Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2009, Article ID 897058, 12 pages
doi:10.1155/2009/897058
Research Article
Oscillation Criteria for a Class of
Second-Order Nonlinear Differential Equations
with Damping Term
Zigen Ouyang,1, 2 Jichao Zhong,1, 2 and Shuliang Zou1
1
Center of Nuclear Energy Economy and Management, University of South China,
Hengyang 421001, China
2
School of Mathematics and Physics, University of South China, Hengyang 421001, China
Correspondence should be addressed to Zigen Ouyang, zigenouyang@yahoo.com.cn
Received 31 August 2009; Accepted 29 September 2009
Recommended by Yong Zhou
A class of second-order nonlinear differential equations with damping term rt|x t|σ−1 x t pt|x t|σ−1 x t qtfxt 0 are investigated in this paper. By using a new method, we obtain
some new sufficient conditions for the oscillation of the above equation, and some references are
extended in this paper. Examples are inserted to illustrate this result.
Copyright q 2009 Zigen Ouyang et al. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
1. Introduction
Consider the following second-order nonlinear differential equations with damping term:
σ−1
σ−1
rtx t x t ptx t x t qtfxt 0,
t ≥ t0 ,
1.1
where rt ∈ C1 t0 , ∞; R , pt, qt ∈ Ct0 , ∞; R, σ is a positive constant, and f is a
continuous real-valued function on the real line R and satisfies xfx ≥ 0 for x /
0. We restrict
our attention to those solutions xt of 1.1 which exist on some half line tx , ∞ and satisfy
sup{|xt| : t ≥ T } > 0 for any T ≥ tx .
Recently, there are many authors who have investigated the oscillation for secondorder differential equations 1–9, Li 10 and Zhao investigated oscillation criteria for the
following equation:
σ σ
pt x t qtfxt 0,
rt x t
t ≥ t0 ,
1.2
2
Abstract and Applied Analysis
where σ is a quotient of odd positive integer. It is obvious that 1.2 is a special case of 1.1.
In fact, the conditions of Theorem 3.2 in 10 are too complex.
More recently, Rogovchenko and Tuncay 11 have obtained oscillation criteria of the
following:
rtx t ptx t qtfxt 0,
t ≥ t0 .
1.3
Motivated by the above discussions, we investigate the oscillation of 1.1 in this
paper; our oscillatory conditions and the proof of the main results are more simple than those
of Theorem 3.2 in 10.
A solution xt of 1.1 is oscillatory if and only if it has arbitrarily large zeros;
otherwise, it is nonoscillatory. Equation 1.1 is oscillatory if and only if every solution of
1.1 is oscillatory.
The paper is arranged as follows. In Section 2, we will establish our main results.
Finally, examples are given to illustrate our results.
2. Main Results
To obtain our results, we introduce a lemma as follows.
Lemma 2.1 see 2, 3. Let the function Kt, s, x : R × R × R → R be such that for each fixed t,
s, the function Kt, s, · is nondecreasing. Further, let ht be a given function and ut satisfies that
ut ≥ ≤ht t
t ≥ t0 ,
Kt, s, usds,
2.1
t0
and v∗ t is the minimal (maximal) solution of
vt ht t
Kt, s, vsds,
t ≥ t0 .
2.2
t0
Then ut ≥ ≤v∗ t for all t ≥ t0 .
Now, we give our main results.
Theorem 2.2. Assume that f x ≥ 0, pt ≤ 0, qt > 0, and
that there exists a positive function ρt such that
∞
∞
t0
1/r 1/σ tdt ∞ hold. Suppose
qtρtdt ∞,
2.3
ptρt ≥ rtρ t.
2.4
t0
Then every solution of 1.1 is oscillatory.
Abstract and Applied Analysis
3
Proof. Assume that 1.1 has a nonoscillatory solution xt. Without loss of generality,
suppose that it is an eventually positive solution if it is an eventually negative solution,
the proof is similar, that is, xt > 0 for all t ≥ t0 .
We consider the following three cases.
Case 1. Suppose that x t is oscillatory. Then there exists t1 ≥ t0 such that x t1 0. From
1.1, we have
ps
ds
t0 rs
t
t
σ−1 σ−1 ps
ps
ds pt x t
ds
rt x t
x t exp
x t exp
t0 rs
t0 rs
t
ps
ds < 0,
−qtfxt exp
t0 rs
σ−1
rtx t x t exp
t
2.5
which means that
σ−1
rtx t x t exp
ps
ds
t0 rs
t
t
σ−1
< rt1 x t1 x t1 exp
ps
ds 0,
t0 rs
2.6
1
t > t1 ,
it follows that x t < 0 for all t > t1 , which contradicts to the assumption that x t is
oscillatory.
Case 2. Suppose that x t < 0. From 1.1, we obtain
σ−1
σ σ
−pt −x t qtfxt ≥ 0,
− rtx t x t rt −x t
2.7
then there exists an M > 0 and a t1 ≥ t0 , such that
σ
rt −x t ≥ M,
t ≥ t1 ,
2.8
it follows
xt ≤ −
∞
t1
1
M1/σ dt xt1 ,
r 1/σ t
t ≥ t1 ,
which means that limt → ∞ xt −∞, this contradicts the assumption that xt > 0.
2.9
4
Abstract and Applied Analysis
Case 3. Suppose that x t > 0. Let wt ρtrtx tσ , then
σ σ
ρt rt x t ρ t,
w t rt x t
2.10
t ≥ t0 ,
in view of 1.1, we obtain
ρtptx tσ ρ trtx tσ
w t
−qtρt −
,
fxt
fxt
fxt
t ≥ t0 ,
2.11
noticing that
wt
fxt
w tfxt − wtf xtx t
f 2 xt
−qtρt −
ρtptx tσ ρ trtx tσ wtf xtx t
−
,
fxt
fxt
f 2 xt
t ≥ t0 ,
2.12
integrating the above from t0 to t, we get
wt0 wt
−
fxt fxt0 t t0
ρsps − ρ srs x sσ wsx sf xs
qsρs
ds.
fxs
f 2 xs
2.13
Using 2.3, 2.4, and x t > 0, we have
wt
−∞,
t → ∞ fxt
0 ≤ lim
2.14
this is a contradiction, the proof is complete.
Remark 2.3. If we replace pt ≤ 0, qt > 0 by pt ≤ 0, qt ≤ 0, limt → ∞ pt/qt M > 0,
Theorem 2.2 holds also.
Theorem 2.4. Assume that f x ≥ 0 holds. Suppose also that
ρ0 t exp
∞
t0
dt
ps
ds ,
t0 rs
2.15
1/σ ∞,
2.16
t
ρ0 trt
and ρ0 t such that 2.3 holds. Then every solution of 1.1 is oscillatory.
Abstract and Applied Analysis
5
Proof. To the contrary, 1.1 has a nonoscillatory solution xt. Without loss of generality, we
assume that xt is an eventually positive solution. Let wt ρ0 trt|x t|σ−1 x t, then
wtx t ρ0 trt|x t|σ−1 x t2 ≥ 0 for t ≥ t0 and
σ−1
σ−1
w t rtx t x t ρ0 t rtx t x tρ0 t,
t ≥ t0 ,
2.17
in view of 1.1 and 2.15, we obtain
w t
−qtρ0 t,
fxt
t ≥ t0 ,
2.18
since
wt
fxt
w tfxt − wtf xtx t
wtf xtx t
−qtρ
,
−
t
0
f 2 xt
f 2 xt
t ≥ t0 ,
2.19
integrating the above from t0 to t, we have
wt0 wt
−
−
fxt
fxt0 t
qsρ0 sds t
wsx sf xs
ds,
f 2 xs
t0
t0
t ≥ t0 .
2.20
In view of 2.3, there exists a constant m > 0 and t1 ≥ t0 such that
−
wt0 fxt0 t
qsρ0 sds t0
t1
wsx sf xs
ds ≥ m,
f 2 xs
t0
2.21
which means that
wt
≥m
−
fxt
t
wsx sf xs
ds.
f 2 xs
t1
2.22
Because that xt is positive, then 2.22 implies −wt > 0, or equivalently x t < 0. Let
σ−1
σ
ut −wt −ρ0 trtx t x t ρ0 trt −x t ,
2.23
thus 2.22 can be changed as
ut ≥ mfxt t
fxtf xs−x s
usds.
f 2 xs
t1
2.24
fxtf xs−x s
u.
f 2 xs
2.25
Define
Kt, s, u 6
Abstract and Applied Analysis
Then, for any fixed t and s, Kt, s, u is nondecreasing in u. Let vt be the minimal solution
of the equation
vt mfxt t
fxtf xs−x s
vsds.
f 2 xs
t1
2.26
Applying Lemma 2.1, we obtain
ut ≥ vt,
2.27
t ≥ t0 .
Dividing both sides of 2.26 by fxt and deriving both sides of 2.26, it follows
vt
fxt
f xs−x s
f xt−x t
vsds
vt.
m
f 2 xs
f 2 xt
t1
t
2.28
On the other hand,
vt
fxt
f xtx t
v t
−
vt.
fxt
f 2 xt
2.29
Combining 2.28 and 2.29, it means
v t ≡ 0.
2.30
So vt vt1 mfxt1 , t ≥ t0 . From 2.27, we obtain
1/σ
−x t ≥ mfxt1 1
ρ0 trt
t ≥ t1 .
1/σ ,
2.31
Integrating both sides of the above from t1 to t, we have
1/σ
−xt xt1 ≥ mfxt1 t
t1
ds
ρ0 srs
1/σ .
2.32
Letting t → ∞ in 2.32, and using 2.16, it follows that limt → ∞ xt ≤ −∞, which contradicts
to that xt is eventually positive. The proof is complete.
In the following, we always suppose that Ht ∈ C2 R; R and it satisfies the following
two conditions:
H1 Ht > 0 for t ≥ t0 , Ht is a bounded function;
H2 H t ht, ht is a bounded function.
Abstract and Applied Analysis
7
Theorem 2.5. Assume that f x ≥ 0,
∞
t0
1/r 1/σ tdt ∞ hold, and
pt ≤ 0,
qt > 0,
2.33
or
pt ≤ 0,
qt ≤ 0,
pt
M > 0.
t → ∞ qt
lim
2.34
Suppose further that there exists a function Ht that satisfies (H1 ), (H2 ), and such that
∞
Htϕtdt ∞,
2.35
lim sup vtrt < ∞,
2.36
t0
t→∞
where
ϕt vt qt − ptht − rtht ,
t ps hs
−
ds .
vt exp
Hs
t0 rs
2.37
2.38
Then every solution of 1.1 is oscillatory.
Proof. For the sake of contradiction, 1.1 has a nonoscillatory solution xt. Without loss of
generality, we may assume that xt > 0 for all t ≥ t0 .
Define
ut vtrt
|x t|σ−1 x t
ht .
fxt
2.39
Deriving 2.39, we get
u t pt ht
−
ut
rt Ht
pt|x t|σ−1 x t
rt|x t|σ−1 x t2 f xt
vt −
− qt −
rtht
fxt
f 2 xt
≤−
ht
ut ptvtht − vtqt vtrtht
Ht
−
ht
ut − ϕt.
Ht
2.40
8
Abstract and Applied Analysis
Multiplying 2.40 by Ht, it follows
ϕtHt ≤ −Htu t − htut.
2.41
We consider the following three cases.
Case 1 ut is oscillatory. Then there exists a sequence {tn }, n 1, 2, . . . , tn → ∞ as n → ∞
and such that utn 0, n 1, 2, . . . . Integrating both sides of 2.41 from t0 to tn , we obtain
tn
Htϕtdt ≤ −
tn
t0
Htu tdt −
t0
−Htut|ttn0
−
tn
tn
htutdt
t0
−H tut htut dt
2.42
t0
Ht0 ut0 − Htn utn Ht0 ut0 ,
that is
tn
lim
tn → ∞
Htϕtdt ≤ Ht0 ut0 ,
2.43
t0
which contradicts 2.35.
Case 2 ut is eventually positive. Integrating both sides of 2.41 from t0 to ∞, we obtain
∞
Htϕtdt ≤ Ht0 ut0 − lim Htut ≤ Ht0 ut0 ,
t→∞
t0
2.44
which also contradicts to 2.35.
Case 3 ut is eventually negative. If lim supt → ∞ ut > −∞, then there exists a sequence
{tn }, n 1, 2, . . ., that satisfies {tn } → ∞ as n → ∞ and such that limtn → ∞ utn lim supt → ∞ ut M1 > −∞. Because Ht is a bounded function, then there exists a M2 > 0
such that Htn ≤ M2 , n 1, 2, . . . . According to 2.41, we obtain
tn
Htϕtdt ≤ Ht0 ut0 − H tn u tn ≤ Ht0 ut0 − M2 u tn .
2.45
t0
Using 2.35 and taking limit as tn → ∞, it is easy to show that
∞ lim
tn → ∞
tn
Htϕtdt ≤ Ht0 ut0 − lim H tn u tn
t0
≤ Ht0 ut0 − M1 M2 < ∞,
which is obviously a contradiction.
tn → ∞
2.46
Abstract and Applied Analysis
9
If lim supt → ∞ ut −∞, limt → ∞ ut −∞. From the definition of ht, combining
2.36 and 2.39, it follows that x t < 0 and limt → ∞ |x t|σ−1 x t/fxt −∞, which
means that limt → ∞ −x tσ /fxt ∞. Owing to pt ≤ 0, qt ≥ 0, or pt ≤ 0,
qt ≤ 0 and limt → ∞ pt/qt M > 0, using the similar method of the proof of Case 2
in Theorem 2.2, we will derive a contradiction. Then the proof is complete.
∞
Theorem 2.6. Assume that 2.36 holds, f x ≥ 0, t0 1/r 1/σ tdt ∞, and
pt ≤ 0,
2.47
qt > 0,
or
pt ≤ 0,
qt ≤ 0,
pt
M > 0.
t → ∞ qt
lim
2.48
Suppose further that there exists a function Ht that satisfies (H1 ), (H2 ) and such that
∞
Htϕtdt ∞,
2.49
t0
where
ϕt vt qt ptht rtht ,
2.50
and vt is defined in 2.38. Then every solution of 1.1 is oscillatory.
Proof. For the sake of contradiction, 1.1 has a nonoscillatory solution. Without loss of
generality, we may assume that 1.1 has an eventually positive solution if it has an
eventually negative solution, the proof is similar, then there exists a t1 > t0 such that xt > 0
for all t ≥ t1 . Define
ut vtrt
|x t|σ−1 x t
− ht .
fxt
2.51
The rest of the proof is similar to Theorem 2.5. The proof is complete.
∞
Theorem 2.7. Assume that 2.36 holds, f x ≥ 0, t0 1/r 1/σ tdt ∞, and
pt ≤ 0,
2.52
qt > 0,
or
pt ≤ 0,
qt ≤ 0,
lim
pt
t → ∞ qt
M > 0.
2.53
10
Abstract and Applied Analysis
Suppose further that there exists a function Ht that satisfies (H1 ), (H2 ) and such that
∞
Htφtdt ∞,
2.54
t0
where
φt vt −ptht − rtht ,
2.55
where vt is defined in 2.38. Then every solution of 1.1 is oscillatory.
Proof. For the sake of contradiction, 1.1 has a nonoscillatory solution xt. Without loss of
generality, we may assume that xt > 0 for all t ≥ t0 .
Define
ut vtrt
|x t|σ−1 x t
ht .
xt
2.56
Noting that xfx ≥ 0 for x /
0, so fx/x ≥ 0 for x /
0. Deriving 2.56, we obtain
pt ht
u t −
ut
rt Ht
pt|x t|σ−1 x t qtfxt rt|x t|σ−1 x t2
vt −
−
−
rtht
xt
xt
x2 t
≤−
ht
ut ptvtht vtrtht
Ht
−
ht
ut − φt.
Ht
2.57
Multiplying 2.57 by Ht, we get
Htφt ≤ −Htu t − htut.
2.58
The rest of the proof is similar to Theorem 2.5; the proof is complete.
∞
Theorem 2.8. Assume that 2.36 holds, f x ≥ 0, t0 1/r 1/σ tdt ∞, and
pt ≤ 0,
2.59
qt > 0,
or
pt ≤ 0,
qt ≤ 0,
lim
pt
t → ∞ qt
M > 0.
2.60
Abstract and Applied Analysis
11
Suppose further that there exists a function Ht satisfies (H1 ), (H2 ) and such that
∞
Htφtdt ∞,
2.61
t0
where
φt vt ptht rtht ,
2.62
where vt is defined in 2.38. Then every solution of 1.1 is oscillatory.
Proof. For the sake of contradiction, 1.1 has a nonoscillatory solution xt. Without loss of
generality, we may assume that xt > 0 for all t ≥ t0 .
Define
ut vtrt
|x t|σ−1 x t
− ht .
xt
2.63
The rest of the proof is similar to Theorem 2.5; the proof is complete.
3. Examples
Example 3.1. Consider the following delay differential equation:
3
xt 0.
tx t − 2x t t 4t
3.1
∞
It is obvious that σ 1, qt t 3/4t > 0, pt −2 < 0, rt t, and t0 1/tdt ∞. It is
difficult to distinguish whether every solution of 3.1 is oscillatory by Theorem 3.2 of 10.
By taking ρt 1/t2 , then
∞
t0
qtρtdt ∞
3 1
dt ∞,
t
4t t2
t0
3.2
2
ptρt − 2 rtρ t.
t
From Theorem 2.2 or Theorem 2.4, it is easy to show that 3.1 is oscillatory.
In fact, xt t1/2 cos t is such an oscillatory solution.
Example 3.2. Consider the following differential equation:
tx t − x t txt 0.
It is obvious that σ 1, rt t, pt −1 < 0, qt t > 0, and
3.3
∞
t0
1/rtdt ∞
t0
1/tdt ∞.
12
Abstract and Applied Analysis
We are taking Ht C > 0, ht 0. By a simple calculation, it is easy to
t
t
show that vt exp t0 ps/rs − hs/Hsds exp t0 −1/s − 0/Cds t0 /t,
lim supt → ∞ vtrt lim supt → ∞ t0 < ∞, ϕt vtqt − ptht − rtht t0 , and
∞
∞
Htϕtdt t0 t0 Cdt ∞.
t0
From Theorem 2.5 or Theorem 2.6, it follows that 3.3 is oscillatory.
In fact, xt sin t is such an oscillatory solution.
Acknowledgments
This work was supported by the Natural Science Foundation of Hunan Province under
Grant no. 07JJ3130, the Doctor Foundation of University of South China under Grant no.
5-XQD-2006-9, and the Subject Lead Foundation of University of South China under Grant
no. 2007XQD13.
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