We investigate the following nonlinear equations with p-Laplacian-like operators (&#x03C6;(x&#x2032;(t)))&#x2032;+f(x(t))x&#x2032;(t)+g(x(t))=e(t): some criteria to guarantee the existence and uniqueness of periodic solutions of the above equation are given by using Mawhin&#39;s continuation theorem. Our results are new and extend some recent results due to Liu (B. Liu, Existence and uniqueness of periodic solutions for a kind of Lienard type p-Laplacian equation, Nonlinear Analysis TMA, 69, 724&#x2013;729, 2008)

Guo-Rong Ye

Hui-Sheng Ding

Wei Long

Advances in Difference Equations

Springer - Publisher Connector

Hindawi Publishing Corporation
Advances in Diﬀerence Equations
Volume 2010, Article ID 197263, 11 pages
doi:10.1155/2010/197263
Research Article
Existence and Uniqueness of Periodic
Solutions for a Class of Nonlinear Equations with
p-Laplacian-Like Operators
Hui-Sheng Ding, Guo-Rong Ye, and Wei Long
College of Mathematics and Information Science, Jiangxi Normal University, Nanchang,
Jiangxi 330022, China
Correspondence should be addressed to Wei Long, hopelw@126.com
Received 1 February 2010; Accepted 19 March 2010
Academic Editor: Gaston Mandata N’Guerekata
Copyright q 2010 Hui-Sheng Ding 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.
We investigate the following nonlinear equations with p-Laplacian-like operators ϕx′t~~′ 
fxt~~x′t  gxt~~  et: some criteria to guarantee the existence and uniqueness of periodic
solutions of the above equation are given by using Mawhin’s continuation theorem. Our results
are new and extend some recent results due to Liu B. Liu, Existence and uniqueness of periodic
solutions for a kind of Lienard type p-Laplacian equation, Nonlinear Analysis TMA, 69, 724–729,
2008.
1. Introduction
In this paper, we deal with the existence and uniqueness of periodic solutions for the
following nonlinear equations with p-Laplacian-like operators:
(
ϕ
(
x′t
))′  fxt~~x′t  gxt~~  et, 1.1
where f , g are continuous functions on R, and e is a continuous function on R with period
T > 0; moreover, ϕ : R → R is a continuous function satisfying the following:
H1 for any x1, x2 ∈ R, x1 /x2, ϕx1 − ϕx2 · x1 − x2 > 0 and ϕ0  0;
H2 there exists a function α : 0,∞ → 0,∞ such that lims→∞αs  ∞ and
ϕx · x ≥ α|x||x|, ∀x ∈ R. 1.2
2 Advances in Diﬀerence Equations
It is obvious that under these two conditions, ϕ is an homeomorphism from R onto R and is
increasing on R.
Recall that p-Laplacian equations have been of great interest for manymathematicians.
Especially, there is a large literature see, e.g., 1–7 and references therein about the existence
of periodic solutions to the following p-Laplacian equation:
(
ϕp
(
x′t
))′  fxt~~x′t  gxt~~  et, 1.3
and its variants, where ϕps  |s|p−2s for s / 0 and ϕp0  0. Obviously, 1.3 is a special case
of 1.1.
However, there are seldom results about the existence of periodic solutions to 1.1.
The main diﬃculty lies in the p-Laplacian-like operator ϕ of 1.1, which is more complicated
than ϕp in 1.3. Since there is no concrete form for the p-Laplacian-like operator ϕ of 1.1, it
is more diﬃcult to prove the existence of periodic solutions to 1.1.
Therefore, in this paper, we will devote ourselves to investigate the existence of
periodic solutions to 1.1. As one will see, our theorem generalizes some recent results even
for the case of ϕs  ϕps see Remark 2.2.
Next, let us recall some notations and basic results. For convenience, we denote
C1T :
{
x ∈ C1R,R : x is T -periodic
}
, 1.4
which is a Banach space endowed with the norm ‖x‖  max{|x|∞, |x′|∞}, where
|x|∞  max
t∈0,T
|xt|, ∣∣x′∣∣∞  maxt∈0,T
∣∣x′t
∣∣. 1.5
In the proof of our main results, we will need the following classical Mawhin’s
continuation theorem.
Lemma 1.1 8. Let (H1), (H2) hold and f˜ is Carathe´odory. Assume thatΩ is an open bounded set
in C1T such that the following conditions hold.
S1 For each λ ∈ 0, 1, the problem
(
ϕ
(
x′t
))′  λf˜
(
t, x, x′
)
, x0  xT, x′0  x′T 1.6
has no solution on ∂Ω.
S2 The equation
Fa :
1
T
∫T
0
f˜t, a, 0dt  0 1.7
has no solution on ∂Ω ∩ R.
S3 The Brouwer degree
degF,Ω ∩ R, 0/ 0. 1.8
Advances in Diﬀerence Equations 3
Then the periodic boundary value problem
(
ϕ
(
x′t
))′  f˜
(
t, x, x′
)
, x0  xT, x′0  x′T 1.9
has at least one T -periodic solution on Ω.
2. Main Results
In this section, we prove an existence and uniqueness theorem for 1.1.
Theorem 2.1. Suppose the following assumptions hold:
A1 g ∈ C1R,R and g ′x < 0 for all x ∈ R;
A2 there exist a constant r ≥ 0 and a function εt ∈ CR,R such that for all t ∈ R and
|x| > r,
x
[
gx − εt] < 0,
∫T
0
εt − etdt ≤ 0,
∫T
0
|εt − et|dt < 2.
2.1
Then 1.1 has a unique T -periodic solution.
Proof
Existence. For the proof of existence, we use Lemma 1.1. First, let us consider the homotopic
equation of 1.1:
(
ϕ
(
x′t
))′  λfxt~~x′t  λgxt~~  λet, λ ∈ 0, 1. 2.2
Let xt ∈ C1T be an arbitrary solution of 2.2. By integrating the two sides of 2.2
over 0, T, and noticing that x′0  x′T and x0  xT, we have
∫T
0
[
gxt~~ − et]dt  0, 2.3
that is,
1
T
∫T
0
gxt~~dt  e :
1
T
∫T
0
etdt. 2.4
Since gx·~~ is continuous, there exists t0 ∈ 0, T such that
gxt0 ≥ e. 2.5
4 Advances in Diﬀerence Equations
In view of A1, we obtain
xt0 ≤ e˜, 2.6
where e˜  g−1e. Then, for each t ∈ t0, t0  T, we have
2xt  xt  xt − T
 xt0 
∫ t
t0
x′sds  xt0 −
∫ t0
t−T
x′sds
≤ 2xt0 
∫ t
t0
∣
∣x′s
∣
∣ds 
∫ t0
t−T
∣
∣x′s
∣
∣ds
≤ 2e˜ 
∫T
0
∣∣x′s
∣∣ds,
2.7
which gives that
|x|∞ ≤ e˜ 
1
2
∫T
0
∣∣x′s
∣∣ds. 2.8
Thus,
|x|∞ ≤ |e˜| 
1
2
∫T
0
∣∣x′s
∣∣ds. 2.9
Since lims→∞αs  ∞, there is a constantM > 0 such that
αs ≥ 1, ∀s ≥ M. 2.10
Set
E1 
{
t : t ∈ 0, T, ∣∣x′t∣∣ > M}, E2 
{
t : t ∈ 0, T, ∣∣x′t∣∣ ≤ M},
F1  {t : t ∈ 0, T, |xt| > r}, F2  {t : t ∈ 0, T, |xt| ≤ r}.
2.11
In view of A2 and
ϕx · x ≥ α|x||x|, ∀x ∈ R, 2.12
Advances in Diﬀerence Equations 5
we get
∫T
0
∣
∣x′t
∣
∣dt 
∫
E1
∣
∣x′t
∣
∣dt 
∫
E2
∣
∣x′t
∣
∣dt
≤
∫
E1
∣
∣x′t
∣
∣dt MT
≤
∫
E1
ϕx′tx′t
α|x′t| dt MT
≤
∫
E1
ϕ
(
x′t
)
x′tdt MT
≤
∫T
0
ϕ
(
x′t
)
x′tdt MT

∫T
0
ϕ
(
x′t
)
dxt MT
 −
∫T
0
(
ϕ
(
x′t
))′
xtdt MT
 λ
∫T
0
[
gxt~~ − et]xtdt  λ
∫T
0
fxt~~x′txtdt MT
 λ
∫T
0
[
gxt~~ − et]xtdt MT
 λ
∫
F1
[
gxt~~ − εt]xtdt  λ
∫
F2
[
gxt − εt]xtdt
 λ
∫T
0
εt − etxtdt MT
≤
∫
F2
∣∣gxt~~ − εt∣∣ · |xt|dt MT 
∫T
0
|εt − et|dt · |x|∞
≤ M′T MT 
∫T
0
|εt − et|dt · |x|∞,
2.13
where
M′ 
[
max
|x|≤r
∣∣gx
∣∣  max
t∈0,T
|εt|
]
· r. 2.14
By 2.9, we have
|x|∞ ≤ |e˜| 
M′ MT
2

1
2
∫T
0
|εt − et|dt · |x|∞. 2.15
6 Advances in Diﬀerence Equations
Noticing that
∫T
0
|εt − et|dt < 2, 2.16
there exists a constantM′′ > |e˜| such that
|x|∞ ≤ M′′. 2.17
On the other hand, it follows from
∣
∣ϕx
∣
∣ · |x|  ϕx · x ≥ α|x||x|, ∀x ∈ R, 2.18
that α|x| ≤ |ϕx| for x / 0. In addition, since x0  xT, there exists t1 ∈ 0, T such that
x′t1  0. Thus ϕx′t1~~  0.
Then, for all t ∈ E : {t ∈ 0, T : x′t/ 0}, we have
α
(∣∣x′t
∣∣) ≤ ∣∣ϕ(x′t)∣∣

∣∣∣∣∣
∫ t
t1
(
ϕ
(
x′s
))′
ds
∣∣∣∣∣
≤
∣∣∣∣∣
∫ t
t1
fxs · x′sds
∣∣∣∣∣

∫T
0
∣∣gxs~~
∣∣ds 
∫T
0
|es|ds
≤
∣∣∣∣∣
∫xt
xt1
fudu
∣∣∣∣∣

[
max
|x|≤M′′
∣∣gx
∣∣  max
t∈0,T
|et|
]
· T
≤
∫M′′
−M′′
∣∣fu
∣∣du 
[
max
|x|≤M′′
∣∣gx
∣∣  max
t∈0,T
|et|
]
· T
≤ max
|x|≤M′′
∣∣fx
∣∣ · 2M′′ 
[
max
|x|≤M′′ ′
∣∣gx
∣∣  max
t∈0,T
|et|
]
· T : M′′′.
2.19
For the aboveM′′′, it follows from lims→∞αs  ∞ that there exists G > M′′ such that
αs > M′′′, s ≥ G. 2.20
Combining this with α|x′t| ≤ M′′′, we get
∣∣x′t
∣∣ < G, t ∈ E, 2.21
which yields that |x′|∞ < G.
Now, we have proved that any solution xt ∈ C1T of 2.2 satisfies
|x|∞ < G,
∣∣x′
∣∣
∞ < G. 2.22
Advances in Diﬀerence Equations 7
Since G > |e˜|, we have
G > e˜  g−1e, −G < e˜  g−1e. 2.23
In view of g being strictly decreasing, we get
gG < e, g−G > e 2.24
Set
Ω 
{
x ∈ C1T : |x|∞ ≤ G,
∣
∣x′
∣
∣
∞ ≤ G
}
. 2.25
Then, we know that 2.2 has no solution on ∂Ω for each λ ∈ 0, 1, that is, the assumption
S1 of Lemma 1.1 holds. In addition, it follows from 2.24 that
− 1
T
∫T
0
[
gG − et]dt  e − gG > 0,
− 1
T
∫T
0
[
g−G − et]dt  e − g−G < 0.
2.26
So the assumption S2 of Lemma 1.1 holds. Let
H
(
x, μ
)
 μx − (1 − μ) 1
T
∫T
0
[
gx − et]dt. 2.27
For x ∈ ∂Ω ∩ R and μ ∈ 0, 1, by 2.24, we have
xH
(
x, μ
)
 μx2 − (1 − μ)x 1
T
∫T
0
[
gx − et]dt
 μx2 
(
1 − μ)x 1
T
∫T
0
[
e − gx]dt > 0.
2.28
Thus,Hx, μ is a homotopic transformation. So
degF,Ω ∩ R, 0  degHx, 0,Ω ∩ R, 0
 degHx, 1,Ω ∩ R, 0
 degI,Ω ∩ R, 0/ 0,
2.29
that is, the assumption S3 of Lemma 1.1 holds. By applying Lemma 1.1, there exists at least
one solution with period T to 1.1.
8 Advances in Diﬀerence Equations
Uniqueness. Let
ψx 
∫x
0
fudu, yt  ϕ
(
x′t
)
 ψxt~~. 2.30
Then 1.1 is transformed into
x′t  ϕ−1
[
yt − ψxt],
y′t  −gxt~~  et.
2.31
Let x1t and x2t being two T -periodic solutions of 1.1; and
yit  ϕ
(
x′it
)
 ψxit~~, i  1, 2. 2.32
Then we obtain
x′it  ϕ−1
[
yit − ψxit~~
]
,
y′it  −gxit~~  et.
i  1, 2. 2.33
Setting
vt  x1t − x2t, ut  y1t − y2t, 2.34
it follows from 2.33 that
v′t  ϕ−1
[
y1t − ψx1t~~
] − ϕ−1[y2t − ψx2t~~
]
,
u′t  −[gx1t~~ − gx2t~~
]
.
2.35
Now, we claim that
ut ≤ 0, ∀t ∈ R. 2.36
If this is not true, we consider the following two cases.
Case 1. There exists t2 ∈ 0, T such that
ut2  max
t∈0,T
ut  max
t∈R
ut > 0, 2.37
which implies that
u′t2  −
[
gx1t2~~ − gx2t2~~
]
 0,
u′′t2  −
[
gx1t~~ − gx2t~~
]′|tt2  −
[
g ′x1t2~~x′1t2 − g ′x2t2~~x′2t2
] ≤ 0.
2.38
Advances in Diﬀerence Equations 9
By A1, g ′x < 0. So it follows from gx1t2~~ − gx2t2~~  0 that x1t2  x2t2. Thus, in
view of
−g ′x1t2~~ > 0, ut2  y1t2 − y2t2 > 0, 2.39
and H1, we obtain
u′′t2  −g ′x1t2~~
[
x′1t2 − x′2t2
]
 −g ′x1t2~~
{
ϕ−1
[
y1t2 − ψx1t2~~
] − ϕ−1[y2t2 − ψx2t2~~
]}
 −g ′x1t2~~
{
ϕ−1
[
y1t2 − ψx1t2~~
] − ϕ−1[y2t2 − ψx1t2~~
]}
> 0,
2.40
which contradicts with u′′t2 ≤ 0.
Case 2.
u0  max
t∈0,T
ut  max
t∈R
ut > 0. 2.41
Also, we have u′0  0 and u′′0 ≤ 0. Then, similar to the proof of Case 1, one can get a
contradiction.
Now, we have proved that
ut ≤ 0, ∀t ∈ R. 2.42
Analogously, one can show that
ut ≥ 0, ∀t ∈ R. 2.43
So we have ut ≡ 0. Then, it follows from 2.35 that
gx1t~~ − gx2t~~ ≡ 0, ∀t ∈ R, 2.44
which implies that
x2t ≡ x1t, ∀t ∈ R. 2.45
Hence, 1.1 has a unique T -periodic solution. The proof of Theorem 2.1 is now completed.
Remark 2.2. In Theorem 2.1, setting εt ≡ et, then A2 becomes as follows:
A2′ there exists a constant r ≥ 0 such that for all t ∈ R and |x| > r,
x
[
gx − et] < 0. 2.46
10 Advances in Diﬀerence Equations
In the case ϕs  ϕps, Liu 7, Theorem 1 proved that 1.1 has a unique T -periodic solution
under the assumptions A1 and A2′. Thus, even for the case of ϕs  ϕps, Theorem 2.1
is a generalization of 7, Theorem 1.
In addition, we have the following interesting corollary.
Corollary 2.3. Suppose A1 and
A2′′ there exist a constant α ≥ 0 such that
∫T
0
[
gα − et]dt ≤ 0,
∫T
0
∣
∣gα − et∣∣dt < 2 2.47
hold. Then 1.1 has a unique T -periodic solution.
Proof. Let εt ≡ gα. Noticing that
x
[
gx − gα] < 0, |x| > α, 2.48
we know that A2 holds with r  α. This completes the proof.
At last, we give two examples to illustrate our results.
Example 2.4. Consider the following nonlinear equation:
(
ϕ
(
x′t
))′  fxtx′t  gxt~~  et, 2.49
where ϕx  2xex
2 − 2x, fx  e−x, gx  −x3  x, and et  sin t. One can easily check
that ϕ satisfy H1 and H2. Obviously, A1 holds. Moreover, since
lim
x→∞
gx  −∞, lim
x→−∞
gx  ∞, 2.50
it is easy to verify that A2 holds. By Theorem 2.1, 2.49 has a unique 2π-periodic solution.
Example 2.5. Consider the following p-Laplacian equation:
(
ϕp
(
x′t
))′  gxt~~  et, 2.51
where gx  −1/2 arctanx, and et  sin2t. Obviously, A1 holds. Moreover, we have
∫π
0
[
g0 − et]dt ≤ 0,
∫π
0
∣∣g0 − et∣∣dt 
∫π
0
sin2t 
π
2
< 2. 2.52
So A2′′ holds. Then, by Corollary 2.3, 2.51 has a unique π-periodic solution.
Advances in Diﬀerence Equations 11
Remark 2.6. In Example 2.5, ∀r > 0, we have
x
(
gx − e
(π
2
))
> r
(
1 − π
4
)
> 0, ∀x < −r. 2.53
Thus, A2′ does not hold. So 7, Theorem 1 cannot be applied to Example 2.5. This means
that our results generalize 7, Theorem 1 in essence even for the case of ϕs  ϕps.
Acknowledgments
The authors are grateful to the referee for valuable suggestions and comments, which
improved the quality of this paper. The work was supported by the NSF of China, the NSF
of Jiangxi Province of China 2008GQS0057, the Youth Foundation of Jiangxi Provincial
Education Department GJJ09456, and the Youth Foundation of Jiangxi Normal University.
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Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with p-Laplacian-Like Operators

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Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with p-Laplacian-Like Operators

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