On Stability of Periodic Solutions of Lienard Type Equations

Abstract

We use the Floquet theory to analyze the stability of periodic solutions of Lienard type equations under the asymptotic linear growth of restoring force in this paper. We find that the existence and the stability of periodic solutions are determined primarily by asymptotic behavior of damping term. For special type of Lienard equation, the uniqueness and stability of periodic solutions are obtained. Furthermore, the sharp rate of exponential decay of the stable periodic solutions is determined under suitable conditions imposed on restoring force.

1. Introduction

This paper is devoted to the stability of large periodic solutions of the following Lienard type equation: where is a continuous function and is a -periodic function. The existence and multiplicity of periodic solutions of (1) or more general types of nonlinear second-order differential equations have been investigated extensively by many authors. For details, we refer the reader to [1–4]. One can mention, for example, the papers by Fonda and Habets [5] or more recent papers by Qian [6] and the literature therein. In these papers, the asymptotic behavior of restoring force and damping term are controlled by the inequalities These tend to keep away the combinational effects of the quotients and from the spectrum of the linear operator as . However, the sign condition on damping term on solvability of (1) seems to be ignored. The aim of the paper is to show that sign condition plays dominate role on the existence of periodic solutions of (1) under the condition that is asymptotically linear. Resonance phenomenon may appear when . In this case, the Landesman-Lazer type solvability condition should be imposed. In this paper, the following hypotheses on and are imposed: orwhere the second limit converges uniformly for .

When the hypotheses are satisfied, which can be treated as the resonance conditions, (1) does not exist with -periodic solution in general, even if does not change sign for all . A counterexample will be given at the end of Section 2. In this case, an additional condition should be imposed to guarantee the existence.

The related results about the stability of periodic solutions were less extensively studied. In [7], Lazer and Mckenna established stability results by converting the equation to a fixed point problem. Recently, more complete results concerning the stability and the sharpness of the rate of decay of periodic solutions were obtained by Chen and Li in [8, 9].

The following notations will be used throughout the rest of the paper:(1): -periodic function with for ;(2): -periodic function , with -norm;(3): if on and on some positive measure subset.

The main results of this paper are the following.

Theorem 1. Assume , , and -periodic in , satisfying (4) or (5). Then (1) has at least one -periodic solution.

Let us introduce the following symbols: where .

Theorem 2. Assume that is bounded, where and (6). Then (2) has a -periodic solution provided that either or for any .

Remark 3. When and are continuous, the conclusion of Theorem 2 still holds if the second and third term in (8) or (9) are replaced by .

In particular, by Theorem 2, we see that the following well known Landesman-Lazer type solvability conditions hold, if both and are all constants.

Theorem 4. Under the same conditions of Theorem 2, where and are constants, then (1) has a -periodic solution if either or

When is a linear function, more refined results can be obtained. Concerning uniqueness and stability as well as the rate of decay to the unique periodic solution, we have the following.

Theorem 5. Assume that where and , such thatThen (1)(1) has a unique -periodic solution;(2)the unique -periodic solution is locally exponential asymptotically stable;(3)the unique -periodic solution is globally asymptotically stable.Furthermore, if the conditions, hold for some , then every solution to (1) that is near the unique -periodic solution decays uniformly at the exponential rate of .

2. Preliminaries

In this section we shall recall some basic results about topological methods.

Consider the periodic boundary value problemwhere is a continuous function and -periodic in . In order to use a homotopic method to compute the degree, we assume that is a continuous function such that where is continuous. The following continuation theorem is due to Mawhin [10].

Lemma 6. Let be an open bounded set such that the following conditions are satisfied. (1)There is no such that .(2). Then (12) has at least one solution.

Next, we consider the homogeneous periodic equation where is constant and .

The following Lemma is crucial to the argument for existence of periodic solutions.

Lemma 7. Assume that satisfies either or . Then (14) does not admit any nontrivial -periodic solutions.

Proof. Suppose on the contrary that there exists a nontrivial -periodic solution . Multiplying both sides of (14) by and , respectively, integrating by parts, and applying the boundary condition, we get thenThis implies we have on positive measure subset of . It follows from Rolle’s theorem that the derivative has a zero, between two zeros of . Let be an accumulation point of zero of , such that exists. Otherwise, the zeros of are isolated; hence the set consisting of zeros of is a zero measure set. Evidently, at such point . According to a theorem concerning the uniqueness of initial value problem, we have on .

Let us give a counterexample, which demonstrates that (1) does not possess any periodic solutions under condition (6).

Consider the following equation: where and satisfies condition (6) but (17) does not admit any -periodic solutions for .

Indeed, if is a -periodic solution of (17), multiplying both sides of (17) by and integrating over a period, we obtain which means that

Thus, the equation does not have any -periodic solutions for . Moreover, according to Massera’s theorem [11], we obtain that any solutions of (17) are unbounded for .

For convenience, we begin with a definition.

Definition 8. Let be a -periodic solution of , where . Then is stable if for each there exists a , such that if is any solution of for which at some , then for any .

And next, we shall recall a principle of linearized stability for periodic systems.

Let be a -periodic solution of (12); then we associate the -periodic solution with the linearized equation

Let be the fundamental matrix of (22) and and the eigenvalues of the matrix . Then is exponential asymptotically stable if and only if , . Otherwise, if there exists an eigenvalue of with modulus greater than one, then is unstable.

In order to show that every solution of the nonlinear equation (1) locally decays at the rate of to the unique -periodic solution, we need the following version of the Hartman-Grobman theorem [12].

Lemma 9. Let be an open neighborhood of 0 and be a function such that is a contraction mapping. Then is conjugate equivalent to .

Consider the following boundary value problem:we have the following lemma which is given by the author in [9]

Lemma 10. Let such that for some . Then (23) does not admit any nontrivial solution.

Proof. Consider the eigenvalue problem we have , the eigenvalue of the above equation. If is a nontrivial solution of (23), then is an eigenfunction associated with eigenvalue for some of the eigenvalue problemSince , it follows from the comparison principle that Thus . On the other hand, it follows from Sturm theory that the eigenfunction corresponding to the eigenvalue of (26) has exact zeros in , which indicates that . Together with the boundary condition of (23), we have . This is a contradiction.

Now we consider the homogeneous periodic equation

We have the following.

Lemma 11. Assume that there exists an integer such that Then (28) does not admit any positive Floquet multipliers. In particular, (28) does not admit any nontrivial -periodic solutions.

Proof. Suppose that there is a nontrivial -periodic solution of (28) such that with .
Set then solves the following equation: with the Floquet multiplier .
Claim. There exists some such thatAssume, by way of contradiction, that (31) does not hold. Then for all . Dividing (30) by and integrating from to by parts, noticing that , we obtain which contradicts condition . This implies that the claim (31) holds.
Since , we see that is a nontrivial solution of the Dirichlet boundary value problemwhere . Multiplying (33) by and integrating from and , we have, by which contradicts the poincaré inequality.

Similarly, consider the following Dirichlet boundary value problem:we can derive the following lemma.

Lemma 12 (see [9]). Let such that for some . Then (35) does not admit any nontrivial solution.

Lemma 13. Suppose that there is an integer such that Then (28) does not admit any negative Floquet multipliers. In particular, (28) does not admit any nontrivial subharmonic periodic solution of order 2.

Combining Lemmas 11 and 13, under the condition of Theorem 5, we can prove that (1) does not admit any real Floquet multiplier.

3. Proof of Main Results

Now we are ready to prove our main results.

3.1. Proof of Theorem 1

Proof. Without loss of generality, we may assume that ; otherwise, we can subtract from both sides of (1).
Consider the parametrized equation, where .
First, we claim that there is an which is independent of such that for any solution of (37).
If there is not such an , let be a sequence such that and , and denote by the ratio . Dividing (37) by , multiplying by , and integrating from to gives The condition of Theorem 1 implies that is bounded. It is precompact in the topology in . Thus there are subsequences such that and . Passing to the limit in (38), we get where .
If (4) holds, then , which satisfies the condition of Lemma 7. Since is a -periodic solution of it follows from Lemma 7 that , which contradicts the fact that . This completes the boundedness of under condition (4).
Next, we show that is bounded which is independent of under condition (5).
If , then satisfiesIt follows from Lemma 7 that , which contradicts .
If , then satisfies By assumption, is not the eigenvalue of . Obviously, ; we reach a contradiction. This shows that the solution of (1) is bounded.
Evidently, the periodic solution of (37) is equivalent to the planar system A natural choice for the homotopy in applying Lemma 6 is to take Let be the -periodic solution of (43); in order to apply Continuation Theorem to (43), we have to show that is bounded. Directly from the first equation of (43) and the periodic condition, we see that there is such that which implies that is bounded which is independent of . Integrating the second equation of (43) yields which is bounded.
Let and be sufficiently large, and set It follows from the estimates obtained above that the equivalent planar system defined in (43) has no solutions on for . Since then condition (2) in Lemma 6 reduces to By applying Lemma 6, we see that (1) has at least one -periodic solution.

3.2. Proof of Theorem 2

Proof. The idea of the proof of Theorem 2 is essentially the same as above, so here we just outline the proof and explain how to use the resonance conditions and (8)-(9) to get desired a priori estimates.
Consider the parametrized equation First, we will show that there is which is independent of such that for any solutions of (50).
If not, let be a sequence of -periodic solutions such that and be the corresponding sequence. Let . Noting the resonance condition (6), the same procedure of the proof of Theorem 1 gives that there exists some subsequences such that weakly and . Passing to the limit, one obtains that satisfies Namely, If , it follows from Lemma 7 that , which contradicts . Hence it remains to focus on the case that and .
In , we introduce the following symbol:Since , we have that .
That is to say, and . Then . Thus, the sign of is the same as that of for large enough. Taking inner product of (50) with and noting that , where is replaced by and by , we have Taking the limits in the above equation, we obtain which contradicts (9). This shows the boundedness of the periodic solutions of (50).
Similarly, by taking one can prove that periodic solutions of (50) are bounded under condition (10).
The rest is along the same line as the proof of Theorem 1; we omit the detail.

3.3. Proof of Theorem 5

Proof. Firstly, we will show that there exists a unique -periodic solution.
The existence of -periodic solution of (1) has been obtained in the proof of Theorem 1. Hence it suffices to investigate the uniqueness of (1).
Let and be two distinct -periodic solutions of (1), and . Then is a nontrivial -periodic solution of the following equation: where .
We claim that, under the assumption of the theorem, there exists no nontrivial -periodic solution of the linearized equation (28) associated with (1).
Supposing the opposite of the previous claim, multiplying (28) by and integrating from 0 to , we havewhich contradicts the fact that both and are positive. In the same manner, we can demonstrate that there exists no nontrivial -periodic solutions.
By the similar argument as above, multiplying (56) by and integrating by parts, a contradiction will be reached directly. Therefore, .
Secondly, we will show that the unique -periodic solution is locally asymptotically stable.
For each with , we have Therefore, from the above conclusion, for each , there exists a unique periodic solution of period of the linear homogeneous differential equation Under the hypothesis of theorem, we can conclude that there exists no nontrivial -periodic solution of (59) for all .
Ifthen is a -periodic solution of if and only if and , where is a nontrivial -periodic solution of (59). Therefore, system (61) has no nontrivial -periodic solution for .
Let denote the fundamental matrix associated with system (61) for ; we have where is the identity.
If for some then is a solution of system (61) which satisfies . So by the fact that and are both solutions of (61) which are equal at and the uniqueness theorem, it follows that . Hence, is a nontrivial -periodic solution. So , and therefore .
It follows that is not an eigenvalue of for all .
A similar argument as above shows that if , then is a -periodic solution, and hence a -periodic solution of (61). Therefore, for , is not an eigenvalue of .
For each with , let and denote the eigenvalues of . By standard results concerning continuous dependence of solutions of differential equations on parameters and Rouche’s theorem of complex analysis, the moduli and depend continuously on . We will show that and since , where is the fundamental matrix for the linear system and is the matrix function Since the trace of is equal to , applying Liouville’s theorem, we have For is equal to the matrix so and , where and are the eigenvalues of . Since , it is easy to verify that both and have negative real parts. Therefore, for .
If (64) were not true for , then, by continuity, there exists a with such that , for either or .
and can not be complex conjugates, for this means , which contradicts with (67). Therefore, both and would be real, so for either or . This contradicts the fact that can not have as an eigenvalue for all .
Hence, the eigenvalues of have moduli less than for . In other words, the unique -periodic solution is locally asymptotically stable.
Thirdly, we will show that the unique -periodic solution is globally asymptotically stable.
We choose as a Lyapunov function where is the unique -periodic solution satisfying the following equation: A direct computation, by , yields Thus we display that the periodic solution is globally asymptotically stable.
Finally, we are in a position to prove the rate of decay of the solution to the unique -periodic solution is .
Consider the planar system associated with (1),Let be the unique -periodic solution determined by the initial condition . Then corresponds to the unique fixed point of the poincaré mapping , where is the initial value solution of (72) with .
Let be the fundamental matrix solution of the linearization (61) of (72), and by the differentiability of with respect to the initial value, the poincaré mapping can be expressed in terms of the initial value by By virtue of Lemmas 11 and 13, we see that has a pair of conjugate eigenvalue and . Thus is a contracting mapping. According to Lemma 9, there is a diffeomorphism which is near enough to the identity that is conjugate equivalent to . There is an invertible constant matrix such that and we may suppose that for small, since is near the identity.
So, the Lyapunov exponent is given by Hence, the rate of decay of the solution to the unique -periodic solution is , independent of the initial value . This completes the proof of Theorem 5.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.