On Stability of Periodic Solutions of Lienard Type Equations

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.

The existence and multiplicity of periodic solutions of (1) or more general types of nonlinear second-order differential equations   () +    (,  ()) +  (,  ()) = ℎ () have been investigated extensively by many authors.For details, we refer the reader to [1][2][3][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  () ≤ lim inf ( 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 lim →∞ ((, )/) ≡ 0. In this case, the Landesman-Lazer type solvability condition should be imposed.In this paper, the following hypotheses on  and  are imposed: where the second limit converges uniformly for  ∈ [0, ].
When the hypotheses lim →∞  (, )  ≡ 0, 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  ∈ R. 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 main results of this paper are the following.
Theorem 2. Assume that (, ), (, ) is bounded, where (, ) = (2/) 2 + (, ) and (6).Then (2) has a -periodic solution provided that either or 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.(2) the unique -periodic solution is locally exponential asymptotically stable; (3) the unique -periodic solution is globally asymptotically stable.
Furthermore, if the conditions,

Preliminaries
In this section we shall recall some basic results about topological methods.Consider the periodic boundary value problem where  : [0, ] × R  → R  is a continuous function and periodic in .In order to use a homotopic method to compute the degree, we assume that ℎ : where () is continuous.The following continuation theorem is due to Mawhin [10].
Lemma 6.Let Ω ⊂   be an open bounded set such that the following conditions are satisfied.
Next, we consider the homogeneous periodic equation where  ̸ = 0 is constant and () ∈  ∞  .The following Lemma is crucial to the argument for existence of 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 This implies we have () ≡ 0 on positive measure subset of [0, ].It follows from Rolle's theorem that the derivative   () has a zero, between two zeros of ().Let  0 be an accumulation point of zero of (), such that  0 exists.Otherwise, the zeros of () are isolated; hence the set consisting of zeros of () is a zero measure set.Evidently, at such point ( 0 ) =   ( 0 ) = 0.According to a theorem concerning the uniqueness of initial value problem, we have () ≡ 0 on [0, ].
Let us give a counterexample, which demonstrates that (1) does not possess any periodic solutions under condition (6).
Indeed, if () is a 2-periodic solution of (17), multiplying both sides of (17) by cos  and integrating over a period, we obtain which means that Thus, the equation does not have any 2-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.
And next, we shall recall a principle of linearized stability for periodic systems.
Let  0 be a -periodic solution of ( 12); then we associate the -periodic solution  0 with the linearized equation Let () be the fundamental matrix of ( 22) and  1 and  2 the eigenvalues of the matrix ().Then (,  0 ) is exponential asymptotically stable if and only if |  | < 1,  = 1, 2. Otherwise, if there exists an eigenvalue of () with modulus greater than one, then (,  0 ) is unstable.
In order to show that every solution of the nonlinear equation ( 1) locally decays at the rate of (1/2) ∫  0    (,  0 ()) to the unique -periodic solution, we need the following  1 version of the Hartman-Grobman theorem [12].Lemma 9. Let  be an open neighborhood of 0 and  :  ⊂ R  → R  be a  1 function such that    (0) : R  → R  is a contraction mapping.Then  is  1 conjugate equivalent to    (0).
Similarly, consider the following Dirichlet boundary value problem: we can derive the following lemma.
Lemma 13.Suppose that there is an integer  ∈ N such that (H7) 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.

Proof of Main Results
Now we are ready to prove our main results.

Proof of Theorem 1
Proof.Without loss of generality, we may assume that (0) = 0; otherwise, we can subtract (0) from both sides of (1).
If  = 1, then  satisfies By assumption,  is not the eigenvalue of  = −  .Obviously, () ≡ 0; 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  ∈ [0, ] such that   () = 0 which implies that () is bounded which is independent of .Integrating the second equation of (43) yields which is bounded.
Let  1 and  2 be sufficiently large, and set It follows from the estimates obtained above that the equivalent planar system defined in ( 43) has no solutions on Ω for  ∈ then condition (2) in Lemma 6 reduces to By applying Lemma 6, we see that (1) has at least one -periodic solution.

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  > 0 which is independent of  ∈ [0, 1] such that ‖‖ ∞ <  for any solutions  of (50).
In  1,2  , we introduce the following symbol: which contradicts (9).This shows the boundedness of the periodic solutions of (50).Similarly, by taking lim sup 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.

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).
Supposing the opposite of the previous claim, multiplying (28) by   () and integrating from 0 to , we have which contradicts the fact that both  and    ((), ) are positive.In the same manner, we can demonstrate that there exists no nontrivial 2-periodic solutions.
By the similar argument as above, multiplying (56) by V  () and integrating by parts, a contradiction will be reached directly.Therefore, V ≡ 0.
Secondly, we will show that the unique -periodic solution is locally asymptotically stable.
For each  with 0 ≤  ≤ 1, let  1 () and  2 () 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

Lemma 11 .
Assume that there exists an integer  ∈ N such that The condition of Theorem 1 implies that (  (,   ) + (1 −   )  )/‖  ‖ is bounded.It is precompact in the weak * topology in  1 [0, ].Thus there are subsequences such that (,   )/  → () and   → .Passing to the limit in (38), we get Thus, the sign of   is the same as that of  for  large enough.Taking inner product of (50) with  and noting that (, ) = (2/) 2 + (, ), where  is replaced by   and  by   , we have