Existence and uniqueness of anti-periodic solutions for prescribed mean curvature Rayleigh equations
© Li and Wang; licensee Springer 2012
Received: 12 May 2012
Accepted: 22 September 2012
Published: 9 October 2012
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.
Keywordsprescribed mean curvature Rayleigh equation anti-periodic solutions Leray-Schauder degree
where is T-periodic, and are T-periodic in the first argument, is a constant.
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.
Obviously, a -anti-periodic function u is a T-periodic function.
The following lemmas will be useful to prove our main results.
Lemma 2.1 
Lemma 2.2 Suppose that the following condition holds:
, for all and .
Then Eq. (1.1) has at most one T-periodic solution.
Therefore, Eq. (1.1) has at most one T-periodic solution. The proof is completed. □
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 3.1 Let hold. Moreover, assume that the following conditions hold:
Then Eq. (1.1) has a unique anti-periodic solution for .
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.
Then Eq. (3.2) has no anti-periodic solution on ∂ Ω for .
Then , and thus L is continuous.
It is easy to see that is a compact homotopy, and the fixed point of on is the anti-periodic of Eq. (3.1).
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 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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