- Open Access
Periodic solutions for prescribed mean curvature Rayleigh equation with a deviating argument
© Li et al.; licensee Springer. 2013
- Received: 14 December 2012
- Accepted: 19 March 2013
- Published: 3 April 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.
- prescribed mean curvature Rayleigh equation
- periodic solutions
- Leray-Schauder degree
where are T-periodic, and are T-periodic in the first argument, is a constant.
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.
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 ()
Lemma 2.3 ()
Lemma 2.4 ()
The proof is completed. □
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).
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.
It follows from (2.2) that N is L-compact on , where Ω is an open, bounded subset of X.
In this section, we will state and prove our main results.
We first give the following assumptions:
(H1) , and (or ), .
(H2) (or ), , and for .
(H4) There exists an integer m such that , and .
then (1) and (2) of Lemma 2.1 are satisfied.
and let , .
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.
In this section, as applications for Theorem 3.1, we list the following example.
where , .
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.
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.
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