Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 972656, 10 pages
doi:10.1155/2011/972656
Research Article
Generalized Variational Principles on
Oscillation for Nonlinear Nonhomogeneous
Differential Equations
Jing Shao1 and Fanwei Meng2
1
2
Department of Mathematics, Jining University, Qufu 273155, China
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Correspondence should be addressed to Jing Shao, shaojing99500@163.com
Received 30 June 2010; Revised 28 December 2010; Accepted 18 January 2011
Academic Editor: Allan C. Peterson
Copyright q 2011 J. Shao and F. Meng. 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.
Using the generalized variational principle and the Riccati technique, new oscillation criteria
are established for the forced second-order nonlinear differential equation, which improves and
generalizes some of the new results in literature.
1. Introduction
In this paper, we consider the oscillatory behavior of the nonlinear nonhomogeneous
differential equation of the form
α−1
ptΨ yt y t y t qtf yt et,
t ≥ t0 ,
1.1
where α is a positive constant, p, q, e ∈ Ct0 , ∞, Ê with pt > 0, Ψ ∈ CÊ, 0, ∞, f ∈
0.
CÊ, Ê satisfying ufu > 0 for u /
As usual, by a solution of 1.1 we mean a function y ∈ C1 Ty , ∞, Ty ≥ t0 , where
Ty ≥ t0 depends on the particular solution, which has the property ptΨyt|y t|α−1 y t ∈
C1 Ty , ∞ and satisfies 1.1. A nontrivial solution of 1.1 is called oscillatory if it has
arbitrarily large zeros; otherwise, it is said to be nonoscillatory. Equation 1.1 is said to be
oscillatory if all its solutions are oscillatory.
Recently, many research works have been done on the oscillatory and asymptotic
behavior of solutions of the nonlinear nonhomogeneous differential equation of the form
2
Abstract and Applied Analysis
1.1 or its special cases see 1–7 and references cited therein. Using the variational method,
oscillation criteria are obtained by Wong 1 for the forced linear differential equation, by Li
and Cheng 2 for the forced half-linear differential equation, by Zheng and Meng 3 for
the forced quasilinear differential equation, by Çakmak and Tiryaki 4 as well as Zheng
and Cheng 5 for the forced nonlinear differential equation some deficiencies in 2, 4 are
pointed out by Zheng and Meng 3, and Erbe et al. 7, as well as Saker 8, 9 for the dynamic
equation on time scales.
Meanwhile, in 6, Komkov gave a generalized Leighton’s variational principle, which
can also be used to obtain the oscillation criterion.
Theorem 1.1. Suppose that there exist a C1 function ut defined on s1 , t1 and a function Gu
such that Gut is not constant on s1 , t1 , Gus1 Gut1 0, gu G u is continuous,
t1 2 qtGut − pt u t dt > 0,
1.2
s1
and gut2 ≤ 4Gut for t ∈ s1 , t1 . Then each solution of the equation
pty t qtyt 0
1.3
vanishes at least once on s1 , t1 .
The purpose of our paper is to use the generalized variational principle to study the
oscillation for 1.1. These oscillation criteria are closely related to the generalized variational
formulae 1.2, which improve the results mentioned above. Examples will also be given to
illustrate the effectiveness of our main results.
Before going into the main results, let us state three sets of conditions commonly used
in the literature which we rely on:
β−1/β
> 0,
S1 0 < Ψu ≤ M, f u ≥ K fu
f u
S2 / 0,
1/β ≥ γ > 0, for u Ψu|fu|β−1
fu
≥ δ > 0, for u /
0.
S3 0 < Ψu ≤ M,
|u|β
for u / 0,
1.4
1.5
1.6
Here, M, K > 0, 0 < α ≤ β, and γ, δ > 0 are constants. It is clear that assumption S1
implies S2, but not conversely. For example, the function fu u3 , Ψu u2 and β 1
do not satisfy S1, but S2 holds. In S1 and S2, we need f to be differentiable. Clearly,
this condition is not required in S3. These differences force us to study 1.1 under the
assumptions S1, S2, and S3 in separate manners.
2. The Case Where β α
Firstly, we give an inequality, which is a transformation of Young’s inequality.
Abstract and Applied Analysis
3
Lemma 2.1 see 10. Suppose that X and Y are nonnegative. Then
λXY λ−1 − X λ ≤ λ − 1Y λ ,
2.1
λ > 1,
where the equality holds if and only if X Y .
Now, we will give our main results.
Theorem 2.2. Assume (S2) holds. Suppose further that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 <
t2 such that
⎧
⎨≤ 0, t ∈ s1 , t1 ,
et
⎩≥ 0, t ∈ s , t .
2 2
2.2
Let u ∈ C1 si , ti , and nonnegative functions G1 , G2 satisfying Gi usi Gi uti 0, gi u Gi u are continuous and gi utα1 ≤ α 1α1 Gαi ut for t ∈ si , ti , i 1, 2. If there exists a
positive function ρ ∈ C1 t0 , ∞, Ê such that
ρ
Qi u
⎡
α1 ⎤
α
G1/α1
utρ t
α
i
⎦dt > 0,
ρt⎣qtGi ut −
pt u t :
γ
α 1ρt
si
ti
2.3
for i 1, 2. Then 1.1 is oscillatory.
Proof. Suppose to the contrary that there is a nonoscillatory solution yt of 1.1. First, we
consider the case when yt > 0 eventually. Assume that yt > 0 on T0 , ∞ for some T0 ≥ t0 .
Set
α−1
ρtptΨ yt y t y t
,
wt f yt
t ≥ T0 .
2.4
Then differentiating 2.4 and making use of 1.1, it follows that for all t ≥ T0 ,
|wt|α1/α f yt
ρtet ρ t
w t −ρtqt wt − .
α−1 1/α
ρt
f yt
ptρtΨ yt f yt
2.5
By assumptions, we can choose s1 , t1 ≥ T0 with s1 < t1 so that et ≤ 0 on the interval I1 s1 , t1 . For t ∈ I1 and in view of 1.5 and 2.5, wt satisfies the inequality
ρtqt ≤ −w t ρ t
|wt|α1/α
wt − γ 1/α
.
ρt
p tρ1/α t
2.6
4
Abstract and Applied Analysis
Multiplying G1 ut through 2.6 and integrating 2.6 from s1 to t1 , using the fact that
G1 us1 G1 ut1 0, we obtain
γ |wt|α1/α
ρ t
dt
wt − 1/α
G1 utρtqtdt ≤
G1 ut −w t ρt
p tρ1/α t
s1
s1
t1
t1
−G1 utwt|ts11 t1
G1 ut
s1
≤
t1 s1
−γ
t1
Gi ut wtdt
s1
ρ t
wtdt −
ρt
t1
G1 ut
s1
γ |wt|α1/α
dt
p1/α tρ1/α t
ρ g1 utu t G1 ut t |wt|dt
ρt
t1
G1 ut
s1
|wt|α1/α
dt
p1/α tρ1/α t
t1 α/α1
≤ α 1
utu t G1 ut
G1
s1
−γ
t1
s1
G1 ut
ρ t
α 1ρt
|wt|dt
|wt|α1/α
dt.
p1/α tρ1/α t
2.7
Let
γ α/α1
1
α/α1
G1
ut|wt|, λ 1 ,
1/α1
1/α1
α
p
tρ
t
α
1/α1
utρ t
αα pα/α1 tρα/α1 t G1
u t Y .
α 1ρt
γ α2 /α1
X
2.8
By Lemma 2.1 and 2.7, we have
α1
t1 α
G1/α1
utρ t
α
1
G1 utρtqtdt ≤
ptρt u t dt,
γ
α 1ρt
s1
s1
t1
2.9
which contradicts 2.3 with i 1.
When yt < 0 holds eventually, we assume yt < 0 for t ≥ T0 > t0 . Defining the
Riccati transformation as 2.4, we get that 2.5 is true. In this case, we choose t2 > s2 ≥ T0
so that et ≥ 0 on the interval I2 s2 , t2 . For a given t ∈ I2 , 1.5 and 2.5 imply 2.6.
Multiplying G2 ut through 2.6 and integrating 2.6 from s2 to t2 , using the fact that
G2 us2 G2 ut2 0, we obtain a similar contradiction to 2.3 with i 2. This completes
the proof.
Abstract and Applied Analysis
5
Corollary 2.3. If ρt ≡ 1 in Theorem 2.2 and 2.3 is replaced by
ti α
α1
α
qtGi ut −
dt > 0,
pt u t
Qi u :
γ
si
2.10
for i 1, 2, then 1.1 is oscillatory.
Remark 2.4. The left side of inequality 2.10 is closely related to the generalized variational
formulae 1.2. Particularly, we obtain α γ for the linear equation; thus, 2.10 reduces
to 1.2 for the linear differential equation. So our Theorem 2.2 and Corollary 2.3 are
generalizations of the paper by Zheng and Cheng 5 and improvement of papers by Li and
Cheng 2 and by Çakmak and Tiryaki 4.
Theorem 2.5. Assume that (S3) holds. Suppose further that for any T ≥ t0 , there exist T ≤ s1 <
t1 ≤ s2 < t2 such that 2.2 holds. Let u ∈ C1 si , ti , and nonnegative functions G1 , G2 satisfying
Gi usi Gi uti 0, gi u Gi u are continuous and gi utα1 ≤ α 1α1 Gαi ut for
t ∈ si , ti , i 1, 2. If there exists a positive function ρ ∈ C1 t0 , ∞, Ê such that
⎡
α1 ⎤
1/α1
ρ t
G
ut
ρ
⎦dt > 0,
ρt⎣δqtGi ut − Mpt u t i
Qi u :
α 1ρt
si
ti
2.11
for i 1, 2. Then 1.1 is oscillatory.
Proof. Suppose to the contrary that there is a nonoscillatory solution yt. Firstly, we assume
that yt > 0 on T0 , ∞ for some T0 ≥ t0 . Set
α−1
ρtptΨ yt y t y t
wt ,
ytα−1 yt
t ≥ T0 .
2.12
Then differentiating 2.12 and making use of 1.1 and S3, we see that for all t ≥ T0 , we
have
−qtρtf yt
ρtet
ρ twt
|wt|α1/α
− α
w t α−1
α−1
1/α
ρt
yt yt
yt yt
ρtptΨ yt
etρt
ρ twt
α
|wt|α1/α
− 1/α 1/α
≤ −δqtρt .
α−1
ρt
yt yt M ρ tp1/α t
2.13
6
Abstract and Applied Analysis
By assumptions, we choose s1 , t1 ≥ T0 so that et ≤ 0 on the interval I1 s1 , t1 . For t ∈ I1 ,
2.13 implies that wt satisfies the inequality
δρtqt ≤ −w t ρ twt
α
|wt|α1/α
− 1/α
.
ρt
M p1/α tρt1/α t
2.14
A similar method as to that of Theorem 2.2 implies a contradiction to 2.11 with i 1.
When yt < 0 holds eventually, we assume yt < 0 for t ≥ T0 > t0 . Defining the Riccati
transformation as 2.12, we get that 2.13 is true. In this case, we choose t2 > s2 ≥ T0 so that
et ≥ 0 on the interval I2 s2 , t2 . For a given t ∈ I2 , we get that 2.14 holds. A similar
method reaches a similar contradiction to 2.11 with i 2. This completes the proof.
Now we give two examples to illustrate the efficiency of our results.
Example 2.6. Consider the following forced half-linear differential equation:
α−1
α−1
tλ y t y t Ktλ yt yt − sin t,
2.15
for t ≥ 1, where K, λ > 0 are constants and α 1. We may show that 2.15 is oscillatory for
K > 2e1 λ/22 using Theorem 2.2. Indeed, since the zeros of the forcing term − sin t are nπ,
the constant γ in 1.5 is α, that is, γ α. For any T ≥ 1, we choose n sufficiently large so that
nπ 2kπ ≥ T and s1 2kπ and t1 2k 1π. Selecting ut sin t ≥ 0, G1 u u2 exp−u
we note that G1 u2 ≤ 4G1 u for u ≥ 0, ρt t−λ , then we have
t1
ρtqtG1 utdt s1
K
π
2k1π
t−λ Ktλ sin t2 exp− sin tdt
2kπ
sin t2 exp− sin tdt
0
K
≥
e
π
0
Kπ
1 − cos 2t
dt ,
2
2e
α1
α
G1/α1
utρ t
α
i
ρt
pt u t dt
γ
α 1ρt
s1
t1
2k1π
2.16
α α λ sin t exp− sin t/2 2
tλ |cos t| dt
α
t
2kπ
π λ 2
λ 2
<
1
dt 1 π.
2
2
0
ρ
t−λ
So we have Q1 u > 0 provided K > 2e1 λ/22 . Similarly, for s2 2k 1π and t2 2k 2π, we select ut sin t ≤ 0, G2 u u2 expu, and we note that G2 u2 ≤ 4G2 u
ρ
for u ≤ 0; we can show that the integral inequality Q2 u > 0 for K > 2e1 λ/22 . So 2.15
is oscillatory for K > 2e1 λ/22 by Theorem 2.2.
Abstract and Applied Analysis
7
Example 2.7. Consider the following forced half-linear differential equation:
α−1
α−1
2 cos tt−λ y t y t Kt−λ expsin tyt yt − sin t,
2.17
for t ≥ 1, where K, λ > 0 are constants and α 1. We may show that 2.17 is oscillatory
for K > 31 λ2 using Theorem 2.2. Indeed, since the zeros of the forcing term − sin t are
nπ, the constant γ in 1.5 is α, that is, γ α. In fact, for any T ≥ 1, we choose n sufficiently
large so that nπ 2kπ ≥ T and s1 2kπ and t1 2k 1π. Selecting ut sin t ≥ 0,
G1 u u2 exp−u we note that G1 u2 ≤ 4G1 u for u ≥ 0, ρt tλ , then we have
t1
ρtqtG1 utdt s1
K
π
2k1π
tλ Kt−λ sin t2 exp− sin tdt
2kπ
sin t2 expsin t − sin tdt
0
≥K
π
0
Kπ
1 − cos 2t
dt ,
2
2
α1
α
G1/α1
utρ t
α
i
ρt
pt u t dt
γ
α 1ρt
s1
t1
2k1π
2kπ
π
<
0
tλ
2.18
λ sin t exp− sin t/2 2
1 α α
dt
2 cos tt−λ |cos t| 2 α
t
3
3
1 λ2 dt 1 λ2 π.
2
2
ρ
So, we have Q1 u > 0 provided K > 31 λ2 . Similarly, for s2 2k 1π and t2 2k 2π,
we select ut sin t ≤ 0, G2 u u2 expu, and we note that G2 u2 ≤ 4G2 u for u ≤ 0; we
ρ
can show that the integral inequality Q2 u > 0 for K > 31 λ2 . So, 2.17 is oscillatory for
K > 31 λ2 by Theorem 2.2.
3. The Case Where β > α
We now handle the case where β > α.
Theorem 3.1. Assume that (S3) holds. Suppose further that for any T ≥ t0 , there exist T ≤ s1 <
t1 ≤ s2 < t2 such that 2.2 holds. Let u ∈ C1 si , ti , and the nonnegative functions G1 , G2 satisfying
Gi usi Gi uti 0, gi u Gi u are continuous and gi utα1 ≤ α 1α1 Gαi ut for
t ∈ si , ti , i 1, 2. If there exists a positive function ρ ∈ C1 t0 , ∞, Ê such that
ρ
Qi u
⎡
α1 ⎤
ρ t
G1/α1
ut
⎦dt > 0,
ρt⎣Qe tGi ut − Mpt u t i
:
α 1ρt
si
ti
3.1
8
Abstract and Applied Analysis
for i 1, 2, then 1.1 is oscillatory, where
α/β
α−β/β δqt
Qe t α−α/β β β − α
|et|β−α/β .
3.2
Proof. Suppose to the contrary that there is a nontrivial nonoscillatory solution. Firstly, we
assume that yt > 0 on T0 , ∞ for some T0 ≥ t0 . Set
α−1
ρtptΨ yt y t y t
wt ,
ytα−1 yt
t ≥ T0 .
3.3
Then differentiating 3.3 and making use of 1.1, it follows that for all t ≥ T0 ,
−qtρtf yt
ρtet
ρ twt
|wt|α1/α
w t −
α
1/α
ρt
ytα−1 yt
ytα−1 yt
ρtptΨ yt
qtρtf yt ρ twt
|wt|α1/α
ytβ−α etρt
− α
− α−1
β−1
1/α
ρt
yt yt
yt yt
ptρtΨ yt
β−α
ρ twt
etρt
α
|wt|α1/α
− 1/α 1/α
≤ −δqtρtyt .
α−1
ρt
yt yt
M p tρ1/α t
3.4
By assumptions, we can choose t1 > s1 ≥ T0 so that et ≤ 0 on the interval I1 s1 , t1 . For
a given t ∈ I1 , set Fx δqtxβ−α − et/xα , and we have F x∗ 0, F x∗ > 0, where
x∗ −αet/β − αδqt1/β . So, Fx attains its minimum at x∗ and
Fx ≥ Fx∗ Qe t.
3.5
So 3.4 and 3.5 imply that wt satisfies
ρtQe t ≤ −w t −
ρ twt
α
|wt|α1/α
.
ρt
M1/α p1/α tρ1/α t
3.6
The remaining argument is the same as in the proof of Theorem 2.2, so we obtain a desired
contradiction to 3.1 with i 1 when yt > 0 eventually.
On the other hand, if yt is a negative solution for t ≥ T0 ≥ t0 , we define the Riccati
transformation 3.3 to yield 3.4. In this case, we choose t2 > s2 ≥ T0 so that et ≥ 0 on the
interval I2 s2 , t2 . For a given t ∈ I2 , set Fx δqtxβ−α − et/xα , and we have Fx ≥
Fx∗ Qe t. The remaining proof is similar to that of Theorem 2.2; a desired contradiction
to 3.1 with i 2 can be obtained. This completes the proof.
Abstract and Applied Analysis
9
Corollary 3.2. If ρt ≡ 1 in Theorem 3.1 and the hypothesis 3.1 is replaced by
i u :
Q
ti α1 Qe tGi ut − Mptu t
dt > 0,
3.7
si
for i 1, 2, then 1.1 is oscillatory.
We remark that Corollary 3.2 is closely related to the generalized variational formulae.
Furthermore, in Theorem 3.1, there is no restriction on the positive constant α, so Theorem 2.5
can be treated as its limiting case when β → α 0 with the convention that 00 1.
Example 3.3. Consider the following forced quasilinear differential equation:
2
γ tλ/3 y t tλ yt yt −sin3 t,
t ≥ 1,
3.8
where γ, λ > 0 are constants. We see that Ψu ≡ 1, which implies that M 1, and α 1,
β 3 in Theorem 3.1. Since α < β, Theorem 2.2 cannot be applied. However, we can obtain
oscillation for 3.8 using Theorem 3.1. For any T ≥ 1, choose n sufficiently large so that
nπ 2kπ ≥ T and s1 2kπ and t1 2k 1π. We choose ut sin t ≥ 0, G1 u u2 exp−u
we note that√G1 u2 ≤ 4G1 u, for u ≥ 0, ρt tλ . In fact, we can easily verify that
Qe t 3/2 3 2tλ/3 sin2 t,
t1
Qe tqtG1 utdt s1
π
2k1π
2kπ
t−λ/3
3√
3
2 sin t4 exp− sin tdt
2
3√
3
2
sin t4 exp− sin tdt
2
0
√
932
≥
π,
16e
α1
t1
α
G1/α1
utρ t
α
i
ρt
Mpt u t dt
γ
α 1ρt
s1
3.9
t1
λ/3 sin t exp− sin t/2 2
t−λ/3 γ tλ/3 |cos t| dt
2t
s1
λ 2
π.
<γ 1
6
√
So, we have that 3.1 is true for i 1 provided 0 < γ < 9 3 2/16e1 λ/62 . Similarly, for
s2 2k 1π
√ and t2 2k 2π, we can show that 3.1 is true for i 2. So 3.8 is oscillatory
for 0 < γ < 9 3 2/16e1 λ/62 by Theorem 3.1.
10
Abstract and Applied Analysis
Acknowledgments
The authors sincerely thank the referees for their constructive suggestions and corrections.
This research was partially supported by the NSF of China Grant 10771118, 10801089, NSF of Shandong Grant ZR2009AM011, ZR2009AQ010, and NSF of Jining University Grant
2010QNKJ09.
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