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
Notice that, when
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
where
with the norm
Notice that,
and
where
and the mapping
It is easy to prove that the mappings
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
Then the equation
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 functions
(H_{2}) there exist nonnegative functions
(H_{3})
Then (1.3) has at least one even antiperiodic solution
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):
Define the operator
Obviously, the operator
By hypothesis (H_{1}), it is easy to see that
Thus, the operator
From hypotheses (H_{2}), (H_{3}) and (5) in [10], for the possible even antiperiodic solution
where
With this in mind, let us define the set as follows:
Obviously, the set
Let us define the completely continuous field
By (3.2), we get that zero element
Consequently, the operator
Theorem 3.2Assume that
(H_{4}) the function
and the assumptions (H_{2}), (H_{3}) are true.
Then (1.3) has at least one odd antiperiodic solution
Proof We consider the homotopic equation (3.1) of (1.3). Define the operator
Let
By hypothesis (H_{4}), it is easy to see that
Thus, the operator
Our problem is reduced to construct one completely continuous operator
Define the completely continuous operator
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 functions
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:
Define the operator
Let
By hypothesis (H_{5}), it is easy to see that
Thus, the operator
Our problem is reduced to construct one completely continuous operator
where
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 function
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
Let
By hypothesis (H_{6}), it is easy to see that
Thus, the operator
Our problem is reduced to construct one completely continuous operator
Define the completely continuous operator
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
Theorem 3.5Assume that
(H_{7})
Then (1.3) (
Theorem 3.6Suppose that the assumptions (H_{4}), (H_{7}) are true. Then (1.3) (
Basing on the proof of Theorem 2 in [10], for the possible antiperiodic solution
where
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 functions
(H_{9})
Then (1.5) has at least one even antiperiodic solution.
Proof We consider the following homotopic equation of (1.5):
Define the operator
where
Let
Obviously, the operator
From hypotheses (H_{8}), (H_{9}) and (3.8) in [15], for the possible even antiperiodic solution
where
With this in mind, let us define the set as follows:
By hypothesis (H_{5}), it is easy to see that
Hence, the operator
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
where
Let
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
By hypothesis (H_{6}), it is easy to see that
Hence, the operator
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 that
such that
(H_{10}) there exist nonnegative constantsγ, rwith
(H_{11}) there are nonnegative functions
(H_{12})
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
where
Remark Assumptions (H_{10}), (H_{11}), (H_{12}) guarantee that the degree with respect to x of
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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