Antiperiodic Solutions for Liénard-Type Differential Equation with -Laplacian Operator

The existence of antiperiodic solutions for Liénard-type and Duffing-type differential equations with -Laplacian operator has been studied by using degree theory. The results obtained improve and enrich some known works to some extent.

Antiperiodic problems arise naturally from the mathematical models of various of 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 travelling-wave tube's theories is an antiperiodic problem (see [6]).

During the past twenty years, antiperiodic 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 antiperiodic solutions was considered in [9–12]. Recently, existence results were extended to antiperiodic boundary value problems for impulsive differential equations (see [13]), and antiperiodic wavelets were discussed in [14].

Wang and Li (see [15]) discussed the existence of solutions of the following antiperiodic boundary value problem for second-order conservative system:

using of the main assumption as follows:

() There exist constants and , such that

The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson (see [16]) introduced the following -Laplacian equation:

where , . Obviously, the inverse operator of is , where is a constant such that .

Notice that, when , the nonlinear operator reduces to the linear operator .

In the past few decades, many important results relative to (1.3) with certain boundary conditions have been obtained. We refer the reader to [17–20] and the references cited therein. However, to the best of our knowledge, there exist relatively few results for the existence of antiperiodic solutions of (1.3). Moreover, it is well known that the existence of antiperiodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [21]). Thus, it is worthwhile to continue to investigate the existence of antiperiodic solutions for (1.3).

A primary purpose of this paper is to study the existence of antiperiodic solutions for the following Liénard-type -Laplacian equation:

and antiperiodic solutions with symmetry for Duffing-type -Laplacian equation as follows:

where , with , , and . That is, we will prove that (1.4) or (1.5) has at least one solution satisfying

Note that, is also a -periodic solution of (1.4) or (1.5) if is a -antiperiodic solution of (1.4) or (1.5). Hence, from the arguments in this paper, we can also obtain the existence results of periodic solutions for above equations.

The rest of this paper is organized as follows. Section 2 contains some necessary preliminaries. In Section 3, we establish some sufficient conditions for the existence of antiperiodic solutions of (1.4), basing on Leray-Schauder principle. Then, in Section 4, we obtain two existence results of antiperiodic solutions with symmetry for (1.5). Finally, in Section 5, some explicit examples are given to illustrate the main results. Our results are different from those of bibliographies listed above.

For convenience, we introduce some notations as follows:

and denotes norm in .

For each , there exists the following Fourier series expansion:

where . Let us define the mapping by

Notice that, may be written as Fourier series as follows:

and may be written as the following Fourier series:

We define the mapping by

and the mapping by

It is easy to prove that the mappings are completely continuous by using Arzelà-Ascoli theorem.

Next, we introduce a Wirtinger inequality (see [22]) and a continuation theorem (see [23, 24]) as follows.

Lemma 2.1 (Wirtinger inequality).

For each such that and , one has

where

Lemma 2.2 (Continuation theorem).

Let be open-bounded in a linear normal space . Suppose that is a completely continuous field on . Moreover, assume that the Leray-Schauder degree

Then equation has at least one solution in .

In this section, an existence result of antiperiodic solutions for (1.4) will be given.

Theorem 3.1.

Assume that

there exists a nonnegative function such that

where

Then (1.4) has at least one antiperiodic solution.

Remark 3.2.

When , is equal to 1. It is easy to see that condition () in [15] is stronger than condition () of Theorem 3.1.

For making use of Leray-Schauder degree theory to prove the existence of antiperiodic solutions for (1.4), we consider the homotopic equation of (1.4) as follows:

Define the operator by

where

Let be the Nemytski operator

Obviously, the operator is invertible and the antiperiodic problem of (3.3) is equivalent to the operator equation

We begin with some lemmas below.

Lemma 3.3.

Suppose that the assumption is true. Then the antiperiodic solution of (3.3) satisfies

where is a positive constant only dependent of and .

Proof.

Multiplying the both sides of (3.3) with and integrating it over , we get

Noting that

and , we have

By hypothesis , there exists a nonnegative constant such that

Thus, from (3.11), we have

That is,

where .

For each , we get

Similarly, we obtain that

Basing on Lemma 2.1, it can be shown from (3.17) and (3.14) that

Let , then

The proof is complete.

Lemma 3.4.

Suppose that the assumption is true. Then, for the possible antiperiodic solution of (3.3), there exists a prior bounds in , that is, satisfies

where is a positive constant independent of .

Proof.

By (3.15), there exists such that . Hence, (3.8) yields that

Letting

From (3.16), there exists such that , which implies that . Therefore, integrating the both sides of (3.3) over , we have

Thus, we get from (3.8) that

Noting that , we obtain that

Combining (3.21) with (3.25), we have

where . The proof is complete.

Now we give the proof of Theorem 3.1.

Proof of Theorem 3.1..

Setting

Obviously, the set is an open-bounded set in and zero element .

From the definition of operator , it is easy to see that

Hence, the operator sends into . Let us define the operator by

Obviously, the operator is completely continuous in and the fixed points of operator are the antiperiodic solutions of (1.4).

With this in mind, let us define the completely continuous field by

By (3.20), we get that zero element for all . So that, the following Leray-Schauder degrees are well defined and

Consequently, the operator has at least one fixed point in by using Lemma 2.2. Namely, (1.4) has at least one antiperiodic solution. The proof is complete.

In this section, we will prove the existence of even antiperiodic solutions or odd antiperiodic solutions for (1.5).

Theorem 4.1.

Assume that

the functions and are even in , that is,

and the assumption () is true. Then (1.5) has at least one even antiperiodic solution , that is, satisfies

Proof.

We consider the homotopic equation of (1.5) as follows:

Define the operator by

Obviously, the operator is continuous.

Basing on the proof of Theorem 3.1, for the possible even antiperiodic solution of (4.3), there exists a prior bounds in , that is, satisfies

where is a positive constant independent of . So that, our problem is reduced to construct one completely continuous operator in 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 following set:

Obviously, the set is an open-bounded set in and zero element .

By hypothesis (), it is easy to see that

Hence, the operator sends into . Let us define the completely continuous operator by

From the similar arguments in the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator in . Thus, (1.5) has at least one even antiperiodic solution. The proof is complete.

Theorem 4.2.

Assume that

the function is odd in and is odd in , that is,

and the assumption () is true. Then (1.5) has at least one odd antiperiodic solution , that is, satisfies

Proof.

We consider the homotopic equation (4.3) of (1.5). Define the operator by

Obviously, the operator is continuous.

Based on the proof of Theorem 3.1, for the possible odd antiperiodic solutions of (4.3), there exists a prior bounds in . Hence, our problem is reduced to construct one completely continuous operator in 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 set as follows:

Obviously, the set is an open-bounded set in and zero element .

From the hypothesis (), it is easy to see that

Thus, the operator sends into . Let us define the completely continuous operator by

By a similar way as the proof of Theorem 3.1, we can prove that there exists at least one fixed point of operator in . So that, (1.5) has at least one odd antiperiodic solution. The proof is complete.

In this section, we will give some examples to illustrate our main results.

Consider the following second-order differential equation with -Laplacian operator:

Example 5.1.

Let

For , by direct calculation, we can get . Choosing , then (5.1) satisfies the condition of Theorem 3.1. So it has at least one antiperiodic solution.

Moreover, the conditions of Theorem 4.1 are also satisfied. Thus (5.1) has at least one even antiperiodic solution.

Example 5.2.

Let

We choose . Obviously, (5.1) satisfies all the conditions of Theorem 4.2. Hence it has at least one odd antiperiodic solution.

The authors would like to thank the referees very much for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China (10771212), the Fundamental Research Funds for the Central Universities, and the Science Foundation of China University of Mining and Technology (2008A037).

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