A Periodic Solution of the Generalized Forced Liénard Equation

Abstract

We consider the generalized forced Liénard equation as follows: . By applying Schauder's fixed point theorem, the existence of at least one periodic solution of this equation is proved.

1. Introduction

Liénard equation is a kind of differential equations which has a broad set of applications in physics, engineering, and so forth. The existence of at least one periodic solution of this equation has been studied by a number of authors (see, e.g., [1–6]). Forced Liénard equation can be considered as an important generalization of Liénard equations. This one appears in a number of physical models such as fluid mechanics and nonlinear elastic mechanical phenomena.

In this paper, we deal with a generalized forced Liénard equation: where , , and are real functions on such that , is a -periodic real function on , , and is an increasing homeomorphism with such that and . Actually, we show that the methodology of [5] can be adopted for more general classes of forced Liénard equations. The application of these equations is in physical models such as classical forced pendulum equations. As an example, consider the following forced pendulum equation: where , , and is -periodic function for , where is the mean value of and has the property . By taking , , and , (2) can be considered as an example of (1). This problem is investigated in [7] (see [5] for another example).

2. Existence of a Periodic Solution

In this section, we state and prove the existence of at least one periodic solution of the generalized forced Liénard equation by using Schauder’s fixed point theorem (see [8–10]). In order to do this, at first consider the following generalized forced Liénard equation: Suppose that , , and are real functions on which are locally Lipschitz such that and . Also, suppose that is a nonconstant, continuous, and real function on . Suppose that is the maximum value of on , and , , and are the maximum values of , , and on , respectively. Also, suppose that . Actually, we describe the set of mean values for which (3) has at least one -periodic solution.

Similar to [5], choose a new variable ; then, (3) can be written as follows: We now integrate (4) to have where is a constant.

In the next step, for all , we define the operator by So by (6), (5) can be written as follows: One now integrates (7) to get where is a constant.

Suppose that , where stands for -periodic continuous functions with zero mean value and Note that is a Banach space. Now, define the set as follows: One can show that is a closed, bounded, and convex subspace of . Define the operator by First, by Lemma 1 of [5], there exists a unique choice of and , such that . Now, we prove that . In order to show this, let be the maximum value of on , and let , , and be the maximum values of , , and on , respectively. Using (6), we have where . Hence, for all so Also, we have Set Hence, for each and , we have Using the same method of [5], one can prove the existence of at least one -periodic solution of (1) as follows.

Theorem 1. Suppose that , , and are real functions on which are locally Lipschitz such that and for real numbers and in which , is a nonconstant, continuous, and -periodic real function on for , and   is an increasing homeomorphism with , and . Let , , , and be defined by (11), (10), and (16), respectively. If then (3) has at least one -periodic solution on .

Proof. According to the definition of and (17), is continuous. Hence, in order to use Schauder’s fixed point theorem, we need to prove that is compact and maps into itself. For and , we have because so hence, Now, by (18) for all , So .
We recall a given sequence of functions from to , which is called equicontinuous if for every there exists such that for all and for all if then . Now, we recall the Ascoli-Arzela theorem as follows.

Theorem 2 (Ascoli-Arzela). Let be a sequence of functions from to which is uniformly bounded and equicontinuous. Then, has a uniformly convergent subsequence.

In the sequel, we show that is a compact operator on . For this, we show that each bounded sequence in has a convergent subsequence in . Suppose that is a sequence in . It is clear that is bounded. Set , for given ; then, there exists such that and, for every , if , then So, Therefore, is an equicontinuous sequence on . By the Ascoli-Arzela theorem, there exists a subsequence of such that is uniformly convergent on . Hence, is a compact operator.

Now, we recall Schauder’s fixed point theorem as follows.

Theorem 3 (Schauder’s fixed point theorem). Let be a Banach space and a closed, bounded, and convex subset of . If is a compact operator, then has at least one fixed point on .

Therefore, by Schauder’s fixed point theorem, there exists such that . This means that is a solution of (1).

When , Theorem 1 is satisfied by the following changes: Suppose that are defined by where is defined by . Now, we have the next corollary.

Corollary 4. Suppose that , , and are real functions on which are locally Lipschitz such that and for real numbers and in which , is a nonconstant, continuous, and -periodic real function on for , and   is an increasing homeomorphism with , and . Let , , , and be defined by (10), (27), and (28), respectively. If then (3) has at least one -periodic solution on .

Proof. Set Now, one can prove the existence of at least one -periodic solution of (3) by the same argument in the proof of Theorem 1.

Now, we consider a special case and prove the existence of at least one -periodic solution of the equation where , , and are real functions on such that , is a -periodic real function on ,  , and . Consider that , , and are real functions on such that and . Suppose that is the maximum value of on and , , and are the maximum values of , , and on . Also, suppose that is defined by (11). In the following, it is shown that (31) has at least one -periodic solution.

Define a new variable ; thus, (31) can be rewritten as follows: Then, where is a constant.

Now, define the operator , for all , by Also, So, where is a constant.

Consider that and is defined by (9). Define by is a closed, bounded, and convex subspace of . Define the operator by Again, by Lemma 1 of [5], there exists a unique choice of and , such that and with the same argument as before, .

Now, the following corollary proves the existence of at least one -periodic solution for (31) on .

Corollary 5. Suppose that , , and are real functions on which are locally Lipschitz such that and for real numbers and in which , is a nonconstant, continuous, and -periodic real function on for , and . Let , , and be defined by (16) and (37), respectively. If then (31) has at least one -periodic solution on .

Proof. Suppose that is defined by (38). Now, one can prove the existence of at least one -periodic solution of (31) by the same method of Theorem 1, (case is the same).

Remark 6. A good question is to study the stability of -periodic solution of (1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referee for the valuable comments and suggestions.