Uniqueness of Periodic Solution for a Class of Liénard Laplacian Equations

doi:10.1155/2010/235749

Research Article

Uniqueness of Periodic Solution for a Class of

Li ´enard

p-Laplacian Equations

Fengjuan Cao,

_{Zhenlai Han,}

_{Ping Zhao,}

_{and Shurong Sun}

1_{School of Science, University of Jinan, Jinan, Shandong 250022, China}

2_{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

Correspondence should be addressed to Zhenlai Han,[email protected]

Received 31 December 2009; Accepted 23 February 2010

Academic Editor: Gaston Mandata N’Guerekata

Copyrightq2010 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 Li´enard typep-Laplacian equations. Upon some suitable assumptions, the existence and uniqueness of periodic solutions for the generalized Li´enard typep-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 Li´enard 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 Man´asevich, 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.

many authors. We mention the works by Man´asevich and Mawhin3and Cheung and Ren

4,8,10.

In3, Man´asevich and Mawhin investigated the existence of periodic solutions to the boundary value problem

φuft, u, u, u0 uT, u0 uT, 1.1

where the functionφ:RN _{→ R}N_{is quite general and satisfies some monotonicity conditions}

which ensure thatφis homeomorphism ontoRN_{.Applying Leray-Schauder degree theory,}

the authors brought us the widely used Man´asevich-Mawhin continuation theorem. When

φ φp : R → R is the so-called one-dimensional p-Laplacian operator given byφps |s|p−2_s.

Recently, by Mawhin’s continuation theorem, Cheung and Ren studied the existence ofT-periodic solutions for ap-Laplacian Li´enard equation with a deviating argument in4 as follows:

ϕp

xt fxtxt gxt−τt et, 1.2

and two resultsTheorems3.1and3.2on the existence of periodic solutions were obtained. Ge and Ren5promoted 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 Li´enard typep-Laplacian differential equation of the form

ϕp

xt fxtxt gxt et 1.3

by using topological degree theory, and one sufficient condition for the existence and uniqueness ofT-periodic solutions of this equation was established.

The aim of this paper is to study the existence of periodic solutions to a class ofp -Laplacian Li´enard equations as follows:

ϕp

xt ft, xtxt gt, xt et, 1.4

where p > 2, ϕp : R → R is given by ϕps |s|p−2s fors /0, ϕp0 0,f ∈ C2R2, R,

g∈C1_R2_{, RandT-periodic in the first variable, whereT >0 is a given constant,e∈CR, R,} andet T et.

2. Preliminary Results

For convenience, we define

C1_T :x∈C1R, R; xisT-periodic, 2.1

and the norm · is defined byxmax{|x|∞,|x|∞}, for allx,

|x|_∞ max

t∈0,T|xt|, x

∞_tmax_∈0,Txt, |x|k

T

0

|xt|k_dt 1/k

. 2.2

Clearly,C1_Tis a Banach space endowed with such norm. For the periodic boundary value problem

ϕp

xtft, x, x, x0 xT, x0 xT, 2.3

wherefis a continuous function andT-periodic in the first variable, we have the following result.

Lemma 2.1see3. LetΩbe an open bounded set inC1

T. If the following conditions hold:

ifor eachλ∈0,1,the problem

ϕp

xtλft, x, x, x0 xT, x0 xT 2.4

has no solution on∂Ω,

iithe equation

Fa: 1

T

T

0

ft, a,0dt0 2.5

has no solution on∂Ω∩R,

iiithe Brouwer degree ofFisdeg{F,Ω∩R,0}/0,

then the periodic boundary value problem2.3has at least oneT-periodic solution onΩ.

Set

Ψt, xt

_xt

0

We can rewrite1.4in the following form:

xt yt−Ψt, xtq−1sgnyt−Ψt, xt,

yt

_xt

0

∂

∂tft, sds−gt, xt et,

2.7

where 1< q <2 and 1/p 1/q1.

Lemma 2.2. Suppose the following condition holds:

A1∂f/∂t−∂g/∂s|t,s>0, for allt∈R.

Then1.4has at most oneT-periodic solution.

Proof. Letx1tandx2tbe twoT-periodic solutions of1.4. Then, from2.7, we obtain

x_it yit−Ψt, xitq−1sgn

yit−Ψt, xit

,

y_it

_xit

0

∂

∂tft, sds−gt, xit et,

i1,2. 2.8

Set

vt x1t−x2t, ut y1t−y2t. 2.9

Then it follows from2.8that

vt y1t−Ψt, x1tq−1sgn

y1t−Ψt, x1t

−y2t−Ψt, x2tq−1sgn

y2t−Ψt, x2t

,

ut

_x1t

x2t

_∂f

∂t −

∂g ∂s

t, sds.

2.10

We claim thatut ≤0 for allt∈ R.By way of contradiction, in view ofu∈C2_{0, T} andut T utfor allt∈R, we obtain maxt∈Rut> 0.Then there must existt∗ ∈R; for

convenience, we can chooset∗∈0, Tsuch that

ut∗ _max

which implies that

Substituting2.15into the second equation of2.12, we have

NoticingA1and that∂f/∂t−∂g/∂x1|t∗_,x1t∗ >0 andut∗ y₁t∗−y₂t∗, from2.16,

we know that

ut∗>0, 2.17

this contradicts the second equation of2.12. So we haveut y1t−y2t≤0, for allt∈R. By using a similar argument, we can also show that

y2t−y1t≤0, ∀t∈R. 2.18

Then, from2.10we obtain

_x2t

x1t

_∂f

∂t −

∂g ∂x

t, sds≡0. 2.19

For everyt ∈ 0, T,if t ∈ Δ1,then it contradicts2.19, soΔ1 ∅; it implies that

0, T Δ2,thenx1t≡x2t, for allt∈0, T.

Hence,1.4has at most oneT-periodic solution. The proof ofLemma 2.2is completed now.

3. Main Results

Theorem 3.1. Let (A1) hold. Suppose that there exists a positive constant d such that

A2xgt, x−et<0, for all|x|> d,

A3limu→ ∞|ft, u/up−1|r >0.

Then1.4has one uniqueT-periodic solution if1/2p−1r ελTp−1<1.

Proof. Consider the homotopic equation of1.4as follows:

ϕp

xt λft, xtxt λgt, xt λet, λ∈0,1. 3.1

ByLemma 2.2, combiningA1,it is easy to see that1.4has at least oneT-periodic solution. For the remainder, we will applyLemma 2.1to study3.1. Firstly, we will verify that all the possibleT-periodic solutions of3.1are bounded.

Letx∈C1

Tbe an arbitrary solution of3.1with periodT. By integrating the two sides

of3.1from 0 toTandx0 xT,we obtain

_T

0

ft, xtx_tdt T

0

Consider x0 xT and x ∈ C_T1, there exists t0 ∈ 0, T such that xt0 0,while for

Combining the above two inequalities, we obtain

|x|_∞ max

ConsideringA3,there exist constantsd1and the sufficiently smallε >0 such that

_{ft, xt≤r ε|xt|}p−1_{, when|xt|> d}

Set

then we consider the following two cases.

Substituting the above inequality into3.10,we get

SincextisT-periodic, multiplyingxtby3.1and then integrating from 0 toT, in view ofA2,we have

Substituting3.14into3.15and since

where

constantM1such that

T

So, there exists a constant such that

|xt| ≤M2, xt≤M2. 3.20

thusHx, μis a homotopic transformation and

deg{F,Ω∩R,0}deg

−1

T

T

0

gt, x−etdt,Ω∩R,0

deg{x,Ω∩R,0}/0.

3.25

So conditioniiiholds. In view ofLemma 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 constantsd1 and d2 satisfying the following conditions:

A4−sgnxgt, x>|e|∞, when|x|> d1;

A5ft, xt>0,t∈R;

A6|gt, x/x| ≤l, when |x|> d1.

Then for1.4there exists one uniqueT-periodic solution whenσ > lT.

Proof. We can rewrite3.1in the following from:

x₁t ϕqx2t,

x₂t −λft, x1tx1t−λgt, x1t λet.

3.26

Letx1t, x2t∈C1_Tbe aT-periodic solution of3.26, thenx1tmust be aT-periodic solution of3.1. First we claim that there is a constantξ∈Rsuch that

|x1ξ| ≤d. 3.27

Taket0, t1as the global maximum point and global minimum point ofxton0, T, respectively, then

x₁t0 x₁t1 0, x1t0 max

t∈R x1t tmax∈0,Tx1t. 3.28

From the first equation of3.26we havex2t ϕpx₁t,sox2t0 ϕpx₁t0 0x2t1. We claim that

x₂t0≤0. 3.29

By way of contradiction,3.29does not hold, thenx₂t0 > 0.So there exists ε > 0 such thatx₂t > 0, fort∈ t0−ε, t0 ε; therefore,x2t > x2t0 0, fort ∈ t0, t0 ε,so

whereσmin{ft, xt},andgA max{|gt, xt|:|xt| ≤d1}.That is,

σ−lT

T

0

_xt2

dt≤ld gA |e|_∞

T

0

_{xtdt. 3.34}

Using H ¨older’s inequality andσ > lT,we have

1

T

T

0

_xtdt 2

≤

T

0

_xt2

dt≤ ld |e|∞ gA

σ−lT

T

0

_{xtdt, 3.35}

so

T

0

_xtdt≤Tld |e|0 gA

σ−lT :M; 3.36

there must be a positive constantM1 > Msuch that

T

0

_xtdt≤M 1,

T

0

|xt|dt≤M1; 3.37

hence together with3.31, we have|x1|∞≤d M1:M2.

This proves the claim and that the rest of the proof of the theorem is identical to that ofTheorem 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 Project20080441126, 200902564, Shandong Postdoctoral Funded Project200802018, and the Natural Scientific Foundation of Shandong ProvinceY2008A28, ZR2009AL003and is also supported by University of Jinan Research Funds for DoctorsXBS0843.

References

1 S. Lu, “Existence of periodic solutions to ap-Laplacian Li´enard differential equation with a deviating argument,”Nonlinear Analysis: Theory, Methods & Applications, vol. 68, no. 6, pp. 1453–1461, 2008.

2 B. Xiao and B. Liu, “Periodic solutions for Rayleigh typep-Laplacian equation with a deviating argument,”Nonlinear Analysis: Real World Applications, vol. 10, no. 1, pp. 16–22, 2009.

3 R. Man´asevich and J. Mawhin, “Periodic solutions for nonlinear systems with p-Laplacian-like operators,”Journal of Differential Equations, vol. 145, no. 2, pp. 367–393, 1998.

4 W.-S. Cheung and J. Ren, “On the existence of periodic solutions forp-Laplacian generalized Li´enard equation,”Nonlinear Analysis: Theory, Methods & Applications, vol. 60, no. 1, pp. 65–75, 2005.

6 F. Zhang and Y. Li, “Existence and uniqueness of periodic solutions for a kind of Duffing typep -Laplacian equation,”Nonlinear Analysis: Real World Applications, vol. 9, no. 3, pp. 985–989, 2008.

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 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.

9 R. E. Gaines and J. L. Mawhin,Coincidence Degree, and Nonlinear Differential Equations, vol. 568 of

Lecture Notes in Mathematics, Springer, Berlin, Germany, 1977.

10 W.-S. Cheung and J. Ren, “Periodic solutions forp-Laplacian Rayleigh equations,”Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 10, pp. 2003–2012, 2006.

11 S. Peng and S. Zhu, “Periodic solutions for p-Laplacian Rayleigh equations with a deviating argument,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 1, pp. 138–146, 2007.

12 F. Cao, Z. Han, and S. Sun, “Existence of periodic solutions forp-Laplacian equations on time scales,”

Advances in Difference Equations, vol. 2010, Article ID 584375, 13 pages, 2010.