Existence of anti-periodic solutions with symmetry for some high-order ordinary differential equations

The existence of anti-periodic solutions with symmetry for high-order Duffing equations and a high-order Duffing type p-Laplacian equation has been studied by using degree theory. The results obtained enrich some known works to some extent.

MSC: 34B15, 34C25.

Anti-periodic 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 [3-5]). For instance, electron beam focusing system in traveling-wave tube theories is an anti-periodic problem (see [6]).

In mechanics, the simplest model of oscillation equation is a single pendulum equation

whose anti-periodic solutions satisfy

During the past twenty years, anti-periodic problems have been studied extensively by numerous scholars. For example, for first-order 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 higher-order ordinary differential equations, the existence of anti-periodic solutions was considered in [9-12]. Recently, existence results were extended to anti-periodic boundary value problems for impulsive differential equations (see [13]), and anti-periodic wavelets were discussed in [14].

It is well known that higher-order p-Laplacian equations are derived from many fields such as fluid mechanics and nonlinear elastic mechanics. In the past few decades, many important results on higher-order p-Laplacian equations with certain boundary conditions have been obtained. We refer the readers to [15-19] and the references cited therein.

In [10], the authors considered the existence of anti-periodic solutions for the high-order Duffing equation as follows:

Moreover, in [15] the authors discussed the existence of anti-periodic solutions for the following higher-order Liénard type p-Laplacian equation:

However, to the best of our knowledge, there exist relatively few results on the existence of anti-periodic solutions with symmetry for (1.1) and (1.2). Thus, it is worthwhile to continue to investigate the existence of anti-periodic solutions with symmetry for (1.1) and (1.2).

Motivated by the works mentioned previously, in this paper, we study the existence of anti-periodic solutions with symmetry for high-order Duffing equations of the forms:

and high-order Duffing type p-Laplacian 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 2π-periodic solution if is a π-anti-periodic 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 Leray-Schauder principle, we establish some existence theorems on anti-periodic solutions with symmetry of (1.3), (1.4) and (1.5). Our results are different from those of bibliographies listed in the previous texts.

For the sake of convenience, we set

with the norm

where , and

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

and the mapping by

It is easy to prove that the mappings , are completely continuous by using the Arzelà-Ascoli 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 Leray-Schauder degree

Then the equationhas at least one solution in Ω.

In this section, some existence results on anti-periodic solutions with symmetry of (1.3) and (1.4) will be given.

Theorem 3.1Assume that

(H₁) the functionsandare odd int, i.e.,

(H₂) there exist non-negative functionssuch that

(H₃) .

Then (1.3) has at least one even anti-periodic solution, i.e., satisfies

Proof For making use of the Leray-Schauder degree theory to prove the existence of even anti-periodic solutions for (1.3), we consider the following homotopic equation of (1.3):

Define the operator by

Obviously, the operator is invertible. Let be the Nemytskii operator

By hypothesis (H₁), it is easy to see that

Thus, the operator sends into . Hence, the problem of even anti-periodic solutions for (3.1) is equivalent to the operator equation

From hypotheses (H₂), (H₃) and (5) in [10], for the possible even anti-periodic 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 anti-periodic 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 Leray-Schauder 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 anti-periodic solution. The proof is complete. □

Theorem 3.2Assume that

(H₄) the functionis even int, xandis even int, i.e.,

and the assumptions (H₂), (H₃) are true.

Then (1.3) has at least one odd anti-periodic solution, i.e., satisfies

Proof We consider the homotopic equation (3.1) of (1.3). Define the operator by

Let be the Nemytskii operator

By hypothesis (H₄), it is easy to see that

Thus, the operator sends into . Hence, the problem of odd anti-periodic 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 anti-periodic 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

(H₅) the functionsandare even int, i.e.,

and the assumptions (H₂), (H₃) are true.

Then (1.4) has at least one even anti-periodic solution.

Proof We consider the homotopic equation of (1.4) as follows:

Define the operator by

Let be the Nemytskii operator

By hypothesis (H₅), it is easy to see that

Thus, the operator sends into . Hence, the problem of even anti-periodic 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 anti-periodic 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

(H₆) the functionis odd int, xandis odd int, i.e.,

and the assumptions (H₂), (H₃) are true.

Then (1.4) has at least one odd anti-periodic solution.

Proof We consider the homotopic equation (3.3) of (1.4). Define the operator by

Let be the Nemytskii operator

By hypothesis (H₆), it is easy to see that

Thus, the operator sends into . Hence, the problem of odd anti-periodic 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 anti-periodic 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 (H₂) in Theorem 3.1, Theorem 3.2 and obtain the following results.

Theorem 3.5Assume that

(H₇) and the assumption (H₁) is true.

Then (1.3) () has at least one even anti-periodic solution.

Theorem 3.6Suppose that the assumptions (H₄), (H₇) are true. Then (1.3) () has at least one odd anti-periodic solution.

Basing on the proof of Theorem 2 in [10], for the possible anti-periodic solution of (3.1) (), the hypothesis (H₇) 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.

In this section, we will give some existence results on anti-periodic solutions with symmetry of (1.5).

Theorem 4.1Assume that

(H₈) there exist non-negative functionssuch that

(H₉) and the assumption (H₅) is true.

Then (1.5) has at least one even anti-periodic solution.

Proof We consider the following homotopic equation of (1.5):

Define the operator by

where

Let be the Nemytskii operator

Obviously, the operator is invertible and the problem of even anti-periodic solutions for (4.1) is equivalent to the operator equation

From hypotheses (H₈), (H₉) and (3.8) in [15], for the possible even anti-periodic 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 anti-periodic solutions of (1.5).

With this in mind, let us define the set as follows:

By hypothesis (H₅), 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 (H₆), (H₈), (H₉) are true. Then (1.5) has at least one odd anti-periodic solution.

Proof We consider the homotopic equation (4.1) of (1.5). Define the operator by

where

Let be the Nemytskii operator

Thus, the problem of odd anti-periodic 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 anti-periodic solutions of (1.5). With this in mind, let us define the following set:

By hypothesis (H₆), 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

(H₁₀) there exist non-negative constantsγ, rwith, such that

(H₁₁) there are non-negative functionssuch that

(H₁₂) and the assumption (H₅) is true.

Then (1.5) has at least one even anti-periodic solution.

Theorem 4.4Suppose that the assumptions (H₆), (H₁₀), (H₁₁), (H₁₂) are true. Then (1.5) has at least one odd anti-periodic solution.

Basing on the proof of Theorem 3.2 in [15], for the possible anti-periodic solution of (4.1), the hypotheses (H₁₀), (H₁₁), (H₁₂) 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 (H₁₀), (H₁₁), (H₁₂) guarantee that the degree with respect to x of is allowed to be greater than , which is different from the hypothesis (H₈) of Theorem 4.1 and Theorem 4.2.

The authors declare that they have no competing interests.

HP carried out the theoretical analysis. JY drafted the manuscript. All authors read and approved the final manuscript.

This work was supported by the National Natural Science Foundations of China (50904065) and the Program for New Century Excellent Talents in University (NCET-09-0728). 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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