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
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
Obviously, a
Throughout this paper, we will adopt the following notations:
which is a linear normal space endowed with the norm
The following lemmas will be useful to prove our main results.
Lemma 2.1[12]
If
(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
It is easy to see that
Set
Otherwise, we have
Then there exists a
which implies that
and
It follows from (2.2), (2.4) and (2.5) that
From
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 that
Then the equation
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 3.1Let
Then Eq. (1.1) has a unique antiperiodic solution for
Proof Rewrite Eq. (1.1) in the equivalent form:
where
where
Then Eq. (3.2) can be reduced to the equation as follows:
By Lemma 2.2 and condition
Let
It follows from Lemma 2.1 that
Obviously, Eq. (3.2) is equivalent to the following equation:
Multiplying (3.3) by
Since
For such a
It follows from (3.4) and (3.6) that
For
Hence,
From (3.5) and (3.7), we know that there exists a constant
By the first equation of (3.2), we have
Then there exists
According to
From the second equation of (3.2), we get
From (3.8), we know that there exists a constant
Thus,
which implies that there exists a constant
Let
Set
Then Eq. (3.2) has no antiperiodic solution on ∂Ω for
Next, we consider the Fourier series expansions of two functions
Define an operator
Then
and
Since
and
we obtain
Define
Then
For any
Therefore,
It is easy to see that
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,
4 An example
In this section, we shall construct an example to show the applications of Theorem 3.1.
Example 4.1 Let
has a unique antiperiodic solution with period 2π.
Proof Let
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.
References

Lu, S, Gui, Z: On the existence of periodic solutions to pLaplacian Rayleigh differential equation with a delay. J. Math. Anal. Appl.. 325, 685–702 (2007). Publisher Full Text

Zong, M, Liang, H: Periodic solutions for Rayleigh type pLaplacian equation with deviating arguments. Appl. Math. Lett.. 20, 43–47 (2007). Publisher Full Text

Gao, H, Liu, B: Existence and uniqueness of periodic solutions for forced Rayleightype equations. Appl. Math. Comput.. 211, 148–154 (2009). Publisher Full Text

Ma, T, Wang, Z: A continuation lemma and its applications to periodic solutions of Rayleigh differential equations with subquadratic potential conditions. J. Math. Anal. Appl.. 385, 1107–1118 (2012). Publisher Full Text

Yu, Y, Shao, J, Yue, G: Existence and uniqueness of antiperiodic solutions for a kind of Rayleigh equation with two deviating arguments. Nonlinear Anal.. 71, 4689–4695 (2009). Publisher Full Text

Lv, X, Yan, P, Liu, D: Antiperiodic solutions for a class of nonlinear secondorder Rayleigh equations with delays. Commun. Nonlinear Sci. Numer. Simul.. 15, 3593–3598 (2010). Publisher Full Text

Liu, B: Antiperiodic solutions for forced Rayleightype equations. Nonlinear Anal., Real World Appl.. 10, 2850–2856 (2009). Publisher Full Text

Feng, M: Periodic solutions for prescribed mean curvature Liénard equation with a deviating argument. Nonlinear Anal., Real World Appl.. 13, 1216–1223 (2012). Publisher Full Text

Bonheure, D, Habets, P, Obersnel, F, Omari, P: Classical and nonclassical solutions of a prescribed curvature equation. J. Differ. Equ.. 243, 208–237 (2007). Publisher Full Text

Pan, H: Onedimensional prescribed mean curvature equation with exponential nonlinearity. Nonlinear Anal.. 70, 999–1010 (2009). Publisher Full Text

Benevieria, P, do Ó, J, Medeiros, E: Periodic solutions for nonlinear systems with mean curvaturelike operators. Nonlinear Anal.. 65, 1462–1475 (2006). Publisher Full Text

Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

Gaines, R, Mawhin, J: Coincidence Degree and Nonlinear Differential Equations, Springer, Berlin (1977)

Deimling, K: Nonlinear Functional Analysis, Springer, Berlin (1985)