Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 235749, 14 pages
doi:10.1155/2010/235749
Research Article
Uniqueness of Periodic Solution for a Class of
Liénard p-Laplacian Equations
Fengjuan Cao,1 Zhenlai Han,1, 2 Ping Zhao,2 and Shurong Sun1, 3
1
School of Science, University of Jinan, Jinan, Shandong 250022, China
School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China
3
Department of Mathematics and Statistics, Missouri University of Science and Technology,
Rolla, MO 65409-0020, USA
2
Correspondence should be addressed to Zhenlai Han, hanzhenlai@163.com
Received 31 December 2009; Accepted 23 February 2010
Academic Editor: Gaston Mandata N’Guerekata
Copyright q 2010 Fengjuan Cao 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.
By topological degree theory and some analysis skills, we consider a class of generalized Liénard
type p-Laplacian equations. Upon some suitable assumptions, the existence and uniqueness of
periodic solutions for the generalized Liénard type p-Laplacian differential equations are obtained.
It is significant that the nonlinear term contains two variables.
1. Introduction
As it is well known, the existence of periodic and almost periodic solutions is the
most attracting topics in the qualitative theory of differential equations due to their vast
applications in physics, mathematical biology, control theory, and others. More general
equations and systems involving periodic boundary conditions have also been considered.
Especially, the existence of periodic solutions for the Duffing equation, Rayleigh equation,
and Liénard type equation, which are derived from many fields, such as fluid mechanics and
nonlinear elastic mechanics, has received a lot of attention.
Many experts and scholars, such as Manásevich, Mawhin, Gaines, Cheung, Ren,
Ge, Lu, and Yu, have contributed a series of existence results to the periodicity theory
of differential equations. Fixed point theory, Mawhin’s continuation theorem, upper and
lower solutions method, and coincidence degree theory are the common tools to study
the periodicity theory of differential equations. Among these approaches, the Mawhin’s
continuation theorem seems to be a very powerful tool to deal with these problems.
Some contributions on periodic solutions to differential equations have been made
in 1–13. Recently, periodic problems involving the scalar p-Laplacian were studied by
2
Advances in Difference Equations
many authors. We mention the works by Manásevich and Mawhin 3 and Cheung and Ren
4, 8, 10.
In 3, Manásevich and Mawhin investigated the existence of periodic solutions to the
boundary value problem
f t, u, u ,
φ u
u0 uT ,
u 0 u T ,
1.1
where the function φ : RN → RN is quite general and satisfies some monotonicity conditions
which ensure that φ is homeomorphism onto RN . Applying Leray-Schauder degree theory,
the authors brought us the widely used Manásevich-Mawhin continuation theorem. When
φ φp : R → R is the so-called one-dimensional p-Laplacian operator given by φp s |s|p−2 s.
Recently, by Mawhin’s continuation theorem, Cheung and Ren studied the existence
of T -periodic solutions for a p-Laplacian Liénard equation with a deviating argument in 4
as follows:
ϕp x t fxtx t gxt − τt et,
1.2
and two results Theorems 3.1 and 3.2 on the existence of periodic solutions were obtained.
Ge and Ren 5 promoted Mawhin’s continuation theorem to the case which involved
the quasilinear operator successfully; this also prepared conditions for using Mawhin’s
continuation theorem to solve nonlinear boundary value problem.
Liu 7 has dealt with the existence and uniqueness of T -periodic solutions of the
Liénard type p-Laplacian differential equation of the form
ϕp x t fxtx t gxt et
1.3
by using topological degree theory, and one sufficient condition for the existence and
uniqueness of T -periodic solutions of this equation was established.
The aim of this paper is to study the existence of periodic solutions to a class of pLaplacian Liénard equations as follows:
ϕp x t ft, xtx t gt, xt et,
1.4
0, ϕp 0 0, f ∈ C2 R2 , R,
where p > 2, ϕp : R → R is given by ϕp s |s|p−2 s for s /
1
2
g ∈ C R , R and T -periodic in the first variable, where T > 0 is a given constant, e ∈ CR, R,
and et T et.
The paper is organized as follows. In Section 2, we give the definition of norm in
Banach space and the main lemma. In Section 3, combining Lemma 2.1 with some analysis
skills, two sufficient conditions about the existence of solutions for 1.4 are obtained. The
nonlinear terms f and g contain two variables in this paper, which is seldom considered in
the other papers, and the results are new.
Advances in Difference Equations
3
2. Preliminary Results
For convenience, we define
CT1 : x ∈ C1 R, R; x is T -periodic ,
2.1
and the norm · is defined by x max{|x|∞ , |x |∞ }, for all x,
x max x t,
∞
|x|∞ max |xt|,
t∈0,T t∈0,T |x|k T
1/k
k
|xt| dt
.
2.2
0
Clearly, CT1 is a Banach space endowed with such norm.
For the periodic boundary value problem
ϕp x t f
t, x, x ,
x0 xT ,
x 0 x T ,
2.3
where f
is a continuous function and T -periodic in the first variable, we have the following
result.
Lemma 2.1 see 3. Let Ω be an open bounded set in CT1 . If the following conditions hold:
i for each λ ∈ 0, 1, the problem
ϕp x t λf
t, x, x ,
x0 xT ,
x 0 x T 2.4
has no solution on ∂Ω,
ii the equation
1
Fa :
T
T
a, 0dt 0
ft,
2.5
0
has no solution on ∂Ω ∩ R,
iii the Brouwer degree of F is deg{F, Ω ∩ R, 0} /
0,
then the periodic boundary value problem 2.3 has at least one T -periodic solution on Ω.
Set
Ψt, xt xt
ft, sds,
0
yt ϕ x t Ψt, xt.
2.6
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Advances in Difference Equations
We can rewrite 1.4 in the following form:
q−1
x t yt − Ψt, xt sgn yt − Ψt, xt ,
y t xt
0
2.7
∂
ft, sds − gt, xt et,
∂t
where 1 < q < 2 and 1/p 1/q 1.
Lemma 2.2. Suppose the following condition holds:
A1 ∂f/∂t − ∂g/∂s|t,s > 0, for all t ∈ R.
Then 1.4 has at most one T -periodic solution.
Proof. Let x1 t and x2 t be two T -periodic solutions of 1.4. Then, from 2.7, we obtain
q−1
xi t yi t − Ψt, xi t sgn yi t − Ψt, xi t ,
yi t xi t
0
∂
ft, sds − gt, xi t et,
∂t
i 1, 2.
2.8
Set
vt x1 t − x2 t,
ut y1 t − y2 t.
2.9
Then it follows from 2.8 that
q−1
v t y1 t − Ψt, x1 t sgn y1 t − Ψt, x1 t
q−1
− y2 t − Ψt, x2 t sgn y2 t − Ψt, x2 t ,
u t x1 t x2 t
2.10
∂f ∂g
−
t, sds.
∂t ∂s
We claim that ut ≤ 0 for all t ∈ R. By way of contradiction, in view of u ∈ C2 0, T and ut T ut for all t ∈ R, we obtain maxt∈R ut > 0. Then there must exist t∗ ∈ R; for
convenience, we can choose t∗ ∈ 0, T such that
ut∗ max ut max ut > 0,
t∈0,T t∈R
2.11
Advances in Difference Equations
5
which implies that
∗
u t ∗
u t x2 t x1 t
x
1 t
x2 t
∂f ∂g
−
t, sds
∂t ∂s
0,
tt∗
∂
ft, sds ∂t
tt∗
∂
∂
∂
∂
gt, x1 t −
gt, x1 tx1 t − gt, x2 t −
gt, x2 tx2 t ≤ 0.
∂t
∂x1
∂t
∂x2
tt∗
2.12
Set
Δ1 {t : t ∈ 0, T , x1 t /
x2 t},
Δ2 {t : t ∈ 0, T , x1 t x2 t}.
2.13
Then, 0, T Δ1 ∪ Δ2 . Since t∗ ∈ 0, T , if t∗ ∈ Δ1 , from the first equation of 2.12, we have
x
2 t
x1 t
∂f ∂g
−
t, sds ∂t ∂s
0,
2.14
tt∗
which contradicts assumption A1 , so t∗ ∈Δ1 ; it implies that t∗ ∈ Δ2 , that is, x1 t∗ x2 t∗ .
Hence we have
∂
∂
∂
∂
ft, x1 t
ft, x2 t ,
gt, x1 t
gt, x2 t ,
∂t
∂t
∂x1
∂x2
tt∗
tt∗
tt∗
tt∗
x t
2
∗
∂
∂
ft, sds ft, x1 t
x1 t − x2 t∗ .
∗
∂t
tt∗
x1 t ∂t
2.15
tt
Substituting 2.15 into the second equation of 2.12, we have
∗
∗
∂
∂
∗
ft, x1 t
gt, x1 t
u t x1 t − x2 t −
x1 t − x2 t∗ ∂t
∂x1
tt∗
tt∗
∗
∂
∂
ft, x1 t −
gt, x1 t x1 t − x2 t∗ ∂t
∂x1
tt∗
∂f
∂g −
∂t ∂x1 t∗ ,x1 t∗ q−1
× y1 t∗ − Ψt∗ , x1 t∗ sgn y1 t∗ − Ψt∗ , x1 t∗ ∗
q−1
−y2 t∗ − Ψt∗ , x2 t∗ sgn y2 t∗ − Ψt∗ , x2 t∗ .
2.16
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Advances in Difference Equations
Noticing A1 and that ∂f/∂t − ∂g/∂x1 |t∗ ,x1 t∗ > 0 and ut∗ y1 t∗ − y2 t∗ , from 2.16,
we know that
u t∗ > 0,
2.17
this contradicts the second equation of 2.12. So we have ut y1 t − y2 t ≤ 0, for all t ∈ R.
By using a similar argument, we can also show that
y2 t − y1 t ≤ 0,
∀t ∈ R.
2.18
Then, from 2.10 we obtain
x2 t x1 t
∂f ∂g
−
t, sds ≡ 0.
∂t ∂x
2.19
For every t ∈ 0, T , if t ∈ Δ1 , then it contradicts 2.19, so Δ1 ∅; it implies that
0, T Δ2 , then x1 t ≡ x2 t, for all t ∈ 0, T .
Hence, 1.4 has at most one T -periodic solution. The proof of Lemma 2.2 is completed
now.
3. Main Results
Theorem 3.1. Let (A1 ) hold. Suppose that there exists a positive constant d such that
A2 xgt, x − et < 0, for all |x| > d,
A3 limu → ∞ |ft, u/up−1 | r > 0.
Then 1.4 has one unique T -periodic solution if 1/2p−1 r ελT p−1 < 1.
Proof. Consider the homotopic equation of 1.4 as follows:
ϕp x t λft, xtx t λgt, xt λet,
λ ∈ 0, 1.
3.1
By Lemma 2.2, combining A1 , it is easy to see that 1.4 has at least one T -periodic
solution. For the remainder, we will apply Lemma 2.1 to study 3.1. Firstly, we will verify
that all the possible T -periodic solutions of 3.1 are bounded.
Let x ∈ CT1 be an arbitrary solution of 3.1 with period T . By integrating the two sides
of 3.1 from 0 to T and x 0 x T , we obtain
T
0
ft, xtx tdt T
0
gt, xt − et dt 0.
3.2
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7
Consider x0 xT and x ∈ CT1 , there exists t0 ∈ 0, T such that x t0 0, while for
ϕp 0 0 we see that
T
T
t ϕp x t ft, xt x t dt λ gt, xtetdt,
ϕ x t ds ≤ λ
t0 p
0
0
3.3
where t ∈ t0 , t0 T . Let t be the global maximum point of xt on 0, T . Then as x t 0,
we claim that
p−2 ϕp x t
x t x t
≤ 0.
3.4
Otherwise, we have ϕp x t > 0, there must exist a constant ε > 0 such that ϕp x t |x t|p−2 x t > 0, for t ∈ t−ε, tε; therefore, ϕp x t is strictly increasing for t ∈ t−ε, tε,
which implies that x t is strictly increasing for t ∈ t − ε, t ε. Thus, 3.4 is true. Then
g t, x t − e t ≥ 0.
3.5
In view of A2 , 3.5 implies that xt ≤ d; similar to the global minimum point of xt
on 0, T . Since xt ∈ CT1 , it follows that there exists a constant ξ ∈ 0, T such that |xξ| ≤ d.
Then we have
t
t
|xt| xξ x sds ≤ d x sds, t ∈ ξ, ξ T ,
ξ
ξ
ξ
ξ
x sds, t ∈ ξ, ξ T .
x sds ≤ d |xt| |xt − T | xξ −
t−T
t−T
3.6
Combining the above two inequalities, we obtain
|x|∞ max |xt| max |xt|
t∈0,T t∈ξ,ξT 1
d
2
≤ max
t∈ξ,ξT 1
≤d
2
T
t
x sds ξ
ξ
x sds
t−T
3.7
x sds.
0
Considering A3 , there exist constants d1 and the sufficiently small ε > 0 such that
ft, xt ≤ r ε|xt|p−1 ,
when |xt| > d1 .
3.8
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Advances in Difference Equations
Set
Δ1 {t : t ∈ 0, T , |xt| > d1 },
Δ2 {t : t ∈ 0, T , |xt| ≤ d1 },
E1 {t : t ∈ 0, T , |xt| > d},
E2 {t : t ∈ 0, T , |xt| ≤ d}.
3.9
From 3.7, we have
T
ft, xtx txtdt 0
Δ1
ft, xtx txtdt ≤ fA |xt|∞
T
T
0
x tdt
0
1
≤ r ε d 2
dfA
ft, xtx txtdt
x tdt r ε|xt|p−1
∞
T
Δ2
T
p−1 T
x tdt
0
x tdt 1 fA
2
0
T
2
x tdt
0
x tdt,
0
3.10
where fA max{|ft, xt| : t ∈ Δ2 }. Combining the classical inequality 1xp ≤ 1p 1x,
when x ∈ 0, hp, where hp is a constant, since
1
d
2
T
p−1
x tdt
0
⎞p−1
p−1 ⎛
T
2d
1 ⎝1 ⎠ ,
x t dt
T 2 0
t|dt
|x
0
3.11
then we consider the following two cases.
Case 1. If 2d/
T
0
|x t|dt ≥ hp, then
T
0
|x t|dt ≤ 2d/hp. Combining 3.7, we know that
T
x sds ≤ d 2d
,
h p
0
⎞p
p T
p ⎛
2d
1 T 1 ⎠ ,
x t dt
x t dt ⎝1 T
d
2 0
2 0
|x t|dt
|x|∞ ≤ d 1
2
3.12
0
when 2d/
T
0
|x t|dt ≥ hp, then we have
Case 2. When 0 < 2d/
d
1
2
T
0
T
0
T
0
|x t|dt ≤ 2d/hp.
|x t|dt < hp, then from the above classical inequality, we obtain
p−1
x tdt
⎞
p−1 ⎛
T
2dp
1 ⎝1 ⎠.
x t dt
T 2 0
t|dt
|x
0
3.13
Advances in Difference Equations
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Substituting the above inequality into 3.10, we get
T
T
1
ft, xtx txtdt ≤
2p−1
0
r ε
p
x tdt
1
2p−2
0
1
fA
2
T
x tdt
2
dfA
T
0
dpr ε
T
p−1
x tdt
0
x tdt.
0
3.14
Since xt is T -periodic, multiplying xt by 3.1 and then integrating from 0 to T , in
view of A2 , we have
T
p
x t dt −
0
T
ϕp x t xtdt
0
λ
T
ft, xtx txtdt λ
0
λ
T
T
gt, xtxtdt − λ
T
etxtdt
0
ft, xtx txtdt λ
T
0
3.15
0
gt, xt − et xtdt.
0
Substituting 3.14 into 3.15 and since
T
1/r
r
|xt| dt
≤T
s−r/s
T
0
1/s
s
xt| dt
,
0 < r ≤ s,
3.16
0
we obtain
T
p
x t dt −
T
0
ϕp x t xtdt
0
rε
≤ p−1 λT p−1
2
T
0
T
p−1/p
p
p
2
1
p−1
/p
x t dt x t dt
λT
dpr ε
2p−2
0
T
T
2/p
1/p
p
p
1 2p−1/p
p−1/p
x t dt
x t dt
λT
fA
λT
dfA
2
0
0
T
1 T x t dt
λ d
max gt, xt − et, t ∈ E2 dt
2 0
0
≤ Q1
T
p
x t dt Q2
T
0
Q4
0
T
p−1/p
p
x t dt
Q3
T
2/p
p
x t dt
0
1/p
p
x t dt
Q5 ,
0
3.17
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Advances in Difference Equations
where
Q1 1
r ελT p−1 ,
2p−1
2
1
1
λdpr εT p−1 /p ,
Q3 λfA T 2p−1/p ,
2
2p−2
T
1
Q4 λdfA T p−1/p λT p−1/p
max gt, xt − et, t ∈ E2 dt,
2
0
T
max gt, xt − et, t ∈ E2 dt.
Q5 λd
Q2 3.18
0
Since p > 2 and 1/2p−1 r ελT p−1 < 1, from 3.17, we know that there exists a
T
constant M1 such that 0 |x t|p dt ≤ M1 . Then,
T
p
x tdt
≤T
p−1
T
0
p
x t dt ≤ T p−1 M1 : N1 .
3.19
0
So, there exists a constant such that
|xt| ≤ M2 ,
x t ≤ M2 .
3.20
Set
Ω x ∈ CT1 : |x|∞ ≤ M2 1, x ∞ ≤ M2 1 .
3.21
Then 3.1 has no solution on ∂Ω as λ ∈ 0, 1, and when xt ∈ ∂Ω ∩ R, xt M2 1 or
xt −M2 − 1; from A2 , we can see that
1 T
1 T
−gt, M2 1 et dt −
gt, M2 1 − et dt > 0,
T 0
T 0
T
1
1 T
−gt, −M2 − 1 et dt −
gt, −M2 − 1 − et dt < 0,
T 0
T 0
3.22
so condition ii holds.
Set
1
H x, μ μx 1 − μ
T
T
−gt, x et dt.
3.23
0
Then, when x ∈ ∂Ω ∩ R, μ ∈ 0, 1, we have
1
xH x, μ μx − 1 − μ x
T
2
T
0
gt, x − et dt > 0,
3.24
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11
thus Hx, μ is a homotopic transformation and
1
deg{F, Ω ∩ R, 0} deg −
T
T
gt, x − et dt, Ω ∩ R, 0
0
3.25
deg{x, Ω ∩ R, 0} / 0.
So condition iii holds. In view of Lemma 2.1, there exists at least one solution with period
T . This completes the proof.
Theorem 3.2. Let (A1 ) hold. Suppose that there exist positive constants d1 and d2 satisfying the
following conditions:
A4 − sgnxgt, x > |e|∞ , when |x| > d1 ;
A5 ft, xt > 0, t ∈ R;
A6 |gt, x/x| ≤ l, when |x| > d1 .
Then for 1.4 there exists one unique T -periodic solution when σ > lT.
Proof. We can rewrite 3.1 in the following from:
x1 t ϕq x2 t,
x2 t −λft, x1 tx1 t − λgt, x1 t λet.
3.26
Let x1 t, x2 t ∈ CT1 be a T -periodic solution of 3.26, then x1 t must be a T -periodic
solution of 3.1. First we claim that there is a constant ξ ∈ R such that
|x1 ξ| ≤ d.
3.27
Take t0 , t1 as the global maximum point and global minimum point of xt on 0, T ,
respectively, then
x1 t0 x1 t1 0,
x1 t0 max x1 t max x1 t.
t∈R
t∈0,T 3.28
From the first equation of 3.26 we have x2 t ϕp x1 t, so x2 t0 ϕp x1 t0 0 x2 t1 .
We claim that
x2 t0 ≤ 0.
3.29
By way of contradiction, 3.29 does not hold, then x2 t0 > 0. So there exists ε > 0
such that x2 t > 0, for t ∈ t0 − ε, t0 ε; therefore, x2 t > x2 t0 0, for t ∈ t0 , t0 ε, so
x1 t ϕq x2 t > 0, t ∈ t0 , t0 ε, that is, x1 t > x1 t0 , for t ∈ t0 , t0 ε. This contradicts
the definition of t0 , so we have x2 t0 ≤ 0.
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Advances in Difference Equations
Substituting t0 into the second equation of 3.26, we obtain
gt0 , x1 t0 − et0 ≥ 0.
3.30
By condition A4 , we have x1 t0 ≤ d. Similarly, we get x1 t1 ≥ −d.
Case 1. If x1 t1 ≤ d, define ξ t1 . Obviously |x1 ξ| ≤ d.
Case 2. If x1 t1 ≥ d, from the fact that x1 is a continuous function in R, there exists a constant
ξ between t0 and t1 such that |x1 ξ| d.
So we have that 3.27 holds. Next, in view of ξ ∈ R, there are integer k and constant
t∗ ∈ 0, T such that ξ kT t∗ , hence |x1 ξ| |x1 t∗ | ≤ d.
So
T
|x1 t| ≤ d x sds.
3.31
0
We claim that all the periodic solutions x of 3.1 are bounded and |xt| ≤ d Let
E1 {t : t ∈ 0, T , |xt| > d1 },
T
0
|x s|ds.
E2 {t : t ∈ 0, T , |xt| ≤ d1 }.
3.32
Multiplying both sides of 3.1 by x t and integrating from 0 to T, together with A5 and
A6 , we have
T
σ
2
x t dt ≤
0
T
2
ft, xtx t dt
0
≤
T
T
gt, xtx tdt 0
0
gt, xtx tdt ≤
E1
≤l
|et|x tdt
T
0
gt, xtx tdt |xt|x tdt |e|∞
≤l d
T
T
x tdt
0
ld gA |e|∞
x tdt gA
0
0
T
x tdt
0
x tdt gA |e|
∞
0
T
x tdt lT
|et|x tdt
0
E2
T
T
T
0
2
x t dt,
T
0
x tdt
3.33
Advances in Difference Equations
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where σ min{ft, xt}, and gA max{|gt, xt| : |xt| ≤ d1 }. That is,
T
σ − lT 2
x t dt ≤ ld gA |e|
∞
0
T
x tdt.
3.34
0
Using Hölder’s inequality and σ > lT, we have
1
T
T
2
x tdt
0
≤
T
0
2
x t dt ≤ ld |e|∞ gA
σ − lT
T
x tdt,
3.35
0
so
T
x tdt ≤ T ld |e|0 gA : M;
σ − lT
0
3.36
there must be a positive constant M1 > M such that
T
0
x tdt ≤ M1 ,
T
|xt|dt ≤ M1 ;
3.37
0
hence together with 3.31, we have |x1 |∞ ≤ d M1 : M2 .
This proves the claim and that the rest of the proof of the theorem is identical to that
of Theorem 3.1.
Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful
comments that have led to the present improved version of the original manuscript. This
research is supported by the Natural Science Foundation of China 60774004, 60904024,
China Postdoctoral Science Foundation Funded Project 20080441126, 200902564, Shandong
Postdoctoral Funded Project 200802018, and the Natural Scientific Foundation of Shandong
Province Y2008A28, ZR2009AL003 and is also supported by University of Jinan Research
Funds for Doctors XBS0843.
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