, let
(1) p =
2k
2k − 1
, where k is a positive integer,
(2) 0 < λ < p− 1 < γ,
(3) K =
(
p− 1
p
)p 1
p− 1
(
1
C
)p−1
(2p+ λ+ 1)p
Γ
(
p−1
2
)
Γ
(
2p−1
2
) ;
for p = 2, let
(1) 0 < λ ≤ 1 < γ,
(2) K =
(5 + λ)2
C
;
for p > 2, let
(1) p be an integer,
(2) 0 < λ ≤ 1 and p− 1 < γ,
(3) K =
(
p− 1
p
)p 1
p− 1
(
1
C
)p−1
(2p+ λ+ 1)p
Γ(p)
Γ
(p
2
) .
Let H(t) = sin t, φ(t) = t and ρ(t) = t−(λ+1). For any T ≥ 1, choose n sufficiently large so
that nπ = 2kπ > T and set a = 2kπ, b = (2k + 1)π. It is easy to verify that
Aba(φq, t) =
b∫
a
H2(p−1)(t)φ(t)q(t)ρ(t)dt =
(2k+1)π∫
2kπ
Ksin2(p−1)tdt =
= K
Γ
(
1
2
)
Γ
(
2p−1
2
)
Γ(p)
(
p− 1
p
)p 1
p− 1
(
1
C
)p−1
×
×Aba
((∣∣∣∣2(p− 1)
H ′
H
+
ρ′
ρ
∣∣∣∣+ ξη
)p
(ν)−(p−1); t
)
≤
≤
(
p− 1
p
)p 1
p− 1
(
1
C
)p−1
(2p+ λ+ 1)p
(2k+1)π∫
2kπ
sin(p−2) tdt =
=
(
p− 1
p
)p 1
p− 1
(
1
C
)p−1
(2p+ λ+ 1)p
Γ
(
1
2
)
Γ
(
p−1
2
)
Γ
(p
2
) . (4.2)
Now, we have three cases depending on the value of p:
When p > 1 and p 6= 2 choosing suitable K value in (4.2), the conditions in Theorem 3.3
are satisfied, hence any solution y(t) of Eq. (4.1) such that y′(t) is bounded, is oscillatory.
When p = 2, choosing suitable K value in (4.2), the conditions in Theorem 3.4 are satisfied,
hence any solution y(t) of Eq. (4.1) is oscillatory. This case is already given in [13].
1. Agarwal R. P., Grace S. R., O’Regan D. Oscillation theory for second order linear, half-linear, superlinear
dynamic equations. — London: Kluwer Acad. Publ., 2002.
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
OSCILLATION CRITERIA FOR CERTAIN SECOND-ORDER . . . 545
2. Çakmak D., Tiryaki A. Oscillation criteria for certain forced second order nonlinear differential equations //
Appl. Math. Lett. — 2004. — 17. — P. 275 – 279.
3. Çakmak D., Tiryaki A. Certain forced second order nonlinear differential equations with delayed argument
// Comput. Math. and Appl. — 2005. — 49. — P. 1647 – 1653.
4. Kiguradze I. T., Chanturia A. T. Asymptotic properties of solutions of the nonautonomous ordinary differenti-
al equations. — Moscow: Nauka, 1990 (in Russian).
5. Kusano T., Kitano M. On a class of second order quasilinear ordinary differential equations // Hiroshima
Math. J. — 1995. — 25. — P. 321 – 355.
6. Li W. T., Yan J. R. An oscillation criteria for second order superlinear differential equations // Indian J. Pure
and Appl. Math. — 1997. — 28. — P. 735 – 740.
7. Lu F., Meng F. W. Oscillation theorems for superlinear second order damped differential equations // Appl.
Math. and Comput. — 2007. — 189. — P. 796 – 804.
8. Manojlovic J. V. Integral averages and oscillation of second-order nonlinear differential equations // Comput.
Math. and Appl. — 2001. — 41, № 12. — P. 1521 – 1534.
9. Meng F. W. An oscillation theorem for second order superlinear differential equations // Indian J. Pure and
Appl. Math. — 1996. — 27. — P. 651 – 658.
10. Ouyang Z., Zhong J., Zou S. Oscillation criteria for a class of second-order nonlinear differential equations
with damping term // Abst. Appl. Anal. — 2009. — Article ID 8970548.
11. Philos Ch. G. Oscillation criteria for second order superlinear differential equations // Can. J. Math. Anal. —
1989. — 41. — P. 321 – 340.
12. Philos Ch. G., Purnaras I. K. Oscillations in superlinear differential equations of second order // J. Math.
Anal. and Appl. — 1992. — 165. — P. 1 – 11.
13. Qing-Hai Hao, Fang Lu. Oscillation theorem for superlinear second order damped differential equations //
Appl. Math. and Comput. — 2011. — 217. — P. 7126 – 7131.
14. Temtek P., Tiryaki A. Oscillation criteria for a certain second order nonlinear perturbed differential equations
// J. Inequal. Appl. — 2013. — 2013, № 524. — P. 1 – 12.
15. Tiryaki A., BaşcıY. Oscillation theorems for certain even order nonlinear damped differential equations //
Rocky Mountain J. Math. — 2008. — 38. — P. 1011 – 1035.
16. Tiryaki A., Çakmak D. Integral averages and oscillation criteria of second-order nonlinear differential equati-
ons // Comput. Math. and Appl. — 2004. — 47. — P. 1495 – 1506.
17. Tiryaki A., Çakmak D., Ayanlar B. On the oscillation certain second order nonlinear differential equations //
J. Math. Anal. and Appl. — 2003. — 281. — P. 565 – 574.
18. Tiryaki A., Zafer A. Interval oscillation of a generaal class of second-order nonlinear differential equations
with nonlinear damping // Nonlinear Anal. Theory, Meth. and Appl. — 2005. — 60. — P. 49 – 63.
19. Wong J. S. W. An oscillation criteria for second order nonlinear differential equations // Proc. Amer. Math.
Soc. — 1986. — 98. — P. 109 – 112.
20. Wong J. S. W. On Kamenev-type oscillation theorems for second order differential equations with damping //
J. Math. Anal. and Appl. — 2001. — 258. — P. 244 – 257.
21. Yu Y. H. Oscillation criteria for second order nonlinear differential equations with damping // Acta Math. and
Appl. — 1993. — 16. — P. 433 – 441 (in Chinese).
Received 30.10.13
ISSN 1562-3076. Нелiнiйнi коливання, 2014, т . 17, N◦ 4
|