Abstract
The existence of antiperiodic solutions with symmetry for highorder Duffing equations and a highorder Duffing type pLaplacian equation has been studied by using degree theory. The results obtained enrich some known works to some extent.
MSC: 34B15, 34C25.
Keywords:
antiperiodic solution with symmetry; highorder ordinary differential equation; pLaplacian operator; LeraySchauder degree theory1 Introduction
Antiperiodic problems arise naturally from the mathematical models of various physical processes (see [1,2]) and also appear in the study of partial differential equations and abstract differential equations (see [35]). For instance, electron beam focusing system in travelingwave tube theories is an antiperiodic problem (see [6]).
In mechanics, the simplest model of oscillation equation is a single pendulum equation
whose antiperiodic solutions satisfy
During the past twenty years, antiperiodic problems have been studied extensively by numerous scholars. For example, for firstorder ordinary differential equations, a Massera’s type criterion was presented in [7] and the validity of the monotone iterative technique was shown in [8]. Moreover, for higherorder ordinary differential equations, the existence of antiperiodic solutions was considered in [912]. Recently, existence results were extended to antiperiodic boundary value problems for impulsive differential equations (see [13]), and antiperiodic wavelets were discussed in [14].
It is well known that higherorder pLaplacian equations are derived from many fields such as fluid mechanics and nonlinear elastic mechanics. In the past few decades, many important results on higherorder pLaplacian equations with certain boundary conditions have been obtained. We refer the readers to [1519] and the references cited therein.
In [10], the authors considered the existence of antiperiodic solutions for the highorder Duffing equation as follows:
Moreover, in [15] the authors discussed the existence of antiperiodic solutions for the following higherorder Liénard type pLaplacian equation:
However, to the best of our knowledge, there exist relatively few results on the existence of antiperiodic solutions with symmetry for (1.1) and (1.2). Thus, it is worthwhile to continue to investigate the existence of antiperiodic solutions with symmetry for (1.1) and (1.2).
Motivated by the works mentioned previously, in this paper, we study the existence of antiperiodic solutions with symmetry for highorder Duffing equations of the forms:
and highorder Duffing type pLaplacian equation of the form:
where is a constant, is an integer, ; , , with , . Obviously, the inverse operator of is , where is a constant such that .
Notice that, when , the nonlinear operator reduces to the linear operator . On the other hand, is also a 2πperiodic solution if is a πantiperiodic solution. Hence, from the arguments in this paper, we can also obtain the existence results on periodic solutions for the above equations.
The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3 and Section 4, basing on the LeraySchauder principle, we establish some existence theorems on antiperiodic solutions with symmetry of (1.3), (1.4) and (1.5). Our results are different from those of bibliographies listed in the previous texts.
2 Preliminaries
For the sake of convenience, we set
with the norm
Notice that, may be written as Fourier series as follows:
and may be written as the following Fourier series:
where . We define the mapping by
It is easy to prove that the mappings , are completely continuous by using the ArzelàAscoli theorem.
Next, we introduce a continuation theorem (see [20]) as follows.
Lemma 2.1 (Continuation theorem)
Let Ω be open bounded in a linear normal spaceX. Suppose thatfis a completely continuous field on. Moreover, assume that the LeraySchauder degree
3 Antiperiodic solutions with symmetry of (1.3) and (1.4)
In this section, some existence results on antiperiodic solutions with symmetry of (1.3) and (1.4) will be given.
Theorem 3.1Assume that
(H_{1}) the functionsandare odd int, i.e.,
(H_{2}) there exist nonnegative functionssuch that
Then (1.3) has at least one even antiperiodic solution, i.e., satisfies
Proof For making use of the LeraySchauder degree theory to prove the existence of even antiperiodic solutions for (1.3), we consider the following homotopic equation of (1.3):
Obviously, the operator is invertible. Let be the Nemytskii operator
By hypothesis (H_{1}), it is easy to see that
Thus, the operator sends into . Hence, the problem of even antiperiodic solutions for (3.1) is equivalent to the operator equation
From hypotheses (H_{2}), (H_{3}) and (5) in [10], for the possible even antiperiodic solution of (3.1), there exists a prior bounds in , i.e., satisfies
where is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the even antiperiodic solutions of (1.3).
With this in mind, let us define the set as follows:
Obviously, the set is a open bounded set in and zero element . Define the completely continuous operator by
Let us define the completely continuous field by
By (3.2), we get that zero element for all . So, the following LeraySchauder degrees are well defined and
Consequently, the operator has at least one fixed point in by using Lemma 2.1. Namely, (1.3) has at least one even antiperiodic solution. The proof is complete. □
Theorem 3.2Assume that
(H_{4}) the functionis even int, xandis even int, i.e.,
and the assumptions (H_{2}), (H_{3}) are true.
Then (1.3) has at least one odd antiperiodic solution, i.e., satisfies
Proof We consider the homotopic equation (3.1) of (1.3). Define the operator by
By hypothesis (H_{4}), it is easy to see that
Thus, the operator sends into . Hence, the problem of odd antiperiodic solutions for (3.1) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the odd antiperiodic solutions of (1.3). With this in mind, let us define the following set:
Define the completely continuous operator by
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
Theorem 3.3Assume that
(H_{5}) the functionsandare even int, i.e.,
and the assumptions (H_{2}), (H_{3}) are true.
Then (1.4) has at least one even antiperiodic solution.
Proof We consider the homotopic equation of (1.4) as follows:
By hypothesis (H_{5}), it is easy to see that
Thus, the operator sends into . Hence, the problem of even antiperiodic solutions for (3.3) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the even antiperiodic solutions of (1.4). With this in mind, let us define the following set:
where is a positive constant independent of λ. Define the completely continuous operator by
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
Theorem 3.4Assume that
(H_{6}) the functionis odd int, xandis odd int, i.e.,
and the assumptions (H_{2}), (H_{3}) are true.
Then (1.4) has at least one odd antiperiodic solution.
Proof We consider the homotopic equation (3.3) of (1.4). Define the operator by
By hypothesis (H_{6}), it is easy to see that
Thus, the operator sends into . Hence, the problem of odd antiperiodic solutions for (3.3) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator which sends into , such that the fixed points of operator in some open bounded set are the odd antiperiodic solutions of (1.4). With this in mind, let us define the set as follows:
Define the completely continuous operator by
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
When , we can remove the assumption (H_{2}) in Theorem 3.1, Theorem 3.2 and obtain the following results.
Theorem 3.5Assume that
(H_{7}) and the assumption (H_{1}) is true.
Then (1.3) () has at least one even antiperiodic solution.
Theorem 3.6Suppose that the assumptions (H_{4}), (H_{7}) are true. Then (1.3) () has at least one odd antiperiodic solution.
Basing on the proof of Theorem 2 in [10], for the possible antiperiodic solution of (3.1) (), the hypothesis (H_{7}) yields that there exists a prior bounds in , i.e., satisfies
where is a positive constant independent of λ. The remainder of the proof work of Theorem 3.5 and Theorem 3.6 is quite similar to the proof of Theorem 3.1 and Theorem 3.2, so we omit the details.
4 Antiperiodic solutions with symmetry of (1.5)
In this section, we will give some existence results on antiperiodic solutions with symmetry of (1.5).
Theorem 4.1Assume that
(H_{8}) there exist nonnegative functionssuch that
(H_{9}) and the assumption (H_{5}) is true.
Then (1.5) has at least one even antiperiodic solution.
Proof We consider the following homotopic equation of (1.5):
where
Obviously, the operator is invertible and the problem of even antiperiodic solutions for (4.1) is equivalent to the operator equation
From hypotheses (H_{8}), (H_{9}) and (3.8) in [15], for the possible even antiperiodic solution of (4.1), there exists a prior bounds in , i.e., satisfies
where is a positive constant independent of λ. So, our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the even antiperiodic solutions of (1.5).
With this in mind, let us define the set as follows:
By hypothesis (H_{5}), it is easy to see that
Hence, the operator sends into . Define the completely continuous operator by
or
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
Theorem 4.2Suppose that the assumptions (H_{6}), (H_{8}), (H_{9}) are true. Then (1.5) has at least one odd antiperiodic solution.
Proof We consider the homotopic equation (4.1) of (1.5). Define the operator by
where
Thus, the problem of odd antiperiodic solutions for (4.1) is equivalent to the operator equation
Our problem is reduced to construct one completely continuous operator , which sends into , such that the fixed points of operator in some open bounded set are the odd antiperiodic solutions of (1.5). With this in mind, let us define the following set:
By hypothesis (H_{6}), it is easy to see that
Hence, the operator sends into . Define the completely continuous operator by
or
The remainder of the proof work is quite similar to the proof of Theorem 3.1, so we omit the details. The proof is complete. □
Theorem 4.3Assume thathas the decomposition
such that
(H_{10}) there exist nonnegative constantsγ, rwith, such that
(H_{11}) there are nonnegative functionssuch that
(H_{12}) and the assumption (H_{5}) is true.
Then (1.5) has at least one even antiperiodic solution.
Theorem 4.4Suppose that the assumptions (H_{6}), (H_{10}), (H_{11}), (H_{12}) are true. Then (1.5) has at least one odd antiperiodic solution.
Basing on the proof of Theorem 3.2 in [15], for the possible antiperiodic solution of (4.1), the hypotheses (H_{10}), (H_{11}), (H_{12}) yield that there exists a prior bounds in , i.e., satisfies
where is a positive constant independent of λ. The remainder of the proof work of Theorem 4.3 and Theorem 4.4 is quite similar to the proof of Theorem 4.1 and Theorem 4.2, so we omit the details.
Remark Assumptions (H_{10}), (H_{11}), (H_{12}) guarantee that the degree with respect to x of is allowed to be greater than , which is different from the hypothesis (H_{8}) of Theorem 4.1 and Theorem 4.2.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
HP carried out the theoretical analysis. JY drafted the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was supported by the National Natural Science Foundations of China (50904065) and the Program for New Century Excellent Talents in University (NCET090728). As well, this work was sponsored by the Qing Lan Project and the Fundamental Research Funds for the Central Universities (China University of Mining and Technology).
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