Uniqueness of Periodic Solution for a Class of Liénard -Laplacian Equations

Abstract

By topological degree theory and some analysis skills, we consider a class of generalized Liénard type -Laplacian equations. Upon some suitable assumptions, the existence and uniqueness of periodic solutions for the generalized Liénard type -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 Liénard 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 Manásevich, 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.

Some contributions on periodic solutions to differential equations have been made in [1–13]. Recently, periodic problems involving the scalar -Laplacian were studied by many authors. We mention the works by Manásevich and Mawhin [3] and Cheung and Ren [4, 8, 10].

In [3], Manásevich and Mawhin investigated the existence of periodic solutions to the boundary value problem

(1.1)

where the function is quite general and satisfies some monotonicity conditions which ensure that is homeomorphism onto Applying Leray-Schauder degree theory, the authors brought us the widely used Manásevich-Mawhin continuation theorem. When is the so-called one-dimensional p-Laplacian operator given by

Recently, by Mawhin's continuation theorem, Cheung and Ren studied the existence of -periodic solutions for a -Laplacian Liénard equation with a deviating argument in [4] as follows:

(1.2)

and two results (Theorems 3.1 and 3.2 ) on the existence of periodic solutions were obtained.

Ge and Ren [5] promoted 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 -periodic solutions of the Liénard type -Laplacian differential equation of the form

(1.3)

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

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

(1.4)

where , is given by for , , , and -periodic in the first variable, where is a given constant, and

The paper is organized as follows. In Section 2, we give the definition of norm in Banach space and the main lemma. In Section 3, combining Lemma 2.1 with some analysis skills, two sufficient conditions about the existence of solutions for (1.4) are obtained. The nonlinear terms and contain two variables in this paper, which is seldom considered in the other papers, and the results are new.

2. Preliminary Results

For convenience, we define

(2.1)

and the norm is defined by , for all

(2.2)

Clearly, is a Banach space endowed with such norm.

For the periodic boundary value problem

(2.3)

where is a continuous function and -periodic in the first variable, we have the following result.

Lemma 2.1 (see [3]).

Let be an open bounded set in . If the following conditions hold:

• (i) for each the problem

  

  (2.4)

  has no solution on ,

• (ii) the equation

  

  (2.5)

  has no solution on ,

• (iii) the Brouwer degree of is ,

then the periodic boundary value problem (2.3) has at least one -periodic solution on

Set

(2.6)

We can rewrite (1.4) in the following form:

(2.7)

where and

Lemma 2.2.

Suppose the following condition holds:

(A₁) , for all .

Then (1.4) has at most one -periodic solution.

Proof.

Let and be two -periodic solutions of (1.4). Then, from (2.7), we obtain

(2.8)

Set

(2.9)

Then it follows from (2.8) that

(2.10)

We claim that for all By way of contradiction, in view of and for all , we obtain Then there must exist ; for convenience, we can choose such that

(2.11)

which implies that

(2.12)

Set

(2.13)

Then, Since , if from the first equation of (2.12), we have

(2.14)

which contradicts assumption , so ; it implies that that is, Hence we have

(2.15)

Substituting (2.15) into the second equation of (2.12), we have

(2.16)

Noticing () and that and , from (2.16), we know that

(2.17)

this contradicts the second equation of (2.12). So we have , for all

By using a similar argument, we can also show that

(2.18)

Then, from (2.10) we obtain

(2.19)

For every if then it contradicts (2.19), so ; it implies that then , for all

Hence, (1.4) has at most one -periodic solution. The proof of Lemma 2.2 is completed now.

3. Main Results

Theorem 3.1.

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

(A₂) , for all

(A₃)

Then (1.4) has one unique -periodic solution if

Proof.

Consider the homotopic equation of (1.4) as follows:

(3.1)

By Lemma 2.2, combining it is easy to see that (1.4) has at least one -periodic solution. For the remainder, we will apply Lemma 2.1 to study (3.1). Firstly, we will verify that all the possible -periodic solutions of (3.1) are bounded.

Let be an arbitrary solution of (3.1) with period . By integrating the two sides of (3.1) from to and we obtain

(3.2)

Consider and , there exists such that while for we see that

(3.3)

where Let be the global maximum point of on Then as we claim that

(3.4)

Otherwise, we have there must exist a constant such that , for ; therefore, is strictly increasing for which implies that is strictly increasing for Thus, (3.4) is true. Then

(3.5)

In view of (), (3.5) implies that ; similar to the global minimum point of on Since it follows that there exists a constant such that Then we have

(3.6)

Combining the above two inequalities, we obtain

(3.7)

Considering () there exist constants and the sufficiently small such that

(3.8)

Set

(3.9)

From (3.7), we have

(3.10)

where Combining the classical inequality when where is a constant, since

(3.11)

then we consider the following two cases.

Case 1.

If then Combining (3.7), we know that

(3.12)

when then we have

Case 2.

When then from the above classical inequality, we obtain

(3.13)

Substituting the above inequality into (3.10) we get

(3.14)

Since is -periodic, multiplying by (3.1) and then integrating from to , in view of () we have

(3.15)

Substituting (3.14) into (3.15) and since

(3.16)

we obtain

(3.17)

where

(3.18)

Since and from (3.17), we know that there exists a constant such that Then,

(3.19)

So, there exists a constant such that

(3.20)

Set

(3.21)

Then (3.1) has no solution on as and when or ; from we can see that

(3.22)

so condition (ii) holds.

Set

(3.23)

Then, when , we have

(3.24)

thus is a homotopic transformation and

(3.25)

So condition (iii) holds. In view of Lemma 2.1, there exists at least one solution with period . This completes the proof.

Theorem 3.2.

Let () hold. Suppose that there exist positive constants and satisfying the following conditions:

(A₄) , when

(A₅) ,

(A₆) , when

Then for (1.4) there exists one unique -periodic solution when

Proof.

We can rewrite (3.1) in the following from:

(3.26)

Let be a -periodic solution of (3.26), then must be a -periodic solution of (3.1). First we claim that there is a constant such that

(3.27)

Take as the global maximum point and global minimum point of on , respectively, then

(3.28)

From the first equation of (3.26) we have so

We claim that

(3.29)

By way of contradiction, (3.29) does not hold, then So there exists such that , for ; therefore, , for so , that is, , for This contradicts the definition of so we have

Substituting into the second equation of (3.26), we obtain

(3.30)

By condition () we have Similarly, we get

Case 1.

If define Obviously

Case 2.

If from the fact that is a continuous function in there exists a constant between and such that

So we have that (3.27) holds. Next, in view of there are integer and constant such that hence

So

(3.31)

We claim that all the periodic solutions of (3.1) are bounded and

Let

(3.32)

Multiplying both sides of (3.1) by and integrating from to together with () and (), we have

(3.33)

where and That is,

(3.34)

Using Hölder's inequality and we have

(3.35)

so

(3.36)

there must be a positive constant such that

(3.37)

hence together with (3.31), we have

This proves the claim and that the rest of the proof of the theorem is identical to that of Theorem 3.1.