# Some Results for Periodic Solutions of a Kind of Liénard Equation

Authors: Yong Wang; Xiangyi Yi
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We investigate the following Liénard-typep-Laplacian equation with a deviating argument (φp(x′t))′+f(xt)x′t+βtgxt-τt=e(t). Some new criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation are given by using the Manásevich-Mawhin continuation theorem and some analysis techniques. Our results hold under weaker conditions than some known results from the literature and are more effective. In the last section, an illustrative example is provided to demonstrate the applications of the theoretical results.


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## Body

## 1. Introduction

In this present paper, we consider the following Liénard-typep-Laplacian equation with a deviating argument:
(1)φpx′t′+fxtx′(t)+β(t)g(x(t-τ(t)))=e(t),
where p>1, φp:R→R, and φp(s)=|s|p-2s is a one-dimensional p-Laplacian; f,e∈C(R,R), β,τ,g∈C1(R,R), and β(t) and τ(t) are two T-periodic functions with ∫0T‍e(t)dt=0, T>0.As is known, the Liénard equation can be derived from many fields, such as physics, mechanics, and engineering technique fields, and an important question is whether this equation can support periodic solutions. In the past few years, a lot of researchers have contributed to the theory of this equation with respect to the existence of periodic solutions. For example, in 1928, Liénard [1] discussed the existence of periodic solutions of the following equation:
(2)x′′(t)+fxtx′(t)+kxtx(t)=0,
where f,k∈C(R,R); some sufficient conditions for securing the existence of periodic solutions were established. Afterwards, Levinson and Smith [2] also studied (2) and obtained some new results on the existence of periodic solutions. In 1977, some continuation theorems in [3] were introduced by Gaines and Mawhin. Applying these continuation theorems, many authors discussed the existence of periodic solutions of (2) and generalized the results obtained in [1, 2] (see, e.g., [4–7]); a few authors studied the existence and uniqueness of periodic solutions of (2) (see [8, 9]). In 1998, Manásevich and Mawhin [10] studied periodic solutions for certain nonlinear systems with p-Laplacian-like operators and provided some new continuation theorems which extended some results in [3]. Subsequently, some authors discussed the existence of periodic solutions of certain Liénard-type p-Laplacian equations (see, e.g., [11–15]) using these generalized continuation theorems. However, as far as we know, there exist much fewer results on the existence and uniqueness of periodic solutions of (1). The main difficulty lies in the first term (φp(x′(t)))′ of (1) (i.e., the p-Laplacian operator φp:R→R; φp(s)=|s|p-2s is nonlinear when p≠2), the existence of which prevents the usual methods of finding some criteria for guaranteeing the uniqueness of periodic solutions of (2) from working. Recently, Gao and Lu [16] discussed the existence and uniqueness of periodic solution of (1) by translating (1) into a two-dimensional system and got some results as follows.Theorem 1 (see [16]).
Assume that the following condition holds:(H0)
β(t)>0, g′(x)<0, and τ(t)≡ɛ (ɛ being sufficiently small constant) for all t,x∈R.
Then (1) has at most one T-periodic solution.Remark 2.
However, upon examining their proof of Theorem1, it was found that if τ(t)≠0∀t∈R, then Theorem 1 does not hold; more precisely, for arbitrarily given ɛ>0, v(t*)=y1(t*)-y2(t*)>0 does not positively imply v(t*-ɛ)=y1(t*-ɛ)-y2(t*-ɛ)>0; thus, in line 3 on page 377 in [16], the inequality v′′(t*)>0 is incorrect. On the other hand, if τ(t)≡0, then Theorem 1 is correct.Theorem 3 (see [16]).
Assume that the following conditions hold.(H1)
There existr1>0, r2>0, m>0, and d≥0 such that
(i)
r1|u|m≤|g(u)|≤r2|u|m for all |u|>d,(ii)
ug(u)<0 for all |u|>d.(H2)
Consider(3)A:=r2Tr1∫0T‍β(t)+1dt1/m2(1-m)/m<1,0<m≤1,r2Tr1∫0T‍β(t)+1dt1/m<1,m>1.(H3)
Suppose one of the following conditions holds:
(i)
m=p-1 and β∞r2Tm+2-(m+1/p)/2(1-A)m+1<1,(ii)
m<p-1,whereβ∞=max⁡t∈[0,T]|β(t)|.
Then (1) has at least one T-periodic solution.Remark 4.
However, upon examining their proof of Theorem 3.2 in [16], we have found that the conditions (H1)(i), (H2), and (H3) can be dropped.In this paper, 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 for determining other important practical aspects of the solution behaviour.We now reconsider the periodic solutions of (1). The main purpose of this paper is to establish some new criteria for guaranteeing the existence and uniqueness of periodic solution of (1). We obtain some new sufficient conditions for securing the existence and uniqueness of periodic solutions of (1) by using the Manásevich-Mawhin continuation theorem and some analysis techniques. Our results extend and improve the above-mentioned Theorems 3.1 and 3.2 in [16] (see Remarks 9 and 13 and Example 1).


## 2. Lemmas

For convenience, define(4)x∞=max⁡t∈0,Txt,x′∞=max⁡t∈[0,T]x′t,xk=∫0T‍xtkdt1/k.
Let
(5)CT1:={x∈C1(R,R):xisT-periodic},
which is a Banach space with the norm
(6)x=max⁡x∞,x′∞.
The following conditions will be used later: (A0)
β(t)>0, g′(x)<0, and τ(t)≡0, for all t,x∈R,(A0′)
β(t)<0, g′(x)>0, and τ(t)≡0, for all t,x∈R,(A1)
there existsd≥0, such that ug(u)<0 for all |u|≥d,(A1′)
there existsd≥0, such that ug(u)>0 for all |u|≥d.For the periodic boundary value problem(7)φpx′t′=h(t,x,x′),x(0)=x(T),x′(0)=x′(T),
where h∈C(R3,R) is T-periodic in the first variable, the following continuation theorem can be induced directly from the theory in [10] and is cited as Lemma 1 in [17].Lemma 5 (Manásevich and Mawhin [10]).
LetB={x∈CT1:x<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 onR by F(a)=(1/T)∫0T‍h(t,a,0)dt is such that F(-r)F(r)<0.
Then the periodic boundary value problem (7) has at least one T-periodic solution on B¯.According to Theorem 3.1 in [16] and the above-mentioned Remark 2, we have the following results.Lemma 6.
Suppose (A0) holds. Then (1) has at most one T-periodic solution.Lemma 7.
Suppose (A0′) holds. Then (1) has at most one T-periodic solution.


## 3. Main Results

Now we are in the position to present our main results.Theorem 8.
Suppose (A1) holds. Then (1) has at least one T-periodic solution.Proof.
Consider the homotopic equation of (1):
(8)φpx′t′+λfxtx′(t)+λβtgxt-τt=λe(t),λ∈(0,1).
First, we prove that the set of T-periodic solutions of (8) are bounded in CT1. Let S⊂CT1 be the set of T-periodic solutions of (8). If S=∅, the proof is ended. Suppose S≠∅, and let x∈S. Noticing that x(0)=x(T), x′(0)=x′(T), φp(0)=0, and ∫0T‍e(t)dt=0, it follows from (8) that
(9)∫0T‍βtgxt-τtdt=0,
which, together with β(t)>0, implies that there exists t0∈[0,T] such that
(10)gxt0-τt0=0.
Denoting t¯0=t0-τ(t0), by (A1), (10) implies
(11)xt¯0<d.
Then, for any t∈[t¯0,t¯0+T],
(12)xt=x(t¯0)+∫t¯0t‍x′(s)ds<d+∫t¯0t¯0+T‍x′sds=d+∫0T‍x′sds,
which leads to
(13)x∞=max⁡t∈[t¯0,t¯0+T]xt<d+x′1.
Define E1={t:t∈[0,T],|x(t-τ(t))|>d} and E2={t:t∈[0,T],|x(t-τ(t))|≤d}. Multiplying x(t) and (8) and then integrating from 0 to T, by (A1), we have(14)∫0T‍x′tpdt=-∫0T‍φpx′t′x(t)dt=λ∫0T‍βtgxt-τtxtdt-λ∫0T‍etxtdt=λ∫E1‍βtgxt-τtxtdt+λ∫E2‍βtgxt-τtxtdt-λ∫0T‍etxtdt≤λ∫E2‍βtgxt-τtxtdt-λ∫0T‍etxtdt≤∫E2‍βtgxt-τtxtdt+∫0Tetxtdt≤max⁡t∈[0,T],|x|≤dβtgx+e∞Tx∞.
Let M0=max⁡t∈[0,T],|x|≤d⁡|β(t)g(x)|+|e|∞T. Then we obtain
(15)x′p≤M01/px∞1/p.
Let q>1 such that 1/p+1/q=1. Then by Hölder inequality we have
(16)x′1≤x′p1q=T1/qx′p.
By (13), (15), and (16), we can get
(17)x′1≤T1/qM01/pd+x′11/p,
which yields that there exists M1>0 such that |x′|1<M1 since p>1, and this, together with (13), implies that |x|∞<d+M1.
Meanwhile, there existst^0∈[0,T] such that x′(t^0)=0 since x(0)=x(T). Then by (8) we have, for t∈[t^0,t^0+T],
(18)φpx′t=∫t^0t‍φpx′s′ds=λ∫t^0t‍fxsx′s+βsgxs-τs+esds≤∫0Tfxsx′s+βsgxs-τshhhhhhhfxsx′s+βsgxs-τs+esds<FM1+(G+e∞)T,
where F=max⁡{|f(x)|:|x|≤d+M1} and G=max⁡{|β(t)g(x)|:t∈[0,T],|x|≤d+M1}. So we obtain
(19)x′∞=max⁡t∈0,Tφpx′t1/(p-1)<FM1+G+e∞T1/(p-1).
Let M=max⁡{d+M1,(FM1+(G+|e|∞)T)1/(p-1)}. Then x<M.
Second, we prove the existence ofT-periodic solutions of (1). Set
(20)h(t,x(t),x′(t))=-fxtx′t-βtgxt-τt+et;
then (8) is equivalent to the following equation:
(21)φpx′t′=λh(t,x(t),x′(t)),λ∈(0,1).
Set
(22)B=x:x∈CT1,x<r,wherer≥M;
by (20), we know that (21) has no solution on ∂B as λ∈(0,1), so condition (i) of Lemma 5 is satisfied. By the definition of F in Lemma 5 we get
(23)Fa=1T∫0T‍h(t,a,0)dt=1T∫0T‍(e(t)-β(t)g(a))dt=-1T∫0T‍β(t)g(a)dt.
This, together with β(t)>0 for all t∈R and (A1), yields that F(r)F(-r)<0; that is, condition (ii) of Lemma 5 is satisfied. Therefore, it follows from Lemma 5 that there exists a T-periodic solution x(t) of (1). This completes the proof.Remark 9.
It is easy to see that Theorem8 in this study holds under weaker conditions than Theorem 3.2 in [16].According to the above discussion, we can also get the following result.Theorem 10.
Suppose (A1′) holds. Then (1) has at least one T-periodic solution.Together with Lemmas6 and 7 and Theorems 8 and 10, we can directly lead to two theorems as follows.Theorem 11.
Suppose (A0) and (A1) hold. Then (1) has a unique T-periodic solution.Theorem 12.
Suppose (A0′) and (A1′) hold. Then (1) has a unique T-periodic solution.


## 4. Example and Remark

In this section, we apply the main results obtained in the previous sections to an example.Example 1.
Consider the existence and uniqueness of a2π-periodic solution of the following Liénard-type p-Laplacian equation:
(24)φpx′t′+fxtx′(t)+β(t)g(x(t))=e(t),
where p>1, f∈C(R,R), β(t)=1+cos⁡2t, g(x)=-x3-2x, e(t)=cos⁡t, and T=2π.Proof.
Ifp<4, condition (H3) in Theorem 3.3 in [16] does not hold any more since m=3>p-1. Therefore, Theorem 3.3 in [16] fails, while our criterion in Theorem 11 in this study remains applicable, as we now show. Let d be an arbitrary positive constant; then we can easily check that conditions (A0) and (A1) in Theorem 11 in this study hold. Hence, Theorem 11 shows that there exists a unique 2π-periodic solution of (24).Remark 13.
This example demonstrates that the conditions in our Theorem11 are weaker than those conditions in Theorem 3.3 in [16] when τ(t)≡0 and are able to demonstrate the existence of a unique periodic solution to certain Liénard-type p-Laplacian equations, which the latter cannot decide about. Therefore, our results extend and improve the results in [16].



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