Volume 2010, Article ID 197263, 11 pagesdoi:10.1155/2010/197263 Research Article Existence and Uniqueness of Periodic Solutions for a Class of Nonlinear Equations with p-Laplacian-Like O
Trang 1Volume 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
Copyrightq 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 ϕxt
f xtxt 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 LiuB 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
xt fxtxt 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 lim s→ ∞α s ∞ and
Trang 2It is obvious that under these two conditions, ϕ is an homeomorphism fromR onto R and is increasing onR.
Recall that p-Laplacian equations have been of great interest for many mathematicians.
Especially, there is a large literaturesee, e.g., 1 7 and references therein about the existence
of periodic solutions to the following p-Laplacian equation:
ϕ p
xt fxtxt gxt et, 1.3
and its variants, where ϕ p s |s| p−2s for s / 0 and ϕ p0 0 Obviously, 1.3 is a special case
of1.1
However, there are seldom results about the existence of periodic solutions to1.1 The main difficulty lies in the p-Laplacian-like operator ϕ of 1.1, which is more complicated
than ϕ pin1.3 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 to1.1 As one will see, our theorem generalizes some recent results even
for the case of ϕs ϕ p s seeRemark 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 normx max{|x|∞, |x|∞}, where
|x|∞ max
t ∈0,T |xt|, x
∞ max
t ∈0,Txt. 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 Carath´eodory Assume that Ω is an open bounded set
in C1
T such that the following conditions hold.
S1 For each λ ∈ 0, 1, the problem
ϕ
xt λ f
t, x, x
, x 0 xT, x0 xT 1.6
has no solution on ∂ Ω.
S2 The equation
F a : 1
T
T
0
has no solution on ∂ Ω ∩ R.
S3 The Brouwer degree
Trang 3Then the periodic boundary value problem
ϕ
xt f
t, x, x
, x 0 xT, x0 xT 1.9
has at least one T-periodic solution on Ω.
2 Main Results
In this section, we prove an existence and uniqueness theorem for1.1
Theorem 2.1 Suppose the following assumptions hold:
A1 g ∈ C1R, R and gx < 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
g x − εt < 0,
T
0
εt − etdt ≤ 0,
T
0
|εt − et|dt < 2. 2.1
Then1.1 has a unique T-periodic solution.
Proof
Existence For the proof of existence, we useLemma 1.1 First, let us consider the homotopic equation of1.1:
ϕ
xt λfxtxt λgxt λet, λ ∈ 0, 1. 2.2
Let xt ∈ C1
T be an arbitrary solution of2.2 By integrating the two sides of 2.2 over0, T, and noticing that x0 xT and x0 xT, we have
T
0
that is,
1
T
T
0
g xtdt e : 1
T
T
0
Since gx· is continuous, there exists t0 ∈ 0, T such that
Trang 4In view ofA1, we obtain
wheree g−1e Then, for each t ∈ t0, t0 T, we have
2xt xt xt − T
xt0
t
t0
xsds xt0 −
t0
t −T xsds
≤ 2xt0
t
t0
xsdst0
t −T
xsds
≤ 2e
T
0
xsds,
2.7
which gives that
|x|∞≤ e 1
2
T
0
Thus,
|x|∞≤ |e| 1
2
T
0
Since lims→ ∞α s ∞, there is a constant M > 0 such that
Set
E1 t : t ∈ 0, T,xt> M
, E2 t : t ∈ 0, T,xt ≤ M ,
F1 {t : t ∈ 0, T, |xt| > r}, F2 {t : t ∈ 0, T, |xt| ≤ r}. 2.11
In view ofA2 and
Trang 5we get
T
0
xtdt
E1
xtdt
E2
xtdt
≤
E1
xtdt MT
≤
E1
ϕ xtxt
α |xt| dt MT
≤
E1
ϕ
xtxtdt MT
≤
T
0
ϕ
xtxtdt MT
T
0
ϕ
xtdx t MT
−
T
0
ϕ
xtx tdt MT
λ
T
0
g xt − et x tdt λ
T
0
f xtxtxtdt MT
λ
T
0
g xt − et x tdt MT
λ
F1
g xt − εt x tdt λ
F2
g xt − εt x tdt
λ
T
0
εt − etxtdt MT
≤
F2
g xt − εt · |xt|dt MT T
0
|εt − et|dt · |x|∞
≤ MT MT
T
0
|εt − et|dt · |x|∞,
2.13
where
M max
|x|≤rg x max
t ∈0,T |εt|
By2.9, we have
|x|∞≤ |e| M MT
2
T
0
|εt − et|dt · |x|∞. 2.15
Trang 6Noticing that
T
0
there exists a constant M > |e| such that
On the other hand, it follows from
ϕ x · |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
xt1 0 Thus ϕxt1 0
Then, for all t ∈ E : {t ∈ 0, T : xt / 0}, we have
αxt ≤ ϕxt
t
t1
ϕ
xsds
≤
t
t1
f xs · xsds
T
0
g xsdsT
0
|es|ds
≤
x t
x t1 f udu
|x|≤Mmaxg x max
t ∈0,T |et|
· T
≤
M
−M
f udu max
|x|≤Mg x max
t ∈0,T |et|
· T
≤ max
|x|≤Mf x · 2M max
|x|≤Mg x max
t ∈0,T |et|
· T : M.
2.19
For the above M, it follows from lims→ ∞α s ∞ that there exists G > Msuch that
Combining this with α|xt| ≤ M, we get
xt< G, t ∈ E, 2.21
which yields that|x|∞< G.
Now, we have proved that any solution xt ∈ C1
T of2.2 satisfies
|x|∞< G, x
∞< G. 2.22
Trang 7Since G > |e|, we have
G > e g−1e, −G < e g−1e. 2.23
In view of g being strictly decreasing, we get
Set
Ω x ∈ C1
T :|x|∞≤ G,x
Then, we know that2.2 has no solution on ∂Ω for each λ ∈ 0, 1, that is, the assumption
S1 ofLemma 1.1holds In addition, it follows from2.24 that
−1
T
T
0
g G − et dt e − gG > 0,
−1
T
T
0
g −G − et dt e − g−G < 0.
2.26
So the assumptionS2 ofLemma 1.1holds Let
H
x, μ
μx −1− μ 1
T
T
0
For x ∈ ∂Ω ∩ R and μ ∈ 0, 1, by 2.24, we have
xH
x, μ
μx2−1− μx1
T
T
0
g x − et dt
μx21− μx1
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 assumptionS3 ofLemma 1.1holds By applyingLemma 1.1, there exists at least
one solution with period T to1.1
Trang 8Uniqueness Let
ψ x
x
0
f udu, y t ϕxt ψxt. 2.30 Then1.1 is transformed into
xt ϕ−1
y t − ψxt ,
Let x1t and x2t being two T-periodic solutions of 1.1; and
y i t ϕxi t ψx i t, i 1, 2. 2.32 Then we obtain
xi t ϕ−1
y i t − ψx i t ,
yi t −gx i t et i 1, 2. 2.33
Setting
v t x1t − x2t, u t y1t − y2t, 2.34
it follows from2.33 that
vt ϕ−1
y1t − ψx1t − ϕ−1
y2t − ψx2t ,
ut − g x1t − gx2t 2.35 Now, we claim that
If this is not true, we consider the following two cases
Case 1 There exists t2 ∈ 0, T such that
u t2 max
t ∈0,T u t max
which implies that
ut2 − g x1t2 − gx2t2 0,
ut2 − g x1t − gx2t |t t − gx1t2x
1t2 − gx2t2x
2t2 ≤ 0. 2.38
Trang 9ByA1, gx < 0 So it follows from gx1t2 − gx2t2 0 that x1t2 x2t2 Thus, in view of
−gx1t2 > 0, ut2 y1t2 − y2t2 > 0, 2.39 andH1, we obtain
ut2 −gx1t2 x1t2 − x
2t2
−gx1t2ϕ−1
y1 t2 − ψx1t2 − ϕ−1
y2 t2 − ψx2t2
−gx1t2ϕ−1
y1t2 − ψx1t2 − ϕ−1
y2t2 − ψx1t2 > 0,
2.40
which contradicts with ut2 ≤ 0
Case 2.
u0 max
t ∈0,T u t max
Also, we have u0 0 and u0 ≤ 0 Then, similar to the proof of Case 1, one can get a contradiction
Now, we have proved that
Analogously, one can show that
So we have ut ≡ 0 Then, it follows from 2.35 that
g x1t − gx2t ≡ 0, ∀t ∈ R, 2.44 which implies that
Hence,1.1 has a unique T-periodic solution The proof ofTheorem 2.1is now completed
Remark 2.2 InTheorem 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
Trang 10In the case ϕs ϕ p s, Liu 7, Theorem 1 proved that 1.1 has a unique T-periodic solution
under the assumptionsA1 and A2 Thus, even for the case of ϕs ϕ p s,Theorem 2.1
is a generalization of7, 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
g α − et dt ≤ 0,
T
0
g α − etdt < 2 2.47
hold Then1.1 has a unique T-periodic solution.
Proof Let ε t ≡ gα Noticing that
x
g x − gα < 0, |x| > α, 2.48
we know thatA2 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
xt fxtxt gxt et, 2.49
where ϕx 2xe x2
− 2x, fx e −x , gx −x3 x, and et sin t One can easily check that ϕ satisfyH1 and H2 Obviously, A1 holds Moreover, since
lim
x→ ∞g x −∞, lim
it is easy to verify thatA2 holds ByTheorem 2.1,2.49 has a unique 2π-periodic solution.
Example 2.5 Consider the following p-Laplacian equation:
ϕ p
xt gxt et, 2.51
where gx −1/2 arctan x, and et sin2t Obviously,A1 holds Moreover, we have
π
0
g 0 − et dt ≤ 0,
π
0
g 0 − etdtπ
0
sin2t π
2 < 2. 2.52
SoA2 holds Then, byCorollary 2.3,2.51 has a unique π-periodic solution.
Trang 11Remark 2.6 InExample 2.5,∀r > 0, we have
x
g x − e π
2
> r
1−π 4
> 0, ∀x < −r. 2.53
Thus,A2 does not hold So 7, Theorem 1 cannot be applied toExample 2.5 This means that our results generalize7, Theorem 1 in essence even for the case of ϕs ϕp s.
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 DepartmentGJJ09456, and the Youth Foundation of Jiangxi Normal University
References
1 S Lu, “New results on the existence of periodic solutions to a p-Laplacian differential equation with
a deviating argument,” Journal of Mathematical Analysis and Applications, vol 336, no 2, pp 1107–1123,
2007
2 S Lu, “Existence of periodic solutions to a p-Laplacian Li´enard differential equation with a deviating argument,” Nonlinear Analysis: Theory, Methods & Applications, vol 68, no 6, pp 1453–1461, 2008.
3 B Liu, “Periodic solutions for Li´enard type p-Laplacian equation with a deviating argument,” Journal
of Computational and Applied Mathematics, vol 214, no 1, pp 13–18, 2008.
4 W.-S Cheung and J Ren, “Periodic solutions for p-Laplacian Li´enard equation with a deviating argument,” Nonlinear Analysis: Theory, Methods & Applications, vol 59, no 1-2, pp 107–120, 2004.
5 W.-S Cheung and J Ren, “Periodic solutions for p-Laplacian differential equation with multiple deviating arguments,” Nonlinear Analysis: Theory, Methods & Applications, vol 62, no 4, pp 727–742,
2005
6 W.-S Cheung and J Ren, “Periodic solutions for p-Laplacian Rayleigh equations,” Nonlinear Analysis:
Theory, Methods & Applications, vol 65, no 10, pp 2003–2012, 2006.
7 B Liu, “Existence and uniqueness of periodic solutions for a kind of Li´enard type p-Laplacian equation,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 2, pp 724–729, 2008.
8 R Man´asevich and J Mawhin, “Periodic solutions for nonlinear systems with p-Laplacian-like operators,” Journal of Di fferential Equations, vol 145, no 2, pp 367–393, 1998.
... “New results on the existence of periodic solutions to a p-Laplacian differential equation with< /i>a deviating argument,” Journal of Mathematical Analysis and Applications, vol 336, no...
2 S Lu, ? ?Existence of periodic solutions to a p-Laplacian Li´enard differential equation with a deviating argument,” Nonlinear Analysis: Theory, Methods & Applications, vol 68, no...
7 B Liu, ? ?Existence and uniqueness of periodic solutions for a kind of Li´enard type p-Laplacian equation,” Nonlinear Analysis: Theory, Methods & Applications, vol 69, no 2, pp