By means of the Leray-Schauder degree theory, we establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations.

MSC: 34C25, 34D40.

We are concerned with the existence and uniqueness of anti-periodic solutions of the following prescribed mean curvature Rayleigh equation:

where is T-periodic, and are T-periodic in the first argument, is a constant.

In recent years, the existence of periodic solutions and anti-periodic solutions for some types of second-order differential equations, especially for the Rayleigh ones, were widely studied (see [1-7]) and the references cited therein). For example, Liu [7] discussed the Rayleigh equation

and established the existence and uniqueness of anti-periodic solutions. At the same time, a kind of prescribed mean curvature equations attracted many people’s attention (see [8-11] and the references cited therein). Feng [8] investigated the prescribed mean curvature Liénard equation

and obtained some existence results on periodic solutions. However, to the best of our knowledge, the existence and uniqueness of anti-periodic solution for Eq. (1.1) have not been investigated till now. Motivated by [7,8], we establish some sufficient conditions for the existence and uniqueness of anti-periodic solutions via the Leray-Schauder degree theory.

The rest of the paper is organized as follows. In Section 2, we shall state and prove some basic lemmas. In Section 3, we shall prove the main result. An example will be given to show the applications of our main result in the final section.

We first give the definition of an anti-periodic function. Assume that N is a positive integer. Let be a continuous function. We call an anti-periodic function on ℝ if u satisfies the following condition:

Obviously, a -anti-periodic function u is a T-periodic function.

Throughout this paper, we will adopt the following notations:

which is a linear normal space endowed with the norm defined by

The following lemmas will be useful to prove our main results.

Lemma 2.1[12]

Ifand, then

(Wirtinger inequality) and

(Sobolev inequality).

Lemma 2.2Suppose that the following condition holds:

, for alland.

Then Eq. (1.1) has at most oneT-periodic solution.

Proof Assume that and are two T-periodic solutions of Eq. (1.1). Then we obtain

It is easy to see that (). From (2.1), we know

Set . Now, we prove

Otherwise, we have

Then there exists a such that

which implies that

and

It follows from (2.2), (2.4) and (2.5) that

From , we get

which contradicts (2.3). Thus,

By using a similar argument, we can also show

Hence,

Therefore, Eq. (1.1) has at most one T-periodic solution. The proof is completed. □

To prove the main result of this paper, we shall use a continuation theorem [13,14] as follows.

Lemma 2.3Let Ω be open bounded in a linear normal spaceX. Suppose thatis a complete continuous field on. Moreover, assume that the Leray-Schauder degree

Then the equationhas at least one solution in Ω.

In this section, we present and prove our main result concerning the existence and uniqueness of anti-periodic solutions of Eq. (1.1).

Theorem 3.1Lethold. Moreover, assume that the following conditions hold:

there existssuch that

there existssuch that

for all,

Then Eq. (1.1) has a unique anti-periodic solution for.

Proof Rewrite Eq. (1.1) in the equivalent form:

where . Now, we consider the auxiliary equation of (3.1),

where is a parameter. Set

Then Eq. (3.2) can be reduced to the equation as follows:

By Lemma 2.2 and condition , it is easy to see that Eq. (1.1) has at most one anti-periodic solution. Thus, to prove Theorem 3.1, it suffices to show that Eq. (1.1) has at least one anti-periodic solution. To do this, we shall apply Lemma 2.3. Firstly, we will prove that the set of all possible anti-periodic solutions of Eq. (3.2) is bounded.

Let be an arbitrary possible anti-periodic solution of Eq. (3.2). Then . Thus, we have

It follows from Lemma 2.1 that

Obviously, Eq. (3.2) is equivalent to the following equation:

Multiplying (3.3) by and integrating from 0 to T, we have

Since , there exists a constant such that

For such a , in view of , there exists such that for all , . Hence,

It follows from (3.4) and (3.6) that

For , we have the Schwarz inequality

Hence,

From (3.5) and (3.7), we know that there exists a constant such that

By the first equation of (3.2), we have

Then there exists such that . It follows that , and so

According to , we know there exists such that for all ,

From the second equation of (3.2), we get

From (3.8), we know that there exists a constant such that

Thus,

which implies that there exists a constant such that

Let

Set

Then Eq. (3.2) has no anti-periodic solution on ∂Ω for .

Next, we consider the Fourier series expansions of two functions (). We have

Define an operator by setting

Then

and

Since

and

we obtain

Define by setting

Then , and thus L is continuous.

For any , we know from that

Therefore, . Define an operator by setting

It is easy to see that is a compact homotopy, and the fixed point of on is the anti-periodic of Eq. (3.1).

Define a homotopic field as follows:

From (3.9), we have

Using the homotopy invariance property of degree, we obtain

Till now, we have proved that Ω satisfies all the requirements in Lemma 2.3. Consequently, has at least one solution in Ω, i.e., has a fixed point on . Therefore, Eq. (1.1) has at least one anti-periodic solution . This completes the proof. □

In this section, we shall construct an example to show the applications of Theorem 3.1.

Example 4.1 Let , . Then the prescribed mean curvature Rayleigh equation

has a unique anti-periodic solution with period 2π.

Proof Let . From the definitions of and , we can easily check that conditions and hold. Moreover, it is easy to see that holds for and holds for , . Since , we know from Theorem 3.1 that Eq. (4.1) has a unique anti-periodic solution with period 2π. □

The authors declare that they have no competing interests.

Both authors, AA and MHA, contributed to each part of this work equally and read and approved the final version of the manuscript.

The authors would like to express their thanks to the Editor of the journal and the anonymous referees for their carefully reading of the first draft of the manuscript and making many helpful comments and suggestions which improved the presentation of the paper. Research supported by National Science foundation of China, No. 10771145 and Beijing Natural Science Foundation (Existence and multiplicity of periodic solutions in nonlinear oscillations), No. 1112006.

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