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:
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:
Throughout this paper, we will adopt the following notations:
The following lemmas will be useful to prove our main results.
(Wirtinger inequality) and
Lemma 2.2Suppose that the following condition holds:
Then Eq. (1.1) has at most oneT-periodic solution.
Otherwise, we have
which implies that
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
Therefore, Eq. (1.1) has at most one T-periodic solution. The proof is completed. □
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).
Proof Rewrite Eq. (1.1) in the equivalent form:
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.
It follows from Lemma 2.1 that
Obviously, Eq. (3.2) is equivalent to the following equation:
It follows from (3.4) and (3.6) that
By the first equation of (3.2), we have
From the second equation of (3.2), we get
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.
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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