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Periodic solutions for prescribed mean curvature Rayleigh equation with a deviating argument
Advances in Difference Equations volume 2013, Article number: 88 (2013)
By using Mawhin’s continuation theorem in the coincidence degree theory, some criteria for guaranteeing the existence of periodic solutions for prescribed mean curvature Rayleigh equation with a deviating argument are provided.
Consider the prescribed mean curvature Rayleigh equation
where are T-periodic, and are T-periodic in the first argument, is a constant.
In recent years, there are many results on the existence of periodic solutions for various types of delay differential equation with deviating arguments, especially for the Liénard equation and Rayleigh equation (see [1–11]). Now as the prescribed mean curvature of a function frequently appears in different geometry and physics (see [12–14]), it is interesting to try to consider the existence of periodic solutions of prescribed mean curvature equations. However, to our best knowledge, the studies of delay equations with prescribed mean curvature is relatively infrequent. The main difficulty lies in the nonlinear term , the existence of which obstructs the usual method of finding a priori bounds for delay Liénard or Rayleigh equations from working. In , Feng discussed a delay prescribed mean curvature Liénard equation of the form
estimated a priori bounds by eliminating the nonlinear term , and established sufficient conditions on the existence of periodic solutions for (1.2) by using Mawhin’s continuation theorem.
The conditions imposed on and in  were such as:
(C1) There exists satisfies .
(C2) There exists such that , , .
It is not difficult to see that the condition (C1) is strong. It is natural to relax the conditions (C1) and (C2). Our purpose is studying the more general equation (1.1) under the more weaker conditions.
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.
In this section, we first recall Mawhin’s continuation theorem, which our study is based upon.
Let X and Y be real Banach spaces and be a linear operator. L is said to be a Fredholm operator with index zero provided that
ImL is closed subset of Y,
Set , . Let and be the nature projections. It is easy to see that . Thus, the restriction is invertible. We denote by k the inverse of .
Let Ω be a open bounded subset of X with . A map is said to be L-compact in if and are compact.
The following lemma due to Mawhin  is fundamental to prove our main result.
Lemma 2.1 Let L be a Fredholm operator of index zero and Let N be L-compact on . If the following conditions hold:
(h1) , ;
(h2) , ;
Then has at least one solution in .
The following lemmas is useful in the proof of our main result.
Lemma 2.2 ()
Let with , . Suppose , and with . Then
Lemma 2.3 ()
If and , then
(Wirtinger inequality) and
Lemma 2.4 ()
Suppose , and . Then
Lemma 2.5 Assume that , and . Then
It is easy to see that
Combining the above inequalities and using Hölder’s inequality,
The proof is completed. □
In order to apply Mawhin’s continuation theorem to study the existence of T-periodic solution of Equation (1.1), we rewrite (1.1) as
Obviously, if is a T-periodic solution of (2.1), then must be a T-periodic solution of (1.1). Hence, the problem of finding a T-periodic solution of (1.1) reduces to finding one of (2.1).
Now, we set
with the norm , where
Clearly, X and Y are Banach spaces. Meanwhile, let
Define a nonlinear operator by
Then the problem (2.1) can be written to .
It is easy to see that and . So, L is a Fredholm operator with index zero.
Let and be defined by
and denote by k the inverse of . Then and
It follows from (2.2) that N is L-compact on , where Ω is an open, bounded subset of X.
3 Main results
In this section, we will state and prove our main results.
We first give the following assumptions:
(H1) , and (or ), .
(H2) (or ), , and for .
(H3) There exists such that
(H4) There exists an integer m such that , and .
Theorem 3.1 Assume (H1)-(H4) hold. Then Equation (1.1) has at least one T-periodic solution provided
Consider the operator equation
Let . If , we have
It follows from the first equation of (3.3) that
Then (3.3) can be written to
Integrating the first equation of (3.3) from 0 to T, we have
Then there exist , such that
Assume that are the maximum and minimum points, respectively. Then
It follows from the second equation of (3.3) that
From (H1) and (H2), without loss of generality, we can assume that and , . Then
If , then , which is a contradiction. It follows that
Similarly, from (3.6), we have
Combining the above, we know that there exists a point such that
Note that there exist and such that . Then we get
By Lemma 2.4, we obtain
Meanwhile, by Lemma 2.3, we have
From (3.1), we have , or . Then there exists such that
For such a , it follows from (H3), there exist such that
Multiplying and (3.4) and integrating from 0 to T, we get
It follows from (H1) and (3.13) that
Substituting (3.16) into (3.15) and from (3.14), we have
Case 1. (3.11) holds. It follows from (3.17) and Hölder inequality that
From (3.9) and (3.11), we obtain that there exists a positive constant such that
Case 2. (3.12) holds. It follows from (3.17), Lemma 2.2 and Hölder inequality that
From (3.9) and (3.12), we know there exists a positive constant such that
Take . Then, if (3.1) holds, we have
By the first equation of (3.3), we have
Then there exists such that . It implies that
From the second equation of (3.3), we get
Noticing that , we have there exists , such that
Then, from (3.13), we have
Let . If , then and . Obviously,
then (1) and (2) of Lemma 2.1 are satisfied.
Next, we claim that (3) of Lemma 2.1 is also satisfied. For this, we define the isomorphism by
and let , .
By simple calculations, we obtain, for ,
Obviously, it follows from (H2) that .
which implies condition (3) of Lemma 2.1 is also satisfied.
Thus has a solution , i.e., Equation (1.1) has a T-periodic solution with . This completes the proof. □
Remark 3.1 If , the condition (H4) can be not assumed, i.e., it follows only from (H1)-(H3) that Equation (1.1) has a T-periodic solution.
4 An example
In this section, as applications for Theorem 3.1, we list the following example.
Consider prescribed mean curvature Rayleigh equation with a deviating argument
where , .
Let . Clearly, , , , , and
From Theorem 3.1, Equation (4.1) has at least one T-periodic solution.
Remark 4.1 If , , Equation (4.1) is a prescribed mean curvature Liénard equation. By using Theorem 3.1, it has at least one 2π-periodic solution, which cannot be obtained by . This implies that the results of this paper are essentially new.
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Research supported by National Science foundation of China, No. 10771145 and the Beijing Natural Science Foundation (Existence and multiplicity of periodic solutions in nonlinear oscillations), No. 1112006.
The authors declare that they have no competing interests.
All authors have equally contributed in obtaining new results in this article and also read and approved the final manuscript.