Electronic Journal of Qualitative Theory of Differential Equations 2011, No. 62, 1-7;http://www.math.u-szeged.hu/ejqtde/
A necessary and sufficient condition for existence and uniqueness of periodic solutions for a p-Laplacian
Li´ enard equation
Liehui Zhang
a, Yong Wang
a,b,∗, Junkang Tian
c, Liang Zhang
daState Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, P. R. China;
bSchool of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, P. R. China;
cSchool of Mathematical Sciences, University of Electronic Science and Technology, Chengdu, Sichuan 611731, P. R. China;
dInstitute of Applied Mathematics, College of Science, Northwest A & F University, Yangling, Shaanxi 712100, P. R. China.
Abstract. In this work, we investigate the followingp-Laplacian Li´enard equation:
(ϕp(x′(t)))′+f(x(t))x′(t) +g(x(t)) =e(t).
Under some assumption, a necessary and sufficient condition for the existence and uniqueness of periodic solutions of this equation is given by using Man´asevich–Mawhin continuation theorem. Our results improve and extend some known results.
Keywords: Periodic solution;p-Laplacian; Li´enard equation; Continuation theorem.
AMS 2000 Mathematics Subject Classification: 34C25.
1 Introduction
In this paper, we investigate the existence and uniqueness of periodic solutions of the followingp-Laplacian Li´enard equation
(ϕp(x′(t)))′+f(x(t))x′(t) +g(x(t)) =e(t), (1.1) where p > 1,ϕp : R→ Ris given by ϕp(s) = |s|p−2s fors 6= 0, ϕp(0) = 0, f, g, e∈ C(R,R) ande is T-periodic withT >0. Ifp= 2, (1.1) becomes the following forced Li´enard equation:
x′′(t) +f(x(t))x′(t) +g(x(t)) =e(t). (1.2)
∗Corresponding author; E-mail address: ywangsc@gmail.com (Y. Wang).
Liénard equation
Mechanical problems and bifurcation theory
Stability problems Existence
and uniqueness of limit cycles
Fixed point theory
Signal processing Vibration
and electronic theory
Biological process
Hilbert’s 16th problem
Quadratic systems
Bogdanov-Takens systems
Chemical reactions
FitzHugh’s nerve conduction
Growth of a single species
Gause-type predator-prey systems
Prigogine's trimolecular model
Fermentation model
Figure 1: Diagrammatic representation for applications of the Li´enard equation (1.1).
Generalized Li´enard equations appear in a number of physical models, and the problem concerning the periodic solutions for these equations has been studied extensively by lots of authors; see for example [1-12] and the references therein.
Here we are keen to dispel any perception that the mathematical proofs of existence and uniqueness that we present are merely verifying facts which might already be obvious in other disciplines, based on purely physical considerations. In particular, in many nonlinear problems arising in practical dynamical systems, physical reasoning alone is not sufficient or fully convincing. In these cases questions of existence and uniqueness are of importance in understanding the full range of solution behaviour possible, and represent a genuine mathematical challenge. The answers to these mathematical questions then provide the basis for obtaining the best numerical solutions to these problems, and determining other important practical aspects of the solution behaviour. Figure 1 shows the various applications of the Li´enard equation (1.1).
The main purpose in this work is to give a necessary and sufficient condition for the existence and uniqueness of T-periodic solutions of (1.1) by using Man´asevich–Mawhin continuation theorem. Our results improve and extend some results in [6] (see Remark 2 and Examples 1 and 2).
2 Lemmas
Let us start with some notations. SetCT1 ={x∈C1(R,R) :xisT-periodic}, which is a Banach space endowed with the normk · kdefined bykxk=max{|x|∞,|x′|∞}, and
|x|∞= max
t∈[0,T]|x(t)|,|x′|∞= max
t∈[0,T]|x′(t)|,|x|k = Z T
0
|x(t)|kdt
!1/k
.
For theT-periodic functione, denote
¯ e= 1
T Z T
0
e(t)dt.
The following conditions will be used later:
(H1) g∈C1(R,R) andg′(x)<0 for allx∈R.
(H2) ¯e∈g(R).
Remark 1. Generally,xrefers to the displacement, f(x)x′ refers to the damping term,g(x) refers to the stiffness term anderefers to the forced term in a vibration system. It implies that the stiffness of the vibration system is monotone decreasing with regard to the displacement ifg′(x)<0.
Lemma 1 ( [6]). Suppose(H1)holds. Then(1.1)has at most one T-periodic solution.
Consider the homotopic equation of (1.1):
(ϕp(x′(t)))′+λf(x(t))x′(t) +λg(x(t)) =λe(t), λ∈(0,1). (2.1) We have the following lemma.
Lemma 2. Suppose(H1) and(H2)hold. Then the set ofT-periodic solutions of (2.1)are bounded inCT1.
Proof. LetS⊂CT1 be the set ofT-periodic solutions of (2.1). IfS=∅, the proof is ended. Suppose S6=∅, and letx∈S. Noticing thatx(0) =x(T),x′(0) =x′(T) andϕp(0) = 0, it follows from (2.1) that
Z T
0
(g(x(t))−e(t))dt= 0, which implies that there existsτ ∈[0, T] such that
g(x(τ)) = ¯e.
By (H1), we know that g(x) is strictly decreasing inR. So we have x(τ) =g−1(¯e) =:C.
Then, fort∈[τ, τ+T],
|x(t)|=
x(τ) + Z t
τ
x′(s)ds
≤ |C|+ Z τ+T
τ
|x′(s)|ds=|C|+ Z T
0
|x′(s)|ds, which leads to
|x|∞≤ |C|+|x′|1. (2.2)
By (H2) we haveg(x0) = ¯efor somex0∈R. Letd=|x0|, sinceg(x) is strictly decreasing inR, we get x(g(x)−e)¯ <0 for|x|> d. (2.3) Define E1 ={t :t ∈ [0, T],|x(t)| > d}, E2 ={t :t ∈ [0, T],|x(t)| ≤ d}. Multiplying x(t) and (2.1) and then integrating from 0 toT, by (2.3) we have
Z T
0
|x′(t)|pdt = − Z T
0
(ϕp(x′(t)))′x(t)dt
= λ
Z T
0
(g(x(t))−¯e)x(t)dt−λ Z T
0
(e(t)−e)x(t)dt¯
= λ
Z
E1
(g(x(t))−e)x(t)dt¯ +λ Z
E2
(g(x(t))−¯e)x(t)dt−λ Z T
0
(e(t)−e)x(t)dt¯
≤ λ
Z
E2
(g(x(t))−e)x(t)dt¯ −λ Z T
0
(e(t)−e)x(t)dt¯
≤
|x|≤dmax|g(x)−e|¯ +|e−e|¯∞
T|x|∞.
LetM0=
|x|≤dmax|g(x)−e|¯ +|e−¯e|∞
T. Then we obtain
|x′|p≤M01/p|x|1/p∞ . (2.4)
Letq >1 such that 1/p+ 1/q= 1. Then by the H¨older inequality we have
|x′|1≤ |x′|p|1|q =T1/q|x′|p. (2.5) By (2.2), (2.4) and (2.5), we can get
|x′|1≤T1/qM01/p(|C|+|x′|1)1/p,
which yields that there exists M1 > 0 such that |x′|1 ≤ M1 since p > 1, and this together with (2.2) implies that|x|∞≤ |C|+M1.
Meanwhile, there existst0∈[0, T] such thatx′(t0) = 0 sincex(0) =x(T). Then by (2.1) we have, for t∈[t0, t0+T],
|ϕp(x′(t))| =
Z t
t0
(ϕp(x′(s)))′ds
= λ
Z t
t0
(f(x(s))x′(s) +g(x(s)) +e(s))ds
≤ Z T
0
(|f(x(s))||x′(s)|+|g(x(s))|+|e(s)|)ds
≤ F M1+ (G+|e|∞)T,
whereF = max{|f(x)|:|x| ≤ |C|+M1},G= max{|g(x)|:|x| ≤ |C|+M1}. So we obtain
|x′|∞= max
t∈[0,T]{|ϕp(x′(t))|1/(p−1)} ≤(F M1+ (G+|e|∞)T)1/(p−1).
LetM = max{|C|+M1,(F M1+ (G+|e|∞)T)1/(p−1)}. Thenkxk ≤M. This completes the proof.
For the periodic boundary value problem
(ϕp(x′(t)))′ =h(t, x, x′), x(0) =x(T), x′(0) =x′(T), (2.6) whereh∈C(R3,R) isT-periodic in the first variable. The following continuation theorem can be induced directly from the theory in [9], and is cited as Lemma 1 in [12].
Lemma 3 (Man´asevich–Mawhin [9]). Let B ={x∈ CT1 : kxk < r} for some r > 0. Suppose the following two conditions hold:
(i)For eachλ∈(0,1)the problem(ϕp(x′(t)))′ =λh(t, x, x′)has no solution on∂B.
(ii)The continuous function F defined onRbyF(a) = T1 RT
0 h(t, a,0)dtis such thatF(−r)F(r)<0.
Then the periodic boundary value problem(2.6)has at least one T-periodic solution onB¯.
3 Main results
We are now in the position to give our main result.
Theorem 1. Suppose(H1)holds. Then (1.1)has a unique T-periodic solution if and only if (H2) holds.
Proof. Letx(t) be aT-periodic solution of (1.1). By integrating the two sides of (1.1) from 0 toT, and noticing thatx(0) =x(T) andx′(0) =x′(T), we have
Z T
0
(g(x(t))−e(t))dt= 0.
Then there existsτ ∈[0, T] such that
T g(x(τ)) = Z T
0
e(t)dt, which implies ¯e∈g(R), and the necessity is proved.
On the other hand, by Lemma 2.2, there existsM >0 such that, for any solutionx(t) of (2.1),
||x|| ≤M. (3.1)
Meanwhile, there existsx0∈Rsuch thatg(x0) = ¯esince ¯e∈g(R). Leth(t, x(t), x′(t)) =e(t)−g(x(t))− f(x(t))x′(t) andB ={x∈CT1 :kxk< r} withr= max{M + 1,|x0|+ 1}. Then (3.1) implies that (2.1) has no solution on∂B for allλ∈(0,1), and condition (i) of Lemma 3 is satisfied. Furthermore, we have
g(r)<¯e < g(−r), (3.2)
sinceg(x) is strictly decreasing inR. By the definition ofF in Lemma 3 we get F(a) = 1
T Z T
0
h(t, a,0)dt= 1 T
Z T
0
(e(t)−g(a))dt= ¯e−g(a).
This together with (3.2) yields thatF(r)F(−r)<0, i.e. condition (ii) of Lemma 3 is satisfied. Therefore, it follows from Lemma 3 that there exists aT-periodic solutionx(t) of (1.1). The uniqueness of thisx(t) is guaranteed by Lemma 1. This completes the proof.
Remark 2. In [6], The sufficient conditions for the existence of T-periodic solutions for (1.1) are (H1) and the following condition:
(A2) there exists constantd >0 such thatx(g(x)−e(t))<0 for|x|> d.
Noticing (2.3), it is easy to verify that the condition (H2) is weaker than the condition (A2) since mint∈Re(t)<e <¯ maxt∈Re(t) when e(t)6= constant. So our results improve and extend the main results in [6].
Finally, we close this work by two examples.
Example 1. Consider the following differential equation:
(ϕp(x′(t)))′+f(x(t))x′(t)−arctan(x(t) + 1) = π
4(1 + 3 cost), (3.3)
wheref ∈C(R,R) andp >1.
In this example, g(x) = −arctan(x+ 1), e(t) = π4(1 + 3 cost) and T = 2π. It is obvious that the condition (A1) of theorem 1 in [6] holds. However, it is easy to verify that the condition (A2) does not hold. Therefore, Theorem 1 in [6] fails, while, our criterion in Theorem 1 in this paper remains applicable, as we now show. According to the above arguments, the condition (H1) holds; since ¯e=π4, it is easy to see that the condition (H2) also holds. Hence, Theorem 1 in this paper shows that (3.3) has a unique 2π-periodic solution.
Example 2. Consider the following differential equation:
(ϕp(x′(t)))′+f(x(t))x′(t)−arctan(x(t) + 1) =π(1 + 3 cost), (3.4) wheref ∈C(R,R) andp >1.
In this example, g(x) =−arctan(x+ 1), e(t) = π(1 + 3 cost) and T = 2π. It is obvious that the condition (A1) of theorem 1 in [6] holds. However, it is easy to verify that the condition (A2) does not hold. Therefore, Theorem 1 in [6] fails, while, our criterion in Theorem 1 in this paper remains applicable, as we now show. According to the above arguments, the condition (H1) holds; since ¯e =π, it is easy to see that the condition (H2) does not hold. Hence, Theorem 1 in this paper shows that (3.4) has no 2π-periodic solutions.
4 Acknowledgements
This work was supported by the Sichuan Youth Science and Technology Fund (No. 2011JQ0044), the National Program on Key Basic Research Project (973 Program, Grant No. 2011CB201005), the Open Fund (PLN1003) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University) and the Scientific Research Fund (No. 10ZB113) of Sichuan Provincial Educational Department. The authors are grateful to the editor and referee for their careful reading of the manuscript and helpful suggestions on this work.
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(Received April 8, 2011)
