Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 539213, 10 pages
doi:10.1155/2012/539213
Research Article
Generalized Variational Oscillation Principles
for Second-Order Differential Equations with
Mixed-Nonlinearities
Jing Shao,1, 2 Fanwei Meng,1 and Xinqin Pang2
1
2
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Department of Mathematics, Jining University, Shandong, Qufu 273155, China
Correspondence should be addressed to Jing Shao, shaojing99500@163.com
Received 14 March 2012; Accepted 4 June 2012
Academic Editor: Mingshu Peng
Copyright q 2012 Jing Shao 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.
Using generalized variational principle and Riccati technique, new oscillation criteria are
established for forced second-order differential equation with mixed nonlinearities, which improve
and generalize some recent papers in the literature.
1. Introduction
In this paper, we consider the second-order forced differential equation with mixed nonlinearities:
m
α−1
β −1
α−1
rty t y t ptyt yt qj tyt j yt et,
t ≥ t0 ,
1.1
j1
where r, p, qj 1 ≤ j ≤ m, e ∈ Ct0 , ∞, R with rt > 0 and 0 < α < β1 < β2 < · · · < βm are
real numbers, p, qj 1 ≤ j ≤ m, and e might change signs.
In this paper, we are concerned with the nonhomogeneous equation 1.1. By a
solution of 1.1, we mean that a function y ∈ C1 Ty , ∞Ty ≥ t0 , where Ty ≥ t0 depends on the
particular solution which has the property pt|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.
2
Discrete Dynamics in Nature and Society
When m 0, we have the following second-order half-linear differential equation
without or with forcing term:
α−1
α−1
rty t y t qtyt yt 0,
α−1
α−1
rty t y t qtyt yt et,
1.2
t ≥ t0 ,
1.3
t ≥ t0 .
There are a lot of papers involved oscillation see 1–6 for these equations since the
foundation work of Elbert 2. In paper 1, using Leighton’s variational principle see 3
for 1.3, the following result was obtained by Li and Cheng.
Theorem 1.1. Suppose that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 < t2 such that et ≤ 0 for
t ∈ s1 , t1 and et ≥ 0 for t ∈ s2 , t2 . Let Dsi , ti {u ∈ C1 si , ti : ut ≡
/ 0, usi uti 0}
for i 1, 2. If there exist H ∈ Dsi , ti and a positive, nondecreasing function ρ ∈ C1 t0 , ∞, R
such that
ti
H 2 tρtqtdt >
si
1
α1
α1 ti
si
rtρt
|Ht|α−1
ρ
2H t |Ht|
ρ
α1
dt
1.4
for i 1, 2. Then, 1.3 is oscillatory.
Unfortunately, Theorem 1.1 cannot be applied to the case where α > 1, since for ρt ≡
1, the term |Ht|α−1 will appear as a denominator in 1.4 so that the requirement Hsi Hti 0 will cause trouble. This certainly calls for investigation of oscillation criteria that
can handle with such cases.
When α 1, 1.2 and 1.3 are reduced to the linear differential equation:
rty t qtyt 0, t ≥ t0 ,
rty t qtyt et, t ≥ t0 .
1.5
1.6
In paper 7, Wong proved the following result for 1.6.
Theorem 1.2. Suppose that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 < t2 such that et ≤ 0 for
t ∈ s1 , t1 and et ≥ 0 for t ∈ s2 , t2 . Let Dsi , ti {u ∈ C1 si , ti : ut / 0, usi uti 0}
for i 1, 2. If there exists u ∈ Dsi , ti such that
Qi u :
ti 2 qtu2 t − rt u t dt > 0,
i 1, 2,
1.7
si
then 1.6 is oscillatory.
On the other hand, among the oscillation criteria, Komkov 8 gave a generalized
Leighton’s variational principle, which also can be applied to oscillation for 1.5.
Discrete Dynamics in Nature and Society
3
Theorem 1.3. 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 − rt u t dt > 0,
1.8
s1
and gut2 ≤ 4Gut for t ∈ s1 , t1 . Then, every solution of 1.5 must vanish on s1 , t1 .
We note that when Gu ≡ u2 , the left-hand side of 1.8 is the energy functional related
to 1.5.
When pt ≡ 0, m 1, 1.1 turns into the quasilinear differential equation:
β−1
α−1
rty t y t qtyt yt et,
t ≥ t0 ,
1.9
where p, q, e ∈ Ct0 , ∞, R with pt > 0 and 0 < α ≤ β being constants. In paper 9, using
the generalized variational principle, Shao proved the following result for 1.9.
Theorem 1.4. Assume that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 < t2 such that
≤ 0,
et
≥ 0,
t ∈ s1 , t1 ,
t ∈ s2 , t2 .
1.10
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 φ ∈ C1 t0 , ∞, R such that
φ
Qi u
⎡
α1 ⎤
G1/α1
utφ t
i
⎦dt > 0
φt⎣Qe tGi ut − rt u t :
α 1φt
si
ti
1.11
for i 1, 2. Then 1.9 is oscillatory, where
α−β/β α/β
qt
Qe t α−α/β β β − α
|et|β−α/β ,
1.12
with the convention that 00 1.
Recently, using Riccati transformation, the following oscillation criteria were given for
1.1 by Zheng et al. 10.
Theorem 1.5. Assume that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 < t2 such that qj t ≥ 01 ≤
j ≤ m for t ∈ s1 , t1 ∪ s2 , t2 and
≤ 0,
et
≥ 0,
t ∈ s1 , t1 ,
t ∈ s2 , t2 .
1.13
4
Discrete Dynamics in Nature and Society
Let Dsi , ti {u ∈ C1 si , ti : uα1 t > 0, t ∈ si , ti , usi uti 0} for i 1, 2. If there exist
H ∈ Dsi , ti and a positive function φ ∈ C1 t0 , ∞, R such that
⎡⎛
⎞
⎤
m
Htφ t α1
⎦dt > 0
φt⎣⎝pt Qj t⎠H α1 t − rt H t α 1φt
si
j1
ti
1.14
for i 1, 2. Then 1.1 is oscillatory, where
α/βj
α−βj /βj qj t
Qj t α−α/βj βj m βj − α
|et|βj −α/βj ,
1 ≤ j ≤ m,
1.15
with the convention that 00 1.
The purpose of this paper is to obtain new oscillation criteria for 1.1 based on
generalized variational principles. Roughly, if the existence of a “positive” solution of
a functional relation implies the “positivity” of an associated functional over a set of
“admissible” functions, then we say that a variational oscillation principle is valid. For
instance, in Theorem 1.1, H ∈ Dsi , ti is admissible, and the functional is
ti si
1
α1
α1
ρ t α1
2
− H tρtqt dt.
2 H t |Ht|
ρt
|Ht|α−1
ptρt
1.16
Our emphasis will be directed towards oscillation criteria that are closely related to the
generalized energy functional the generalization of α 1-degree energy functional for
half-linear equations see 4, 11–13 for more details on these functionals, which improve
the results mentioned above. Examples will also be given to illustrate the effectiveness of our
main results.
2. Main Results
Firstly, we give an inequality, which is a transformation of Young’s inequality.
Lemma 2.1 see 14. Suppose that X and Y are nonnegative, then
γXY γ−1 − X γ ≤ γ − 1 Y γ ,
γ > 1,
2.1
where equality holds if and only if X Y .
Now, we will give our main results.
Theorem 2.2. Assume that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 < t2 such that
≤ 0,
et
≥ 0,
t ∈ s1 , t1 ,
t ∈ s2 , t2 .
2.2
Discrete Dynamics in Nature and Society
5
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 φ ∈ C1 t0 , ∞, R such that
φ
Qi u :
ti
⎛
⎞
m
φt⎣Gi ut⎝pt Qj t⎠
⎡
si
2.3
j1
α1 ⎤
G1/α1
utφ t
i
⎦dt > 0
−rt u t α 1φt
2.4
for i 1, 2, where Qj t is defined as 1.15 with the convention that 00 1. Then, 1.1 is oscillatory.
Proof. Suppose to the contrary that there is a nontrivial nonoscillatory solution y yt. We
assume that yt /
0 on T0 , ∞ for some T0 ≥ t0 . Set
α−1
rty t y t
,
wt φt ytα−1 yt
t ≥ T0 .
2.5
Then differentiating 2.5 and making use of 1.1, it follows that for all t ≥ T0 ,
w t m
β −α
φtet
φ t
|wt|α1/α
wt − φtpt − α
−
φt
qj ty j .
α−1
1/α
φt
yt yt
rtφt
j1
2.6
By the assumptions, we can choose si , ti ≥ T0 for i 1, 2 so that et ≤ 0 on the interval
I1 s1 , t1 , with s1 < t1 and yt ≥ 0, or et ≥ 0 on the interval I2 s2 , t2 , with s2 < t2
and yt ≤ 0. For given t ∈ I1 or t ∈ I2 , set Fj x qj txβj −α − et/mxα , 1 ≤ j ≤ m, we have
Fj xj∗ 0, Fj xj∗ > 0, where xj∗ −αet/mβj − αqj t1/βj . So, Fj x obtains it minimum
on xj∗ and
Fj x ≥ Fj xj∗ Qj t.
2.7
So on the interval I1 or I2 , 2.6 and 2.2 imply that wt satisfies
⎛
φt⎝pt m
j1
⎞
Qj t⎠ ≤ −w t φ t
|wt|α1/α
wt − α 1/α .
φt
rtφt
2.8
6
Discrete Dynamics in Nature and Society
Multiplying Gi ut through 2.8 and integrating 2.8 from si to ti , using the fact that
Gi us1 Gi ut1 0, we obtain
ti
⎛
φt⎝pt si
m
⎞
Qj t⎠Gi utdt
j1
≤
φ t
|wt|α1/α
wt − α Gi ut −w t 1/α dt
φt
si
rtφt
ti
−Gi utwt|tsii ti
gi utu twtdt
si
φ t
|wt|α1/α
wt − α Gi ut
1/α dt
φt
si
rtφt
ti
ti φ t
wtdt
gi utu t Gi ut
φt
si
−α
ti
2.9
α1/α
|wt|
Gi ut 1/α dt
si
rtφt
ti φ t
gi utu t Gi ut
≤
|wt|dt
φt
si
−α
ti
|wt|α1/α
Gi ut 1/α dt
si
rtφt
ti α/α1
≤ α 1
Gi
utu t Gi ut
si
−α
φ t
α 1φt
|wt|dt
ti
|wt|α1/α
Gi ut 1/α dt.
si
rtφt
Let
X α
α/α1
α/α1
Gi
1/α
rtφt
|wt|,
α/α1 u t Y αφtrt
by Lemma 2.1 and 2.9, we have
γ 1
α
1/α1 φ t
Gi
α 1φt
1
,
α
2.10
,
Discrete Dynamics in Nature and Society
7
⎛
⎞
α1
ti
m
G1/α1
utφ t
i
⎝
⎠
φt pt Qj t Gi utdt ≤
φtrt u t dt,
α 1φt
si
si
j1
ti
2.11
which contradicts with 2.3. This completes the proof of Theorem 2.2.
Corollary 2.3. If φt ≡ 1 in Theorem 2.2, and 2.3 is replaced by
Qi u :
ti
⎡⎛
⎣⎝pt si
m
⎞
⎤
α1
Qj t⎠Gi ut − rtu t ⎦dt > 0,
2.12
j1
for i 1, 2. Then, 1.1 is oscillatory.
If we choose G1 u G2 u uα1 in Corollary 2.3, then we have the following
corollary.
Corollary 2.4. Suppose that for any T ≥ t0 , there exist T ≤ s1 < t1 ≤ s2 < t2 such that 2.2 is true.
≡ 0, usi uti 0} for i 1, 2. If there exist u ∈ Dsi , ti Let Dsi , ti {u ∈ C1 si , ti : ut /
such that
i u :
Q
ti
⎡⎛
⎣⎝pt si
m
⎞
Qj t⎠|ut|
α1
⎤
α1
− rtu t ⎦dt > 0,
2.13
j1
for i 1, 2. Then, 1.3 is oscillatory.
Remark 2.5. Corollary 2.4 is closely related to the α 1-degree functional 1.8, so
Theorem 2.2, Corollaries 2.3, and 2.4 are generalizations of Theorem 1.2, and improvement
of Theorem 1.1 since the positive constant α in Theorem 2.2 and Corollary 2.3 can be selected
as any number lying in 0, ∞. We note further that in most cases, oscillation criteria are
obtained using the same auxiliary function on s1 , t1 and s2 , t2 , we note that such functions
can be selected differently.
Remark 2.6. If Gu ≡ uα1 , then Theorem 2.2 reduces to Theorem 1.5, and if pt ≡ 0, j 1,
Theorem 2.2 reduces to Theorem 1.4. So Theorem 2.2 and Corollary 2.3 are generalizations of
the papers by Zheng et al. 10 and Shao 9.
Remark 2.7. The hypothesis 2.2 in Theorem 2.2 and Corollary 2.3 can be replaced by the
following condition:
et
≥ 0,
t ∈ s1 , t1 ,
≤ 0,
t ∈ s2 , t2 .
The conclusion is still true for these cases.
2.14
8
Discrete Dynamics in Nature and Society
Example 2.8. Consider the following forced mixed nonlinearities differential equation:
γt−λ/3 y t
2
ptyt qtyt yt −sin3 t,
t ≥ 2π,
2.15
where γ, λ > 0 are constants, qt t−λ exp3 sin t, pt t−λ/3 expsin t, for t ∈ 2nπ, 2n 1π, and qt t−λ exp−3 sin t, pt t−λ/3 exp− sin t, for t ∈ 2n 1π, 2n 2π,
n > 0 is an integer, Shao 9 obtain oscillation
for 2.15 when Kt ≡ 0. Using Theorem 2.2,
√
we can easily√verify that Q1 t 3/2 3 2t−λ/3 expsin tsin2 t for t ∈ 2nπ, 2n 1π, and
Q1 t 3/2 3 2t−λ/3 exp− sin tsin2 t for t ∈ 2n 1π, 2n 2π. For any T ≥ 1, we choose
n sufficiently large so that nπ 2kπ ≥ T and s1 2kπ and t1 2k 1π, we select
ut sin t ≥ 0, G1 u u2 exp−u we note that G1 u2 ≤ 4G1 u for u ≥ 0, φt tλ/3 ,
then we have
t1
φt pt Q1 t G1 utdt π
0
s1
sin2 t dt 3√
3
2
2
π
0
sin4 t dt π 9√
3
2,
2 8
α1
t1
G1/α1
utφ t
1
φtpt u t dt
α 1φt
s1
2k1π λ|sin t| exp3 sin t/2 2
dt
γ
|cos t| 2t
2kπ
2
2
2k1π λe3/2
λe3/2
1
<γ
dt γ 1 π.
2
2
2kπ
2.16
√
φ
So we have Q1 u > 0 provided, 0 < γ < 4π 9 3 2/22 λe3/2 2 π. Similarly, for s2 2k 1π and t2 2k 2π, we select ut sin t ≤ 0, G2 u u2 expu we note that
φ
G2 u2 ≤ 4G2 u for u ≤ 0, we can show that the integral inequality Q2 u > 0 for 0 < γ <
√
√
4π 9 3 2/22 λe3/2 2 π. So 2.15 is oscillatory for 0 < γ < 4π 9 3 2/22 λe3/2 2 π by
Theorem 2.2.
Example 2.9. Consider the following forced mixed nonlinearities differential equation:
α−1
α−1
t−λ y t y t ptyt yt qty3 t −sin1/3 t,
2.17
for t ≥ 2π, where pt Kt−λ expsin t, qt t−9λ/5 exp9 sin t/5, for t ∈ 2nπ, 2n 1π,
and pt Kt−λ exp− sin t, qt t−9λ/5 exp−9 sin t/5, for t ∈ 2n 1π, 2n 2π, n > 0
is an integer, K, λ > 0 are constants and α 5/3 > 1, β 3. Obviously, Theorem 1.1 cannot
be applied to this case. However, we conclude that 2.17 is oscillatory for K > 3/41 3λe/88/3 π − 9/55/9 44/9 . Since the zeros of the forcing term −sin1/3 t are nπ, let ut sin t and
φt tλ . Using Theorem 2.2, we can easily verify that Qt 9/55/9 44/9 t−λ expsin tsin4/27 t
for t ∈ 2nπ, 2n 1π, and Qt 9/55/9 44/9 t−λ exp− sin t sin4/27 t for t ∈ 2n 1π, 2n 2π. For any T ≥ 1, choose n sufficiently large so that nπ 2kπ ≥ T and s1 2kπ and
Discrete Dynamics in Nature and Society
9
t1 2k 1π. For t ∈ s1 , t1 , we select G1 u u8/3 exp−u we note that G1 u8/3 ≤
8/38/3 G1 u5/3 for u ≥ 0. It is easy to verify the following estimations:
t1
φt pt Qt G1 utdt
s1
2k1π
9
4/27
sin
t
dt
55/9 44/9
2kπ
2k1π
9
4
9
K 5/9 4/9 ,
sin3 t dt > K 5/9 4/9
3
5 4
5 4
2kπ
sin8/3 t K 2.18
α1
G1/α1
utφ t
1
φtrt u t dt
α 1φt
s1
t1
2k1π 2kπ
3λe−3 sin t/8 |sin t|
|cos t| 8t
2k1π <
2kπ
1
3λe
8
8/3
dt 8/3
dt
3λe 8/3
1
π.
8
φ
So we have Q1 u > 0. Similarly, for s2 2k1π and t2 2k2π, we select ut sin t < 0,
G2 u u8/3 expu we note that G2 u8/3 ≤ 8/38/3 G2 u5/3 for u ≤ 0, we can show
φ
that the integral inequality Q2 u > 0. So 2.17 is oscillatory for K > 3/41 3λe/88/3 π −
9/55/9 44/9 by Theorem 2.2.
Acknowledgment
This research was partially supported by the NSF of China Grants nos. 11171178 and
11271225 and Science and Technology Project of High Schools of Shandong Province Grant
no. J12LI52.
References
1 W.-T. Li and S. S. Cheng, “An oscillation criterion for nonhomogeneous half-linear differential
equations,” Applied Mathematics Letters, vol. 15, no. 3, pp. 259–263, 2002.
2 Á. Elbert, “A half-linear second order differential equation,” in Qualitative Theory of Differential
Equations, Szeged, Ed., vol. 30 of Colloquia Mathematica Societatis János Bolyai, pp. 153–180, NorthHolland, Amsterdam, The Netherlands, 1979.
3 W. Leighton, “Comparison theorems for linear differential equations of second order,” Proceedings of
the American Mathematical Society, vol. 13, pp. 603–610, 1962.
4 J. Jaroš and T. Kusano, “A Picone type identity for second order half-linear differential equations,”
Acta Mathematica Universitatis Comenianae, vol. 68, no. 1, pp. 137–151, 1999.
5 H. J. Li and C. C. Yeh, “Sturmian comparison theorem for half-linear second-order differential
equations,” Proceedings of the Royal Society of Edinburgh A, vol. 125, no. 6, pp. 1193–1204, 1995.
6 J. V. Manojlović, “Oscillation criteria for second-order half-linear differential equations,” Mathematical
and Computer Modelling, vol. 30, no. 5-6, pp. 109–119, 1999.
10
Discrete Dynamics in Nature and Society
7 J. S. W. Wong, “Oscillation criteria for a forced second-order linear differential equation,” Journal of
Mathematical Analysis and Applications, vol. 231, no. 1, pp. 235–240, 1999.
8 V. Komkov, “A generalization of Leighton’s variational theorem,” Applicable Analysis, vol. 2, pp. 377–
383, 1972.
9 J. Shao, “A new oscillation criterion for forced second-order quasilinear differential equations,”
Discrete Dynamics in Nature and Society, vol. 2011, Article ID 428976, 8 pages, 2011.
10 Z. Zheng, X. Wang, and H. Han, “Oscillation criteria for forced second order differential equations
with mixed nonlinearities,” Applied Mathematics Letters, vol. 22, no. 7, pp. 1096–1101, 2009.
11 J. Jaroš, T. Kusano, and N. Yoshida, “Forced superlinear oscillations via Picone’s identity,” Acta
Mathematica Universitatis Comenianae, vol. 69, no. 1, pp. 107–113, 2000.
12 J. Jaroš, T. Kusano, and N. Yoshida, “Generalized Picone’s formula and forced oscillations in
quasilinear differential equations of the second order,” Archivum Mathematicum, vol. 38, no. 1, pp.
53–59, 2002.
13 O. Došlý, Half-Linear Differential Equations, vol. 202 of North-Holland Mathematics Studies, NorthHolland, Amsterdam, The Netherlands, 2005.
14 G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Mathematical Library, Cambridge
University Press, Cambridge, UK, 2nd edition, 1988.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014