Non-degeneracy and Uniqueness of Periodic Solutions for 2n-order Differential Equations

We analyze the non-degeneracy of the linear 2n-order differential equation u (2n) + 2n−1 m=1 amu (m) = q(t)u with potential q(t) ∈ L p (R/T Z), by means of new forms of the optimal Sobolev and Wirtinger inequalities. The results is applied to obtain existence and uniqueness of periodic solution for the prescribed nonlinear problem in the semilinear and superlinear case.

The periodic solution problem for the high-order differential equations has attracted much attention (see for instance [1]- [3], [10]- [14]), however, the study on non-degenerate problems for high-order differential equation is not adequately covered in the related literature. The main objective of this paper is to contribute to the literature with a new criterium of non-degeneracy in the general case.
The interest of a good understanding of the non-degeneracy problem is twofold. Besides the intrinsic theoretical interest, generally speaking a concrete non-degeneracy result can be applied to obtain existence and uniqueness results for a nonlinear problem. For the second order equation, such techniques have been widely developed for the semilinear case. This line of research can be traced back at least to the seminal paper of Lasota and Opial [6] a present a number of variants, see for instance [4,8,15] and the references therein. The superlinear case has been considered in [9]. The analysis of higher-order problems with this technique is more rare. Just recently, Li and Zhang [7] have used some Sobolev constants to explicitly characterize a class of potentials q(t) ∈ L p (0, T ) for which the beam equation with periodic boundary conditions u (4) (t) = q(t)u(t), t ∈ (0, T ), admits only the trivial solution. As an application of non-degeneracy, they obtain the uniqueness of periodic solutions of a certain class of superlinear beam equations.
In this paper, we develop a novel non-degeneracy criterium for problem (1)- (2). Later, inspired in the cited papers [7,9,15], such criterium is applied to the existence and uniqueness of periodic solutions of the related nonlinear differential equation. In section 2, we present new forms of optimal Sobolev and Wirtinger inequalities recently developed in [5]. In section 3, by using the previous optimal Sobolev and Wirtinger inequalities, we get sufficient conditions for a potential to be non-degenerate for (1)- (2). Section 4 and 5 are devoted to applications of the main result for non-degenerate potentials to the nonlinear problem. Section 4 deals with the semilinear case and applies the technique developed in [15]. In section 5, firstly, the classes C(σ; A, B) of nonlinearities to be considered are given in Definition 5.2. These nonlinearities f (x) can grow superlinearly as x → ∞. Besides the existence for equations of Landesman-Lazer type [14] where the nonlinearities are monotone, by mimicking the technique employed in [7] it is shown in Theorem 5.3 that, for those classes of nonlinear equations, the periodic solution is unique.
We fix some notations. For a function h(t) in the Lebesgue space L 1 (S T ) of Tperiodic function, S T = R/T Z, the mean value of h(t) ish(t) = 1 T T 0 h(t)dt. Then L 1 (S T ) can be decomposed as L 1 (S T ) = R ⊕L 1 (S T ), whereL 1 (S T ) = {h ∈ L 1 (S T ) : h = 0} and R is identified as the set of constant functions of L 1 (S T ). Analogously, the Hilbert space H n (S T ) can be decomposed as H n (S T ) = R ⊕H n (S T ), wherẽ H n (S T ) = H n (S T ) ∩L 1 (S T ). The uniform norm is as usual ||x|| ∞ = max |x(t)|. Finally, the positive and negative part of a function q(t) are given by q + (t) = max{q(t), 0}, q − (t) = max{−q(t), 0}.

Optimal Sobolev and Wirtinger inequalities.
In this section, we recall some novel Sobolev and Wirtinger inequalities recently proved in [5].
As a preparation, we explain briefly about Riemann zeta function, Bernoulli polynomial and Bernoulli number. Riemann zeta function is a meromorphic function defined by Bernoulli polynomial b n (x) is defined by the following recurrence relation.

Bernoulli number is defined by
It can be obtained by the following recurrence relate Bernoulli numbers are positive rational numbers. Next lemmas have been proved in [5]. Such inequalities are directly generalized to T -periodic functions through a time rescalling. If φ(t) ∈H n (S T ), we know that ψ(t) := φ(T t) ∈H n (S 1 ). Since the previous inequalities are readily generalized as follows. where is the best constant for this inequality.
where C M := T 2π 2M is the best constant for this inequality.
3. Sufficient conditions for a potential to be non-degenerate. In this section the main result is stated and proved. To this purpose, let us define σ = {1, 2, . . . , n− 1} and the subsets Of course, one (or both) of these subsets can be empty. In this case, the usual convention ∅ = 0 is used.
, let us assume that one of the following conditions holds (1) n is even,q > 0 and Then Proof. We argue by contradiction. Assume that (1)-(2) has a non-trivial solution Integrating this equation over one period, we have, by the T -periodicity ofx, Integrating this equation over one period and making use of the T -periodicity of Note that integrating by parts one gets T 0x (t)x (m) dt = 0 for every odd m. Then, by reindexing m = 2k, (9) reads

NON-DEGENERACY AND UNIQUENESS FOR DIFFERENTIAL EQUATIONS 2159
First, let us assume that (1) holds. Since n is even, we have Using Wirtinger inequality in left-hand side of (11), we have where C n−k are the optimal constants defined in Lemma 2.4. On the other hand, by using now Sobolev inequality andq > 0, the right-hand side of (11) can be bounded above as follows Therefore, Under assumption (6), it is necessary that ||x (n) || 2 = 0. Thusx (n−1) is constant.

PEDRO J. TORRES, ZHIBO CHENG AND JINGLI REN
Under assumption (2), an analogous argument can be done. As n is odd, then (11) reads and the proof follows the same steps as before.
4. Semilinear case. As a direct application of general non-degenerate potentials, one can obtain reasonable existence results for periodic solutions of nonlinear beam equation here h(t, u) grows semilinearly when |u| → ∞. Denote and ϕ ∈ L 1 (S T ). The proof of the main result of this section follows the strategy adopted by [15] for the second-order equation. Let us consider an m-th order systems of the form where g kx, kx , . . . , kx (m−1) = kg x, x , . . . , x (m−1) for all k > 0, x, x , . . . , x (m−1) ∈ R mn , and suppose that exists and ϕ * ∈ L p (S T ). The main result of this section is as follows.
Theorem 4.2. Let us assume that one of the following conditions holds (1) n is even, p > 0 and (2) n is odd, p < 0 and Then there is a constant c 0 > 0 such that if ||ϕ|| < c 0 , the problem (14) has at least one T -periodic solution.
Proof. Comparing (14) to (15), we have Obviously, it is easy to see that Besides, Firstly, let us consider the linear problem From Theorem 3.1, we know that if n is even, p > 0 and or alternatively if n is odd, p < 0 and then (18) is non-degenerate, therefore condition (H 1 ) holds. On the other hand,g(u) = g(u, 0, . . . , 0) = pu. Therefore, we have trivially deg(g(u), B(0, r), 0) = 0. Then, condition (H 2 ) holds and the result is a direct consequence of Lemma 4.1.

Superlinear case.
In this section, we will give an application of the class of nondegenerate potentials constructed above to the study of existence and uniqueness of T -periodic solution for equations with superlinear term. We will combine techniques from [14] and [7,9]. Let us consider the nonlinear differential equation where s ∈ R,h ∈L 1 (S T ), and the nonlinearity f : R → R is a continuous and monotone function. The parameter s is the mean value of the external term −s + h(t).
It is easy to find a necessary condition for existence of T -periodic solutions. In fact, integrating (19) on [0, T ], we have The proof of the existence of periodic solution of (19) follows the strategy adopted by [14]. Let us consider an m-th order equation of the form y (m) + a m−1 y (m−1) + · · · + a 1 y + g(t, y) = p(t) (m > 1).
where a 1 , · · · , a m−1 is real constants. g : R × R → R be continuous and T -periodic in its first variable; i.e., g(t + T, y) = g(t, y) for all t, y. We define two measurable Let us denote Ly ≡ y (m) + a m−1 y (m−1) + · · · + a 1 y .
The following lemma is the main result of [14].

Lemma 5.1 ([14]
). Assume that g(t, y) is bounded below for y ≥ 0 and bounded above for y ≤ 0, and the following conditions hold: (c 1 ) The only T -periodic solutions to the equation Ly = 0 are the constants. (c 2 ) There are numbers α 1 and β 1 such that for all (t, y) ∈ R × R, |g(t, y)| ≤ g(t, y) Then there is a number ε > 0 such that (21) has a T -periodic solution provided α 1 ≤ ε.
Our existence result is the following one. Proposition 1. Suppose that f : R → R is bounded below for u ≥ 0 and bounded above for u ≤ 0, s ∈ intR(f ), and there are two non-negative constants α and β such that |f (u)| ≤ f (u) + α|u| + β.
Assume that one of the following conditions holds (1) n is even, and k∈σ1 |a 2k | C n−k < 1.
On the other hand, if n is odd, the proof follows the similar steps as before.
In the following, we will consider the uniqueness problem. Let us introduce the following definition from [9]. We say that f satisfies the condition C(σ; A, B) if for every x 1 , x 2 ∈ R, and x 1 = x 2 . Here ϕ + = (ϕ) + = max(ϕ, 0) for ϕ ∈ R.
The main result for uniqueness is as follows.
Proof. Firstly, assume that n is odd. Let x 1 (t) and x 2 (t) be two different T -solutions of (19), we have Let x(t) := x 1 (t) − x 2 (t) be the difference of two solutions. Then x(t) ≡ 0. The difference of (27) gives Let I := {t ∈ R : x(t) = 0}, which is a non-empty open subset of R. The function is well defined for all t ∈ I. It is easy to see that q(t) ∈ C(I). For the sake of convenience, we define q(t) = 0 on the complement J := R \ I. Then q(t) is well defined on R. Obviously, q(t) is measurable. As f (x) is non-increasing in x, one has q(t) ≤ 0 for all t. Moreover, for all t ∈ I, we have from (24) that where C is a constant and C ≥ 0, since f (x) is continuous and the x i (t) are Tperiodic. Therefore, q(t) ≤ 0 for all t and q ∈ L ∞ (S T ). From (29), we have and then ||q|| σ ≤ ((As + B)T ) 1 σ . From (25), we get ||q|| σ < M (σ * , n). Under assumption (25), if we haveq < 0, by Theorem 3.1, we have x(t) ≡ 0, contradicting with the assumption x 1 = x 2 . Thenq = 0. As q(t) ≤ 0, we know that q(t) ≡ 0. Therefore, Multiplying both sides of (30) by x(t) and integrating over [0, T ], we have a m x (m) (t)x(t)dt = 0.
Next, we prove that C ≡ 0. By contradiction, C = 0. Without loss of generality, we assume that C > 0, and we have for all t, f is non-increasing. Now we have T 0 f (x 1 (t))dt > T 0 f (x 2 (t))dt, which contradict (28). Therefore, we get x(t) ≡ 0 and the proof is done.
On the other hand, if n is even, the proof follows the similar steps as before.
In the following we consider equations of the Landesman-Lazer type.
Theorem 5.3. Suppose that f : R → R is bounded below for u ≥ 0 and bounded above for u ≤ 0, and there are two non-negative constants α and β such that Assume that one of the following conditions holds (1) n is odd, f ∈ C * (σ; A, B) is strictly decreasing and s ∈ R(f ) satisfies (25).
(2) n is even, f ∈ C(σ; A, B) is strictly increasing and s ∈ R(f ) satisfies (26). Then there exists a positive constant α 0 such that (19) has exactly one T -periodic solution provided that α ≤ α 0 .
Proof. It follows directly from Propositions 1 and 2.
In order to obtain reasonable conditions, T should be satisfy 129600 × (2π − T ) 2 > π 2 × T 8 . We conclude that when Then (36) has exactly one T -periodic solution for eachh ∈L 1 (S T ). Different from the case for (32) and (34), we have now a restriction on the period T in (37).