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.
Keywords:prescribed mean curvature Rayleigh equation; anti-periodic solutions; Leray-Schauder degree
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  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  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.
If and , then
(Wirtinger inequality) and
Lemma 2.2Suppose that the following condition holds:
, for all and .
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
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
Therefore, Eq. (1.1) has at most one T-periodic solution. The proof is completed. □
Lemma 2.3Let Ω be open bounded in a linear normal spaceX. Suppose that is a complete continuous field on . Moreover, assume that the Leray-Schauder degree
Then the equation has at least one solution in Ω.
3 Main result
In this section, we present and prove our main result concerning the existence and uniqueness of anti-periodic solutions of Eq. (1.1).
Theorem 3.1Let hold. Moreover, assume that the following conditions hold:
there exists such that
there exists such 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
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
which implies that there exists a constant such that
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
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. □
4 An example
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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