Abstract and Applied Analysis

Volume 2013 (2013), Article ID 906972, 10 pages

http://dx.doi.org/10.1155/2013/906972

Research Article

Lazer-Leach Type Conditions on Periodic Solutions of Liénard Equation with a Deviating Argument at Resonance

Zaihong Wang

School of Mathematical Sciences, Capital Normal University, Beijing 100048, China

Received 28 February 2013; Accepted 15 April 2013

Academic Editor: Wing-Sum Cheung

Copyright © 2013 Zaihong Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We study the existence of periodic solutions of Liénard equation with a deviating argument where are continuous and is -periodic, is a constant, and is a positive integer. Assume that the limits and exist and are finite, where . We prove that the given equation has at least one -periodic solution provided that one of the following conditions holds: , for all , for all , for all , for all

1. Introduction

We are concerned with the existence of periodic solutions of Liénard equation with a deviating argument as follows:

where are continuous and is -periodic, is a constant, and is a positive integer.

In recent years, the periodic problem of Liénard equations with a deviating argument has been widely studied because of its background in applied sciences (see [1–8] and the references cited therein).

In the case when , for all and , (1) becomes

Assume that limits

exist and are finite. Lazer and Leach [9] proved that (2) has at least one -periodic solution provided that the following condition holds:

Assume that, besides , the limits

exist and are finite, where . It was proved in [10] that the following equation:

has at least -periodic solution provided that one of the following conditions holds:

In the case when , for all and , (1) becomes as follows:

When the condition holds, it was proved in [5] that (6) has at least one -periodic solution provided that the condition (3) holds.

In the present paper, we deal with the existence of periodic solutions of (1) by assuming and . By using the continuation theorem [11], we prove the following result.

Theorem 1. Assume that the conditions and hold. Then (1) has at least one -periodic solution provided that one of the following conditions is satisfied:

Remark 2. Let us denote by the function on the right-hand side of four inequalities above, namely,

Then can be expressed in the following form:

where

Obviously, the value of forms a closed interval with . Therefore, the four conditions in Theorem 1 are equivalent to the following conditions, respectively:

Remark 3. In the case when , the four conditions in Theorem 1 reduce to the conditions (5). Therefore, Theorem 1 generalizes the result in [10].

Throughout this paper, we always use to denote the real number set. For a multivariate function depending on , the notation always means that, for , holds uniformly with respect to other variables, whereas (or ) always means that (or ) is bounded for large enough. For any continuous -periodic function , we always set .

2. Basic Lemmas

It is well known that continuation theorems play an important role in studying the existence of periodic solutions of differential equations. We now introduce a continuation theorem which will be used to prove the existence of periodic solutions of (1).

Let and be two real Banach spaces and let be a Fredholm operator with index zero, where denotes the domain of . This means that is a closed subspace of and . Let , be two linear continuous projectors satisfying the following:

Then we have the following:

Clearly, is invertible. Denote by the inverse of . Let be an open bounded set. A continuous map is said to be -compact on if both and are compact.

Lemma 4 (see [11]). Let X and Y be two real Banach spaces. Suppose that is a Fredholm operator with index zero and is -compact on , where is an open bounded subset of . Moreover, assume that all the following conditions are satisfied:

(1), for all , ;(2), for all ;(3)The Brouwer degree , where is an isomorphism.

Then equation has at least one solution on .

3. Main Results

In this section, we will use the continuation theorem introduced in Section 2 to prove the existence of periodic solutions of (1). To this end, we first quote some notations and definitions.

Let and be two Banach spaces defined by the following:

with the following norms

Define a linear operator

where , and a nonlinear operator

It is easy to see that

On the other hand, for any , we can write the following:

where is defined by the following:

with

whereas satisfies the following:

Therefore,

It follows that is a Fredholm map of index zero.

Let us define two continuous projectors and by setting the following:

for any and , where constants and are defined as constants and . Obviously, .

Set . Then is an algebraic isomorphism and we define by the following:

Clearly, we have that, for any ,

For any open bounded set , we can prove by standard arguments that and are compact on the closure . Therefore, is -compact on .

It is noted that (1) is equivalent to the operator equation

To use Lemma 4, we embed this operator equation into an equation family with a parameter ,

which is equivalent to the equation as follows:

In the following, we will prove some new results on the existence of periodic solutions of (1) by using the continuation theorem. Consider the equivalent system of (29):

Let be any (possible) -periodic solution of (29). Set . Then is a -periodic solution of (30).

Now, let us introduce a transformation with ,

Under the transformation , if , for , then the -periodic solution of (30) can be expressed in the form satisfying the equations as follows:

Let us set with . Without loss of generality, we always assume . Dividing the second equation of (32) by , we get the following:

Integrating (33) and applying conditions and , we get the following:

Furthermore,

On the other hand, it follows from the first equation of (32) and (35) that

Therefore, we get the following:

The estimations (34)–(37) will be used to obtain apriori bounds of -periodic solutions of (29). Multiplying both sides of (29) by and , respectively, and integrating over the interval , we obtain the following:

Hence,

Multiplying both sides of (29) by and , respectively, and integrating over the interval , we obtain the following:

Hence,

Proof of Theorem 1. We shall prove the existence of periodic solutions of (1) provided that either

or

holds by using (39) and (40). The other cases can be handled similarly by using (43) and (44). We proceed in three steps.

(1) We prove that there exist positive constants and such that, for any -periodic solution of (29),

Assume by contradiction that (45) does not hold. Then there exists a sequence of -periodic solutions of (29) with such that

Write . Since is bounded on the interval , we have the following:

Let be the -periodic solution of (32) related to . Obviously, . Then we have the following:

Without loss of generality, we also assume . It follows from (39) that

From (34) and (37) we get that, for ,

Therefore,

Obviously, we have the following:

Since , there exists a subsequence such that , . By using Lebesgue dominated convergent theorem and the condition , we get the following:

Therefore,

Similarly, we can get the following:

Hence, we obtain the following:

which contradicts with (43).

On the other hand, it follows from (40) that

According to (50), we have the following:

From (53) we obtain the following:

Similarly, we have the following:

It follows from (57)–(60) that

which contradicts with (44). Therefore, there exist positive constants and such that (45) holds.

(2) Let , where is an arbitrary constant. We will prove that there exists such that, for , . Otherwise, there exits a sequence satisfying such that with . We will prove the following:

In fact, since , we have the following:

Using the same method as in step 1, we have the following:

As a consequence, (62) holds. Thus, we get a contradiction.

(3) Let be a sufficiently large constant (if it is necessary, can be enlarged). Set

From the conclusion in step 1 we know that

From the conclusion in step 2 we know that

which implies for any . Since , we can take an isomorphism . In what follows, we will prove the following:

To this end, let us define , , namely,

Obviously, is a linear isomorphism. For any , set

where

Define as follows:

Then we have the following:

To calculate , we first estimate and as follows:

Write with , or . Then we have that, for ,

On the other hand, we have that, for ,

Therefore, we get the following:

To estimate , we have that, for ,

Meanwhile, we get that, for ,

Hence, we obtain the following:

Set

Replacing in with , we get the following:

As a consequence,

We note that, for ,

where

Let us consider the map with

Obviously, is continuous. Next, we shall prove that, for any ,

Otherwise, there exists some such that

Then we have the following:

Therefore, we get the following:

Since , we know from (90) that

where is given in Remark 2. From Remark 2 we know that (91) and (92) contradict with (43) and (44).

In particular, we have that, for ,

Since is odd and (93) holds, we know from Borsuk Theorem [12] that

where is an integer.

On the other hand, we know from (94) and the expressions of and that there exists a positive constant , which is independent of and , such that, for

Consequently, we infer from the homotopy invariance of degree that, if is large enough; then

Therefore, all conditions of Lemma 4 are satisfied. Thus, (1) has at least one -periodic solution.

4. Remarks

We can use the method developed in Section 3 to deal with the existence of -periodic solutions of the following equation:

Assume that the limits

exist and are finite. We can prove the following theorem.

Theorem 5. Assume that the conditions and hold. Then (98) has at least one -periodic solution provided that one of the following conditions holds:

Remark 6. In the case when , the third and the fourth condition in Theorem 5 are identical to the related conditions in [6]. But the first and the second condition in Theorem 5 did not appear in [6].

Acknowledgments

The research was supported by Research Fund for the Doctoral Program of Higher Education of China, no. 11AA0013, Beijing Natural Science Foundation (Existence and multiplicity of periodic solutions in nonlinear oscillations), no. 1112006, and the Grant of Beijing Education Committee Key Project, no. KZ201310028031.

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