Some Results for Periodic Solutions of a Kind of Liénard Equation

τ(t))) = e(t). Some new criteria for guaranteeing the existence and uniqueness of periodic solutions of this equation are given by using theManásevich-Mawhin continuation theorem and some analysis techniques. Our results hold under weaker conditions than some known results from the literature and are more effective. In the last section, an illustrative example is provided to demonstrate the applications of the theoretical results.

As is known, the Liénard equation can be derived from many fields, such as physics, mechanics, and engineering technique fields, and an important question is whether this equation can support periodic solutions.In the past few years, a lot of researchers have contributed to the theory of this equation with respect to the existence of periodic solutions.For example, in 1928, Liénard [1] discussed the existence of periodic solutions of the following equation:   () +  ( ())   () +  ( ())  () = 0, where ,  ∈ (R, R); some sufficient conditions for securing the existence of periodic solutions were established.Afterwards, Levinson and Smith [2] also studied (2) and obtained some new results on the existence of periodic solutions.In 1977, some continuation theorems in [3] were introduced by Gaines and Mawhin.Applying these continuation theorems, many authors discussed the existence of periodic solutions of (2) and generalized the results obtained in [1,2] (see, e.g., [4][5][6][7]); a few authors studied the existence and uniqueness of periodic solutions of (2) (see [8,9]).In 1998, Manásevich and Mawhin [10] studied periodic solutions for certain nonlinear systems with -Laplacian-like operators and provided some new continuation theorems which extended some results in [3].Subsequently, some authors discussed the existence of periodic solutions of certain Liénard-type -Laplacian equations (see, e.g., [11][12][13][14][15]) using these generalized continuation theorems.However, as far as we know, there exist much fewer results on the existence and uniqueness of periodic solutions of (1).The main difficulty lies in the first term (  (  ()))  of (1) (i.e., the -Laplacian operator   : R → R;   () = || −2  is nonlinear when  ̸ = 2), the existence of which prevents the usual methods of finding some criteria for 2 Journal of Function Spaces guaranteeing the uniqueness of periodic solutions of (2) from working.Recently, Gao and Lu [16] discussed the existence and uniqueness of periodic solution of (1) by translating (1) into a two-dimensional system and got some results as follows.
Then (1) has at most one -periodic solution.
Remark 4.However, upon examining their proof of Theorem 3.2 in [16], we have found that the conditions (H 1 )(i), (H 2 ), and (H 3 ) can be dropped.
In this paper, we are keen to dispel any perception that the mathematical proofs of existence and uniqueness that we present are merely verifying facts which might already be obvious in other disciplines, based on purely physical considerations.In particular, in many nonlinear problems arising in practical dynamical systems, physical reasoning alone is not sufficient or fully convincing.In these cases, questions of existence and uniqueness are of importance in understanding the full range of solution behaviour possible and represent a genuine mathematical challenge.The answers to these mathematical questions then provide the basis for obtaining the best numerical solutions to these problems and for determining other important practical aspects of the solution behaviour.
We now reconsider the periodic solutions of (1).The main purpose of this paper is to establish some new criteria for guaranteeing the existence and uniqueness of periodic solution of (1).We obtain some new sufficient conditions for securing the existence and uniqueness of periodic solutions of (1) by using the Manásevich-Mawhin continuation theorem and some analysis techniques.Our results extend and improve the above-mentioned Theorems 3.1 and 3.2 in [16] (see Remarks 9 and 14 and Example 13).
Then the periodic boundary value problem (7) has at least one -periodic solution on .
According to Theorem 3.1 in [16] and the abovementioned Remark 2, we have the following results.Lemma 6. Suppose (A 0 ) holds.Then (1) has at most one periodic solution.

Main Results
Now we are in the position to present our main results.
Second, we prove the existence of -periodic solutions of (1).Set This, together with () > 0 for all  ∈ R and (A 1 ), yields that ()(−) < 0; that is, condition (ii) of Lemma 5 is satisfied.Therefore, it follows from Lemma 5 that there exists a periodic solution () of ( 1).This completes the proof.
Remark 9.It is easy to see that Theorem 8 in this study holds under weaker conditions than Theorem 3.2 in [16].
According to the above discussion, we can also get the following result.Theorem 10.Suppose (A  1 ) holds.Then (1) has at least one periodic solution.
Together with Lemmas 6 and 7 and Theorems 8 and 10, we can directly lead to two theorems as follows.

Example and Remark
In this section, we apply the main results obtained in the previous sections to an example.Proof.If  < 4, condition (H 3 ) in Theorem 3.3 in [16] does not hold any more since  = 3 >  − 1.Therefore, Theorem 3.3 in [16] fails, while our criterion in Theorem 11 in this study remains applicable, as we now show.Let  be an arbitrary positive constant; then we can easily check that conditions (A 0 ) and (A 1 ) in Theorem 11 in this study hold.Hence, Theorem 11 shows that there exists a unique 2periodic solution of (24).Remark 14.This example demonstrates that the conditions in our Theorem 11 are weaker than those conditions in Theorem 3.3 in [16] when () ≡ 0 and are able to demonstrate the existence of a unique periodic solution to certain Liénardtype -Laplacian equations, which the latter cannot decide about.Therefore, our results extend and improve the results in [16].