Electronic Journal of Qualitative Theory of Differential Equations

In this work, we investigate the following p-Laplacian Lienard equation: ('p(x 0 (t))) 0 + f(x(t))x 0 (t) + g(x(t)) = e(t). Under some assumption, a necessary and sufficient condition f or the existence and uniqueness of periodic solutions of this equation is given by using Manasevich-Mawhin continuation theorem. Our results improve and extend some known results.

Here 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 determining other important practical aspects of the solution behaviour.Figure 1 shows the various applications of the Liénard equation (1.1).
The main purpose in this work is to give a necessary and sufficient condition for the existence and uniqueness of T -periodic solutions of (1.1) by using Manásevich-Mawhin continuation theorem.Our results improve and extend some results in [6] The following conditions will be used later: Remark 1. Generally, x refers to the displacement, f (x)x ′ refers to the damping term, g(x) refers to the stiffness term and e refers to the forced term in a vibration system.It implies that the stiffness of the vibration system is monotone decreasing with regard to the displacement if g ′ (x) < 0.
Consider the homotopic equation of (1.1): We have the following lemma.Proof.Let S ⊂ C 1 T be the set of T -periodic solutions of (2.1).If S = ∅, the proof is ended.Suppose S = ∅, and let x ∈ S. Noticing that x(0) = x(T ), x ′ (0) = x ′ (T ) and ϕ p (0) = 0, it follows from (2.1) that T 0 (g(x(t)) − e(t))dt = 0, which implies that there exists τ ∈ [0, T ] such that By (H 1 ), we know that g(x) is strictly decreasing in R.So we have ( Let q > 1 such that 1/p + 1/q = 1.Then by the Hölder inequality we have By (2.2), (2.4) and (2.5), we can get which yields that there exists M 1 > 0 such that |x ′ | 1 ≤ M 1 since p > 1, and this together with (2.2) Meanwhile, there exists t 0 ∈ [0, T ] such that x ′ (t 0 ) = 0 since x(0) = x(T ).Then by (2.1) we have, for where 1) .For the periodic boundary value problem where h ∈ C(R 3 , R) is T -periodic in the first variable.The following continuation theorem can be induced directly from the theory in [9], and is cited as Lemma 1 in [12].
Lemma 3 (Manásevich-Mawhin [9]).Let B = {x ∈ C 1 T : x < r} for some r > 0. Suppose the following two conditions hold: Then the periodic boundary value problem (2.6) has at least one T-periodic solution on B.

Main results
We are now in the position to give our main result.Proof.Let x(t) be a T -periodic solution of (1.1).By integrating the two sides of (1.1) from 0 to T , and noticing that x(0) = x(T ) and x ′ (0) = x ′ (T ), we have On the other hand, by Lemma 2.2, there exists M > 0 such that, for any solution x(t) of (2.1), Meanwhile, there exists 1) has no solution on ∂B for all λ ∈ (0, 1), and condition (i) of Lemma 3 is satisfied.Furthermore, we have since g(x) is strictly decreasing in R. By the definition of F in Lemma 3 we get .
This together with (3.2) yields that F (r)F (−r) < 0, i.e. condition (ii) of Lemma 3 is satisfied.Therefore, it follows from Lemma 3 that there exists a T -periodic solution x(t) of (1.1).The uniqueness of this x(t) is guaranteed by Lemma 1.This completes the proof.Noticing (2.3), it is easy to verify that the condition (H 2 ) is weaker than the condition (A 2 ) since min t∈R e(t) < ē < max t∈R e(t) when e(t) = constant.So our results improve and extend the main results in [6].
Finally, we close this work by two examples.
Example 1.Consider the following differential equation: where f ∈ C(R, R) and p > 1.
In this example, g(x) = − arctan(x + 1), e(t) = π 4 (1 + 3 cos t) and T = 2π.It is obvious that the condition (A 1 ) of theorem 1 in [6] holds.However, it is easy to verify that the condition (A 2 ) does not hold.Therefore, Theorem 1 in [6] fails, while, our criterion in Theorem 1 in this paper remains applicable, as we now show.According to the above arguments, the condition (H 1 ) holds; since ē = π 4 , it is easy to see that the condition (H 2 ) also holds.Hence, Theorem 1 in this paper shows that (3.3) has a unique 2π-periodic solution.
In this example, g(x) = − arctan(x + 1), e(t) = π(1 + 3 cos t) and T = 2π.It is obvious that the condition (A 1 ) of theorem 1 in [6] holds.However, it is easy to verify that the condition (A 2 ) does not hold.Therefore, Theorem 1 in [6] fails, while, our criterion in Theorem 1 in this paper remains applicable, as we now show.According to the above arguments, the condition (H 1 ) holds; since ē = π, it is easy to see that the condition (H 2 ) does not hold.Hence, Theorem 1 in this paper shows that (3.4) has no 2π-periodic solutions.

Lemma 2 .
Suppose (H 1 ) and (H 2 ) hold.Then the set of T -periodic solutions of (2.1) are bounded in C 1 T .

Theorem 1 .
Suppose (H 1 ) holds.Then (1.1) has a unique T -periodic solution if and only if (H 2 ) holds.