Periodic solution for ϕ-Laplacian neutral differential equation

Abstract This paper is devoted to the existence of a periodic solution for ϕ-Laplacian neutral differential equation as follows (ϕ(x(t)−cx(t−τ))′)′=f(t,x(t),x′(t)). $$\begin{array}{} (\phi(x(t)-cx(t-\tau))')'=f(t,x(t),x'(t)). \end{array}$$ By applications of an extension of Mawhin’s continuous theorem due to Ge and Ren, we obtain that given equation has at least one periodic solution. Meanwhile, the approaches to estimate a priori bounds of periodic solutions are different from the corresponding ones of the known literature.

It is easy to see that ϕ represents a large class of nonlinear operator, including ϕp : R → R is a p-Laplacian, i.e., ϕp(x) = |x| p− x for x ∈ R.
The study of p-Laplacian neutral di erential equations began with the paper of Zhu and Lu. In 2007, Zhu and Lu [1] discussed the existence of a periodic solution for a kind of p-Laplacian neutral di erential equation as follows (ϕp(x(t) − cx(t − τ)) ) + g(t, where c is a constant and |c| ≠ . Since (ϕp(x (t))) is nonlinear (i.e. quasilinear), Mawhin's continuous theorem [2] can not be apply directly. In order to get around this di culty, Zhu and Lu translated the p-where p + q = , for which Mawhin's continuation theorem can be applied. Zhu and Lu's work attracted the attention of many scholars in neutral di erential equation and they have contributed to the research of p-Laplacian neutral di erential equation (see [3]- [12]). Besides, a good deal of work has been performed on the existence of periodic solutions to ϕ-Laplacian di erential equation. Manásevich and Mawhin [13] in 1998 investigated ϕ-Laplacian di erential equation (ϕ(x(t) )) = f (t, x(t), x (t)).
Applying Leray-Schauder degree theory, the authors proved that the above equation has at least one periodic solution.
All the aforementioned results are related to p-Laplacian neutral equations [1], [3]- [12] or ϕ-Lpalacian di erential equation [13]. Naturally, a new question arises: how neutral di erential equation works on ϕ-Laplacian operator? Besides practical interests, the topic has obvious intrinsic theoretical signi cance. To answer this question, in this paper, we try to ll the gap and establish the existence of periodic solutions of (1.1) by employing the extension of Mawhin's continuation theorem due to Ge and Ren. The obvious di culty lies in the following two aspects. The rst is that since the leading term contains a ϕ-Laplacian neutral operator, the operator is much more than the corresponding p-Laplacian neutral operator; the second is that a priori bounds of periodic solutions are not easy to estimate. For example, the key step for ϕp to get the priori bounds of periodic solution, is no longer available for general ϕ-Laplacian. So we need to nd a new method to solve that problem. The remaining part of the paper is organized as follows. In section 2, we give some preliminary lemmas. In Section 3, by employing the extension of Mawhin's continuation theorem, we state and prove the existence of periodic solution for (1.1) in |c| ≠ and |c| = (critical) cases. In Section 4, we investigate the existence of the result for a kind of ϕ-Laplacian neutral Liénard equation in |c| ≠ case by applications of the Theorem 3.1. In Section 5, we consider the existence of periodic solution for a kind of p-Laplacian neutral Liénard equation in |c| ≠ and |c| = cases by applications of the Theorem 3.1. In Section 6, two numerical examples demonstrate the validity of the method.
Throughout this paper, we will denote by Z the set of integers, Z the set of odd integers, Z the set of even integers, N the set of positive integers, N the set of odd positive integers and N the set of even positive integers. Let Clearly, L π , L − π and L + π are all Banach spaces.

Preliminaries
In order to use the extension of Mawhin's continuous theorem [14] due to Ge and Ren, we rst recall it. Let X and Z be Banach spaces with norms · X and · Y , respectively. A continuous operator M : Let X =ker M and X be the complement space of X in X, then X = X ⊕ X . On the other hand, Z is a subspace of Z and Z is the complement space of Z in Z, so that Z = Z ⊕ Z . Suppose that P : X → X and Q : Z → Z two projects and Ω ⊂ X is an open and bounded set with the origin θ ∈ Ω.
Let ) Let X and Z be Banach spaces with norm · X and · Y , respectively, and Ω ⊂ X be an open and bounded set with origin θ ∈ Ω. Suppose that M : X ∩ domM → Z is a quasi − linear operator and is an M-compact mapping. In addition, if where N = N , then the abstract equation Mx = Nx has at least one solution inΩ.
has no solution on ∂Ω; Then (1.1) has at least one periodic solution onΩ.
Proof. In order to use Lemma 2.1 studying the existence of a periodic solution to (3.1), we set X : Clearly, dim X = dim Z = , and X = X ⊕ X , P : X → X , Q : Z → Z , are de ned by Hence, we have ( Moreover, for any x ∈ Z, we have So, we have (I − Q)Z ⊂ ImM. On the other hand, x ∈ ImM and T x(t)dt = , then we have Since λ ∈ ( , ), then we have T T f (t, x(t), x (t))dt = . Therefore, we can get QNx = , then, (2.4) also holds.
So by applications of Lemma 2.1, we see that (3.1) has a T-periodic solution.
In the following, applying Lemma 2.3 and Theorem 3.1, we consider the existence of a periodic solution to (1.1) in the case that |c| = . (v) c = and |τ| = π. Then (1.1) has at least one periodic solution onΩ.
Proof. We follow the same strategy and notation as in the proof of Theorem 3.1. Next, we consider R(x, λ)(t).
Case (i) c = − and |τ| = (m/n)π, with m, n are coprime positive integers with m even. Take T = π, from (3.3) and (3.4), applying Lemma 2.3, we know that there exist a continuous inverse operator A − of neutral operator A in the case that c = − such that where a ∈ R is a constant such that Similarly, we can get Case (ii)-Case (v). This proves the claim and the rest of the proof of the theorem is identical to that of Theorem 3.1.

Application of Theorem 3.1: ϕ-Laplacian operator
As an application, we consider the following ϕ-Laplacian neutral Liénard equation where g is a continuous function de ned on R and periodic in t with g(t, ·) = g(t + T, ·), f ∈ C(R, R), e is a continuous periodic function de ned on R with period T and T e(t) dt = . Next, by applications of Theorem 3.1, we investigate the existence of a periodic solution for (4.1) in the case that |c| ≠ . (H ) There exist two positive constants σ * , σ * such that σ * ≤ |f (x(t))| ≤ σ * , ∀ t ∈ R.
Proof. Consider the homotopic equation Firstly, we will claim that the set of all T-periodic solutions of (4.2) is bounded. Let x(t) ∈ C T be an arbitrary T-periodic solution of (4.  From the mean value theorem, there is a constant ξ ∈ ( , T) such that g(ξ , x(ξ )) = .
We claim that there exists a positive constant M * > M + such that, for all t ∈ R, we have (Ax) ≤ M * . (4.12) In fact, if (Ax) (t) is not bounded, then from the de nition of α, there exists a positive constant M such that α(|(Ax) |) > M for all (Ax) ∈ R. However, from (A ), we have Then, we can get which is a contradiction. So, (4.12) holds. By Lemma 2.2 and (4.12), we have Obviously, from (H ), we can get xH(x, µ) > and thus H(x, µ) is a homotopic transformation and So condition (C ) of Theorem 3.1 is satis ed. In view of the Theorem 3.1, there exists a solution with period T.
Remark 4.1. When |c| = , from Theorem 4.1, we know that σ * − (σ * + √ aT) > does not hold. Therefore, by applications of the above method, we do not obtain the existence of periodic solution for (4.1) in critical case (|c| = ).
(H ) There exist positive constants γ, η, B * such that Then (5.1) has at least one solution with period Proof. Consider the homotopic equation We follow the same strategy and notation as in the proof of Theorem 4.1. From (H ), we know that there exists a constant D > such that Multiplying both sides of (5.2) by (Ax)(t) and integrating over the interval [ , T], we get Then, we can get where N = |c|βD + ( + |c|)T(η + g M + e ), N = ( + |c|)TD(η + g M + e ). By application of Lemma 2.2, we have