Abstract
By means of the LeraySchauder degree theory, we establish some sufficient conditions for the existence and uniqueness of antiperiodic solutions for prescribed mean curvature Rayleigh equations.
MSC: 34C25, 34D40.
Keywords:
prescribed mean curvature Rayleigh equation; antiperiodic solutions; LeraySchauder degree1 Introduction
We are concerned with the existence and uniqueness of antiperiodic solutions of the following prescribed mean curvature Rayleigh equation:
where is Tperiodic, and are Tperiodic in the first argument, is a constant.
In recent years, the existence of periodic solutions and antiperiodic solutions for some types of secondorder differential equations, especially for the Rayleigh ones, were widely studied (see [17]) and the references cited therein). For example, Liu [7] discussed the Rayleigh equation
and established the existence and uniqueness of antiperiodic solutions. At the same time, a kind of prescribed mean curvature equations attracted many people’s attention (see [811] 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 antiperiodic 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 antiperiodic solutions via the LeraySchauder 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.
2 Preliminaries
We first give the definition of an antiperiodic function. Assume that N is a positive integer. Let be a continuous function. We call an antiperiodic function on ℝ if u satisfies the following condition:
Obviously, a antiperiodic function u is a Tperiodic 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]
(Wirtinger inequality) and
(Sobolev inequality).
Lemma 2.2Suppose that the following condition holds:
Then Eq. (1.1) has at most oneTperiodic solution.
Proof Assume that and are two Tperiodic solutions of Eq. (1.1). Then we obtain
It is easy to see that (). From (2.1), we know
Otherwise, we have
which implies that
and
It follows from (2.2), (2.4) and (2.5) that
which contradicts (2.3). Thus,
By using a similar argument, we can also show
Hence,
Therefore, Eq. (1.1) has at most one Tperiodic 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 LeraySchauder degree
3 Main result
In this section, we present and prove our main result concerning the existence and uniqueness of antiperiodic solutions of Eq. (1.1).
Theorem 3.1Lethold. Moreover, assume that the following conditions hold:
Then Eq. (1.1) has a unique antiperiodic solution for.
Proof Rewrite Eq. (1.1) in the equivalent form:
where . Now, we consider the auxiliary equation of (3.1),
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 antiperiodic solution. Thus, to prove Theorem 3.1, it suffices to show that Eq. (1.1) has at least one antiperiodic solution. To do this, we shall apply Lemma 2.3. Firstly, we will prove that the set of all possible antiperiodic solutions of Eq. (3.2) is bounded.
Let be an arbitrary possible antiperiodic 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 antiperiodic solution on ∂Ω for .
Next, we consider the Fourier series expansions of two functions (). We have
Then
and
Since
and
we obtain
Then , and thus L is continuous.
Therefore, . Define an operator by setting
It is easy to see that is a compact homotopy, and the fixed point of on is the antiperiodic 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 antiperiodic 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 antiperiodic 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 antiperiodic solution with period 2π. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors, AA and MHA, contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
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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