Oscillation criteria for nonlinear second-order differential equations with damping
Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x?+?p?(?t?)?x?+?q?(?t?)?f?(?x?) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng.
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| author | Çakmak, D. Чакмак, Д. |
| author_facet | Çakmak, D. Чакмак, Д. |
| author_sort | Çakmak, D. |
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| description | Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x?+?p?(?t?)?x?+?q?(?t?)?f?(?x?) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. |
| first_indexed | 2026-03-24T02:37:50Z |
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UDC 517.9
D. Çakmak (Gazi Univ., Turkey)
OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER
DIFFERENTIAL EQUATIONS WITH DAMPING
OSCYLQCIJNI KRYTERI}
DLQ NELINIJNYX DYFERENCIAL|NYX RIVNQN|
DRUHOHO PORQDKU IZ ZATUXANNQM
Some new oscillation criteria are given for general nonlinear second order ordinary differential equations
with damping of the form x′′ + p ( t ) x′ + q ( t ) f ( x ) = 0, where f is with or without monotonicity. Our
results generalize and extend some earlier results of Deng.
Navedeno deqki novi oscylqcijni kryteri] dlq zahal\nyx nelinijnyx zvyçajnyx dyferencial\nyx
rivnqn\ druhoho porqdku iz zatuxannqm vyhlqdu x′′ + p ( t ) x′ + q ( t ) f ( x ) = 0, de funkciq f abo mo-
notonna, abo nemonotonna. Navedeni rezul\taty uzahal\nggt\ ta rozßyrggt\ deqki rezul\ta-
ty, otrymani raniße Denhom.
1. Introduction. Consider the second order linear differential equation with damped
term
x′′ + p ( t ) x′ + q ( t ) x = 0, t ≥ t0 > 0, (1.1)
and the more general nonlinear equation
x′′ + p ( t ) x′ + q ( t ) f ( x ) = 0, t ≥ t0 > 0, (1.2)
where p ∈ C1
( [ t0 , ∞ ), R ), q ∈ C ( [ t0 , ∞ ), R ), f ∈ C ( R, R ) is to be specified in the
subsequent text, and x f ( x ) > 0 whenever x ≠ 0.
As usual, a nontrivial solution of (1.1) or (1.2) is called oscillatory if it has
arbitrarily large zeros; otherwise, it is said to be nonoscillatory. Equation (1.1) or (1.2)
is said to be oscillatory if all of its solutions are oscillatory.
In the last decades, there has been an increasing interest in obtaining sufficient
conditions for the oscillation and/or nonoscillation of solutions for different classes of
second order differential equations [1 – 9]. In the absence of damping, many results
have been obtained for particular cases of (1.1), such as the linear equation
x′′ + q ( t ) x = 0 (1.3)
and the quasilinear equation
′ ′( )′ + ( )− −x x q t x xγ γ1 1 = 0, (1.4)
where γ > 0 is a constant.
It is well known that Hille [4] studied the linear equation (1.3) and obtained that
equation (1.3) is oscillatory if
q s ds
t
t
( ) ≥ +∞
∫ 1
4
δ
, (1.5)
where δ is any small positive number. If introducing the transformation u =
= xe
p s ds
t− ( )∫1
2 for the damping equation (1.1), we have
′′ + ( ) − ( ) − ′( )
u q t
p t p t
u
2
4 2
= 0, (1.6)
which has the same form as equation (1.3). Hence applying condition (1.5) we can
© D. ÇAKMAK, 2008
694 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER … 695
easily know that equation (1.6) is oscillatory if
q s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥ +∞
∫
2
4 2
1
4
δ
. (1.7)
In an early paper [5], Chunchao Huang has established interesting oscillation and
nonoscillation criteria for the equation (1.3) with q ∈ C [ 0, ∞ ) and q ( t ) ≥ 0, where
conditions about the integrals of q ( t ) on every interval [ 2n
t0 , 2n
+
1
t0 ], n = 1, 2, … ,
for some fixed t0 > 0 are used in the results. Since that time, many authors have also
investigated the oscillatory and nonoscillatory behavior of equation (1.3) by using
Huang’s technique, such as papers [3, 8].
Recently, by using the similar method in the proof of [6] (Lemma 3), Deng [2]
presented the following result for the oscillation of equation (1.3) with q ∈ L1
[ t0 , ∞ ).
Theorem A. If for large t ∈ R,
q s ds
t
t
( ) ≥
∞
∫ α0 , (1.8)
where α0 > 1 / 4, then equation (1.3) is oscillatory.
More recently, inspired by the recent work of Deng [2], Yang [9] obtain following
oscillation result.
Theorem B. If q ∈ L1
[ t0 , ∞ ) and for large t > t0 ,
t q s ds
t
γ ( )
∞
∫ ≥ α0 , (1.9)
where α0 >
γ
γ
γ
γ( + ) +1 1 , then equation (1.4) is oscillatory.
It is obvious that (1.8) or (1.9) is condition on the integral of q ( s ) in [ t, ∞ ) for
arbitrarily large value of t, while the conditions of Huang [5] and Elbert [3] concern
the integral of q ( s ) in [ 2n
t0 , 2n
+
1
t0 ] for every n ∈ N, and the condition of Yang [8]
concern the integral of q ( s ) in [ t0 / ε, t0 / εn
+
1
] for every n ∈ N and 0 < ε < 1.
Therefore, they are different kinds of condition.
Motivated by the idea of Deng [2], in this paper we are study the more general
equation (1.2), and obtain oscillation criteria which contain Theorems A and B as a
special case, and establish oscillation criteria for equation (1.2) when f ( x ) = | x | λ sgn x
with λ > 1 or 0 < λ < 1.
This paper is organized as follows. In Section 2, we shall present oscillation criteria
for equation (1.2) when q ( t ) ≥ 0 and the function f ( x ) is not monotonous. Section 3
contains also oscillation criteria for equation (1.2) when q ( t ) changes its sign and the
function f ( x ) is monotonous.
2. Oscillation results for f (((( x )))) without monotonicity. In this section, we
consider the oscillation of equation (1.2) when q ( t ) ≥ 0 and the function f ( x ) is not
monotonous.
Theorem 2.1. Let q ( t ) ≥ 0 and
f x
x
( )
≥ M > 0 for x ≠ 0, where M is a
constant. If for large t ∈ R,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
696 D. ÇAKMAK
p ( t ) = O ( 1 ) and Mq s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
, (2.1)
where α0 > 1 / 4, then equation (1.2) is oscillatory.
Proof. Let x( t ) be a nonoscillatory solution of equation (1.2), which, without loss
of generality, can be assumed to be x ( t ) > 0, f ( x ( t ) ) > 0 for t ≥ t0 > 0. By (2.1), it is
easy to see that there exists t1 ≥ t0 such that
Mq s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
for t ≥ t1 ,
which yields that there exists an integer n ( t ) ≥ t such that
Mq s
p s p s
ds
t
t
t
( ) − ( ) − ′( )
≥
′
∫
2
1
4 2
α
and
α α α1
2
1
2 2
t t t
−
′
≥ for t′ ≥ n ( t ),
(2.2)
where α0 ≥ α1 ≥ α > 1 / 4.
Define v ( t ) = x′ ( t ) / x ( t ) for t ≥ t0 . By equation (1.2), v ( t ) satisfies the equation
′ + + + ( )
v v v2 p q
f x
x
= 0. (2.3)
Because
f x
x
( )
≥ M > 0 and q ≥ 0, (2.3) can be rewritten as
′ + + +v v v2 p Mq ≤ 0.
Let w ( t ) = v ( t ) +
p t( )
2
. Now, we have
′ + + − − ′
w w Mq
p p2
2
4 2
≤ 0. (2.4)
Integrating (2.4) from t to t ′, by (2.2) we obtain
w t w t w s ds Mq s
p s p s
ds
t
t
t
t
( ) − ( ′) ≥ ( ) + ( ) − ( ) − ′( )
′ ′
∫ ∫2
2
4 2
≥ 0 (2.5)
for t′ ≥ n ( t ) and t ≥ t1 .
If there exists t2 ≥ t1 such that w ( t2 ) < 0, then from (2.5), w ( t ) < 0 for t ≥ n ( t2 ).
Therefore, w ( t ) is either eventually positive or eventually negative.
If w ( t ) is eventually negative, then there exists t3 ≥ t1 such that w ( t ) < 0 for t ≥
≥ t3 and
w t w s ds Mq s
p s p s
ds
t
t
t
t
( ) ≥ ( ) + ( ) − ( ) − ′( )
∫ ∫2
2
3 3
4 2
for t ≥ n ( t3 ) which, with (2.2), yields
w t w s ds
t t t
t
t
( ) ≥ ( ) + ≥ ≥∫ 2 1
3
1
3
1
3
α α α
for t ≥ n ( t3 ). (2.6)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER … 697
Substituting this into (2.6), we obtain
w t
s
ds
n t t
n t
t
( ) ≥ +
( )
≥ +
( )
∫α α τ τ
1
2
2
1
3
0
2
01
3
for t ≥ n ( n ( t3 ) ), (2.7)
where τ0 = α > 1 / 4.
If w ( t ) is eventually positive, then there exists t4 ≥ t1 such that w ( t ) > 0 for t ≥
≥ t4 and from (2.2) and (2.5), we have
w t w s ds Mq s
p s p s
ds
t
t
t
t
t
( ) ≥ ( ) + ( ) − ( ) − ′( )
≥
′ ′
∫ ∫2
2
1
4 2
α
(2.8)
for t′ ≥ n ( t ) and t ≥ t4 .
Using similar methods to those in the proof of (2.7), we get
w t
t
( ) ≥ +τ τ0
2
0 for t ≥ n ( t4 ). (2.9)
Setting τi = τ τi − +1
2
0, i = 1, 2, … , and taking t5 = max { n ( n ( t3 ) ), n ( t4 ) }, we
obtain
w t
t
( ) ≥ τ1 for t ≥ t5
from (2.7) and (2.9). By induction, from (2.6) and (2.8), we can prove that
w t
t
i( ) ≥ τ
for t ≥ t5
, i = 1, 2, … .
It is easy to see that
w t( ) → ∞ for t ≥ t5
.
Using p ( t ) = O ( 1 ) for large t ∈ R, we obtain v( )t → ∞ . So, this contradiction
completes the proof.
Thus, f ( x ) acts a linear function for sufficiently large x, we have the following
result for the equation (1.2).
Corollary 2.1. Let f ( x ) behaves like M x for sufficiently large x. If for large
t ∈ R,
Mq s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
, (2.10)
where α0 > 1 / 4, then equation (1.2) is oscillatory.
Remark 2.1. If α 0 > 1 / 4, then α0 ≥
1
4
+ δ
where δ is any small positive
number. Thus, condition (2.10) with M = 1 reduces to (1.7).
The following theorem is concerned with the oscillatory behavior of a special case
of equation (1.2), namely, the equation
x′′ + p ( t ) x′ + q ( t ) | x | λ sgn x = 0, (2.11)
where λ > 0 is a real constant.
Theorem 2.2. Let q ( t ) ≥ 0. If for large t ∈ R and every constant c > 0,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
698 D. ÇAKMAK
p ( t ) = O ( 1 ) and cq s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
, (2.12)
where α0 > 1 / 4, then:
(i) every unbounded solution of equation (2.11) with λ > 1 is oscillatory,
(ii) every bounded solution of equation (2.11) with 0 < λ < 1 is oscillatory.
Proof. Without loss of generality, we assume that x ( t ) is a nonoscillatory
solution of equation (2.11) such that x ( t ) > 0 for t ≥ t0 > 0. By (2.12), there exists
t1 ≥ t0 such that
cq s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
for t ≥ t1
so there exists an integer n ( t ) ≥ t such that
cq s
p s p s
ds
t
t
t
( ) − ( ) − ′( )
≥
′
∫
2
1
4 2
α
and
α α α1
2
1
2 2
t t t
−
′
≥ for t′ ≥ n ( t ),
where α0 ≥ α1 ≥ α > 1 / 4.
Define v ( t ) = x′ ( t ) / x ( t ) for t ≥ t0 . By equation (2.11), v ( t ) satisfies the
equation
′ + + + −v v v2 1p qxλ = 0. (2.13)
Next, we consider the following two cases:
(i) If x ( t ) is an unbounded nonoscillatory solution of equation (2.11) with λ > 1
for t ≥ t0 , then there exist a constant k1 > 0 and t2 ≥ t1 ≥ t0 such that x ( t ) ≥ k1 for
t ≥ t2 . Therefore
x t kλ λ− −( ) ≥1
1
1 = c1 for t ≥ t2 , (2.14)
where c1 is a constant. Using (2.14) and q ( t ) ≥ 0 in (2.13), and proceeding as in the
proof of Theorem 2.1, we arrive at the desired contradiction.
(ii) If x ( t ) is a bounded nonoscillatory solution of equation (2.11) with 0 < λ < 1
for t ≥ t0 , then there exist a constant k2 > 0 and t2 ≥ t1 ≥ t0 such that x ( t ) ≤ k2 for
t ≥ t2 . Therefore
x t kλ λ− −( ) ≥1
2
1 = c2 for t ≥ t2 ,
where c2 is a constant. The rest of the proof is similar to that in the previous case and
hence is omitted.
3. Oscillation results for f (((( x )))) with monotonicity. In this section, we establish
the oscillation of equation (1.2) under the assumption that q ( t ) changes its sign and
the function f ( x ) is monotonous.
Theorem 3.1. Assume that f ∈ C1
( R, R ) and f ′ ( x ) ≥ K > 0 for all x ∈ R,
where K is a constant. If for large t ∈ R,
p ( t ) = O ( 1 ) and q s
p s
K
p s
K
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
, (3.1)
where α0 > 1 / 4, then equation (1.2) is oscillatory.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
OSCILLATION CRITERIA FOR NONLINEAR SECOND ORDER … 699
Proof. Suppose that x ( t ) is a nonoscillatory solution of equation (1.2), say,
x ( t ) > 0 when t ≥ t0 > 0 for some t0 depending on the solution x ( t ). By condition
(3.1), there exists t1 ≥ t0 such that
q s
p s
K
p s
K
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2
α
for t ≥ t1
,
which yields that there exists an integer n ( t ) ≥ t such that
q s
p s
K
p s
K
ds
t
t
t
( ) − ( ) − ′( )
≥
′
∫
2
1
4 2
α
and
K
t
K
t t
α α α1
2
1
2 2
−
′
≥ for t′ ≥ n ( t ),
where α0 ≥ α1 ≥ α > 1 / 4.
Define v ( t ) = x′ ( t ) / f ( x ( t ) ) for t ≥ t0 . By equation (1.2), v ( t ) satisfies the
equation
′ + ′( ) + +v v vf x p q2 = 0. (3.2)
Because f ′ ( x ) ≥ K > 0, and setting w ( t ) = v ( t ) +
p t
K
( )
2
, (3.2) can be rewritten as
′ + + − − ′
w Kw q
p
K
p
K
2
2
4 2
≤ 0.
The rest of the proof is similar to that of Theorem 2.1, and is omitted.
Now, by combining some ingredients of the proofs of Theorem 2.2 and of Theorem
3.1, we give the following theorem, whose proof is similar to that of Theorem 2.2, for
the equation (2.11).
Theorem 3.2. If for large t ∈ R and every constant β > 0,
p ( t ) = O ( 1 ) and q s
p s p s
ds
t
t
( ) − ( ) − ′( )
≥
∞
∫
2
0
4 2β β
α
,
where α0 > 1 / 4, then:
(i) every unbounded solution of equation (2.11) with λ > 1 is oscillatory,
(ii) every bounded solution of equation (2.11) with 0 < λ < 1 is oscillatory.
Remark 3.1. Theorems 2.1 and 3.1 extend and improve Theorem A for the
nonlinear equation (1.2). In addition to this, they are true for the linear equation (1.1).
Also note that when f ( x ) ≡ x, it is not necessary to assume q ( t ) ≥ 0 in Theorem 2.1.
So, if p ( t ) ≡ 0 then Theorems 2.1 and 3.1 reduce to Theorem A, Theorem B with γ =
= 1 and the well known Hille’s result.
Remark 3.2. If we compare Theorem 2.2 or Theorem 3.2 with Theorem 2.2 given
in [1], it is easy to see that these results include different types of sufficient conditions
for the oscillation.
Finally, we give an example to illustrate the efficiency of our results. The example
is not covered by any of the results of Deng [2] and Yang [9].
Example. Consider the equations
′′ + ′ + [ + ]x
t
x
t
x x
2
2
3θ
= 0, (3.3)
′′ + ′ +x
t
x
t
x x
2
2
θ λ sgn = 0, (3.4)
and
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
700 D. ÇAKMAK
′′ + ′ + +
+
x
t
x
t
x
x
1
1
1
12 2
µ
= 0, (3.5)
where t > 0, θ >
1
4
, λ > 0 and µ > 0. Note that, for the equation (3.3)
f x
x
( )
= 1 +
+ x2 ≥ 1 = M or f ′ ( x ) = 1 + 3x2 ≥ 1 = K for all x, and
p t
p t
2
2
( ) + ′( ) = 0 when p ( t ) =
=
2
t
. Applying Theorem 2.1 or Theorem 3.1 for the equation (3.3), and Theorem 3.2
for the equation (3.4), it is easy to verify that
θ θ
s
ds
t t
t
2
1
4
∞
∫ = > . Hence, equation (3.3),
every unbounded solution of equation (3.4) with λ > 1 and every bounded solution of
equation (3.4) with 0 < λ < 1 are oscillatory for θ >
1
4
. Note that f ( x ) =
= x
x
1
1
1 2+
+
and f ′ ( x ) = 1 +
1
1
2
2 2
−
( + )
x
x
for the equation (3.5). It is clear that
Theorem 3.1 cannot be applied to equation (3.5). Nevertheless, we can prove the
oscillatory character of equation (3.5) by using Theorem 2.1 or Corollary 2.1. Taking
into account that
f x
x
( )
= 1 +
1
1 2+ x
≥ 1 = M for x ≠ 0, we get
µ µ
s s s
ds
t t
t
2 2 2
1
4
1
2
1
4
1 1
4
− +
= +
>
∞
∫ .
Hence, equation (3.5) is oscillatory for µ > 0.
Acknowledgement. The author thanks the referee for his valuable suggestions.
1. Agarwal R. P., Grace S. R. Second order nonlinear forced oscillations // Dynam. Systems and
Appl. – 2001. – 10. – P. 455 – 464.
2. Deng J. Oscillation criteria for second-order linear differential equations // J. Math. Anal. and
Appl. – 2002. – 271. – P. 283 – 287.
3. Elbert A. Oscillation/nonoscillation criteria for linear second order differential equations // Ibid. –
1998.– 226. – P. 207 – 219.
4. Hille E. Nonoscillation theorems // Trans. Amer. Math. Soc. – 1948. – 64. – P. 234 – 252.
5. Huang C. Oscillation and nonoscillation for second order linear differential equations // J. Math.
Anal. and Appl. – 1997. – 210. – P. 712 – 723.
6. Tang X. H., Yu J. S., Peng D. H. Oscillation and nonoscillation of neutral difference equations with
positive and negative coefficients // Comput. Math. Appl. – 2000. – 39. – P. 169 – 181.
7. 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.
8. Yang X. A note on „Oscillation and nonoscillation for second-order linear differential equations” //
Ibid. – 1999. – 238. – P. 587 – 590.
9. Yang X. Oscillation criterion for a class of quasilinear differential equations // Appl. Math.
Comput. – 2004. – 153. – P. 225 – 229.
Received 06.03.06,
after revision — 02.06.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 5
|
| id | umjimathkievua-article-3186 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:37:50Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/f5/11147a82e15cce5655ad0b9da05a8af5.pdf |
| spelling | umjimathkievua-article-31862020-03-18T19:47:45Z Oscillation criteria for nonlinear second-order differential equations with damping Осциляційні критерії для нелінійних диференціальних рівнянь другого порядку із затуханням Çakmak, D. Чакмак, Д. Some new oscillation criteria are given for general nonlinear second-order ordinary differential equations with damping of the form x?+?p?(?t?)?x?+?q?(?t?)?f?(?x?) = 0, where f is monotone or nonmonotone. Our results generalize and extend some earlier results of Deng. Наведено деякі нові осцнляційні критерії для загальних нелінійних звичайних диференціальних рівнянь другого порядку із затуханням вигляду x" + p(t)x' + q(t)f(x) = 0, де функція f або монотонна, або немонотонна. Наведені результати узагальнюють та розширюють деякі результати, отримані раніше Денгом. Institute of Mathematics, NAS of Ukraine 2008-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3186 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 5 (2008); 694–700 Український математичний журнал; Том 60 № 5 (2008); 694–700 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3186/3119 https://umj.imath.kiev.ua/index.php/umj/article/view/3186/3120 Copyright (c) 2008 Çakmak D. |
| spellingShingle | Çakmak, D. Чакмак, Д. Oscillation criteria for nonlinear second-order differential equations with damping |
| title | Oscillation criteria for nonlinear second-order differential equations with damping |
| title_alt | Осциляційні критерії для нелінійних диференціальних рівнянь другого порядку із затуханням |
| title_full | Oscillation criteria for nonlinear second-order differential equations with damping |
| title_fullStr | Oscillation criteria for nonlinear second-order differential equations with damping |
| title_full_unstemmed | Oscillation criteria for nonlinear second-order differential equations with damping |
| title_short | Oscillation criteria for nonlinear second-order differential equations with damping |
| title_sort | oscillation criteria for nonlinear second-order differential equations with damping |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3186 |
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