Variational Approach to Quasi-Periodic Solution of Nonautonomous Second-Order Hamiltonian Systems

and Applied Analysis 3 consider the almost periodicity in the sense of Besicovitch 8 , they assume thatDkL satisfies a Lipschitiz condition and L is convex and obtain the existence of Besicovitch almost periodic solution by the least action principle. A special case of the above equation is the system 1.1 ; our main purpose is to apply Minmax method to study the existence of quasi periodic solutions to the system 1.1 , and we do not assume that ∇F t, · is Lipschitzian, but we assume that F satisfies some growth conditions, then we obtain results of existence of quasi periodic solution to the system 1.1 . Moreover, when we consider only a frequency 2π/T , our results will cover the results of periodic solutions to the system 1.1 . The present paper is organized as follows. In Section 1 we review some notations and definitions of almost periodic functions. In Section 2, we state our main theorems. In Section 3, in order to prove our main results, we will state our basic lemmas. In Section 4, we prove our main results and give an example. Now we give some notations and definitions of almost periodic functions. Definition 1.1 Fink 9 . A function f t is said to be Bohr almost periodic, if for any > 0, there is a constant l > 0, such that in any interval of length l , there exists τ such that the inequality |f t τ − f t | < is satisfied for all t ∈ R. Definition 1.2 He 10 . f ∈ C0 R × R,R is so called almost periodic in t uniformly for x ∈ R when, for each compact subset K in R, for each ε > 0, there exists l > 0, and for each α ∈ R, there exists τ ∈ α, α l such that sup t∈R sup x∈K ∥f t τ, x − f t, x ∥∥RN < ε. 1.6 AP 0 R is the space of the Bohr almost periodic functions from R to R , endowed with the norm ‖x‖∞ supt∈R|x t | and it is a Banach space. AP 1 R {x ∈ AP 0 R C1 R,R | x′ t ∈ AP 0 R } ; endowed with the norm ‖x‖ ‖x‖∞ ‖x‖∞, it is a Banach space. A fundamental property of almost periodic functions is that such functions have convergent means, that is, the following limit exists: lim T →∞ 1 2T ∫T −T x t dt. 1.7 The Fourier-Bohr coefficients of AP 0 R are the complex vectors a x, λ lim T →∞ 1 2T ∫T −T e−iλtx t dt, 1.8 and Λ x {λ ∈ R | a x, λ / 0} and it is a countable set, when p ∈ Z , B R is the completion of AP 0 R with respect to the norm ‖u‖p { lim T →∞ 1 2T ∫T −T |u|dt }1/p . 1.9 4 Abstract and Applied Analysis When p 2, B2 R is a Hilbert spaces and its norm ‖ · ‖2 is associated to the inner product 〈u, v〉2 lim T →∞ 1 2T ∫T −T u, v dt, 1.10 the elements of these spaces B R are called Besicovitch almost periodic functions. We use the generalized derivative ∇u ∈ B2 R of u ∈ B2 R defined by ‖∇u − 1/s u · s − u ‖ → 0 s → 0 , and we will identify the equivalence class u and its continuous representant u t ∫ t 0 ∇u t dt c. 1.11 Then we define B1,2 R {u ∈ B2 R | ∇u ∈ B2 R }, endowed with the norm ‖u‖ { lim T →∞ 1 2T ∫T −T ( |u t | |∇u t | ) dt }1/2 . 1.12 Its norm is associated to the inner product 〈u, v〉 〈u, v〉2 〈∇u,∇v〉2, and B1,2 R is a Hilbert space. For convenience, we denoteΛ { 2kπ/Tj | k ∈ Z, j 1, . . . , p} T1, . . . , Tp is rationally independent , Λ x is the set of all Fourier exponents {λk} of x, which is called the spectrum of x;V {x ∈ B1,2 R | Λ x ⊆ Λ}, it is easily obtained that V is a linear subspace of B1,2 R and V is a Hilbert space. 2. Main Theorems In this section, we state our main results. First, we give the following list of assumptions on F: f1 F t, · ∈ C1 R × R,R , and F t, · is almost periodic in t uniformly for x ∈ R , f2 ∇F t, · is almost periodic in t uniformly for x ∈ R , f3 for any λ ∈ R \Λ, x ∈ V , lim T →∞ 1 2T ∫T −T ∇F t, x e−iλtdt 0. 2.1 Theorem 2.1. Suppose F satisfies (f1)–(f3), the functional I : V → R, defined by I x lim T →∞ 1 2T ∫T −T [ 1 2 |∇x| F t, x ] dt 2.2 Abstract and Applied Analysis 5 is continuously differentiable on V , and I ′ x is defined by I ′ x h lim T →∞ 1 2T ∫T −T ∇x∇h ∇F t, x h dt. 2.3and Applied Analysis 5 is continuously differentiable on V , and I ′ x is defined by I ′ x h lim T →∞ 1 2T ∫T −T ∇x∇h ∇F t, x h dt. 2.3 Moreover, if x is a critical point of I in V , then ∇F t, x ∇ ∇x . 2.4 Definition 2.2. When x satisfies 2.4 in Theorem 2.1, we say that x is a weak solution of 1.1 . Theorem 2.3. Suppose that F satisfies (f1)–(f3), and f4 there exists g ∈ Lloc R , for a.e. t ∈ R and all x ∈ R , such that |∇F t, x | ≤ g t , 2.5


Introduction
In this paper, we consider the quasi-periodic solutions of the following second-order Hamiltonian system:ẍ where x t x 1 t , . . . , x N t , ∇F t, x ∂F/∂x 1 , . . . , ∂F/∂x N and ∂F/∂x k ∈ C R × R N , R , k 1, . . . , N.
A special class of the system 1.1 is the following autonomous second-order Hamilton system with convex potential Φ:ẍ Abstract and Applied Analysis For the scalar case 1 and for the vectorial case 2 , Berger and Chen have established the existence and uniqueness of almost periodic solution of 1.2 . In 2 , Berger and Chen assume that e is almost periodic, and the potential Φ is of the form where A is a symmetric positive-definite matrix and U ∈ C 2 R N , R is a convex function. They also need the growth condition. In 3 , Carminati states a local version of the results of Berger and Chen, assuming that Φ is convex only near the minimum of Φ. The above growth condition is not used by Carminati. To prove the existence and uniqueness of bounded or almost periodic solution of 1.2 , Carminati assumes that e is bounded or almost periodic and the potential Φ is of form 1.3 , where A is a symmetric positive-definite matrix and U is a convex function of class C 1 on the ball B x 0 , ρ ρ > 0 , where Φ reaches its minimum in this ball at x 0 .
When F is autonomous in the system 1.1 , Padilla 4 states the existence of the quasi periodic solution by using critical point theory, but it assumes that the Diophantine condition is satisfied. As to the system 1.1 , using a variational method, Zakharin and Parasyuk 5 have studied the existence of almost quasi periodic solutions for the system where Φ ∈ C 0 R × K, R , K is a compact convex subset of R N and Φ t, · is convex and differentiable on K, for each t ∈ R. The authors use a variational method on a Hilbert space of Besicovitch almost periodic functions which looks like a Sobolev space. By this method, the authors establish the existence of generalized solutions and in the quasi-periodic case, they prove that these solutions are classical. To prove the existence of quasi-periodic solutions, Zakharin and Parasyuk 5 assume that ∇ x Φ is quasi-periodic in t, and ∇ x Φ t, · is strongly monotone on K with positive modulus c. They also assume that the boundary ∂K of K is a differentiable manifold of class C 1 such that, for each x ∈ ∂K, the gradient ∇ x Φ t, x makes an acute angle with an external normal unit vector to ∂K at the point x in the Theorem 4.3 of 5 or a similar condition using the projection operator on the closed convex in the Theorem 4.2 of 5 . More recently in 6 , Ayachi and Blot provided new variational settings to study the almost periodic solutions of a class of nonlinear neutral delay equation 1.5 where L : R n 4 × R → R is a differentiable function, D j denotes the partial differential with respect to the jth vector variable, and r ∈ 0, ∞ is fixed. When they consider the almost periodicity in the sense of Corduneau 7 , they obtain some results on the structure of the set of Bohr almost periodic solutions in the case that L is autonomous and convex. When they consider the almost periodicity in the sense of Besicovitch 8 , they assume that D k L satisfies a Lipschitiz condition and L is convex and obtain the existence of Besicovitch almost periodic solution by the least action principle. A special case of the above equation is the system 1.1 ; our main purpose is to apply Minmax method to study the existence of quasi periodic solutions to the system 1.1 , and we do not assume that ∇F t, · is Lipschitzian, but we assume that F satisfies some growth conditions, then we obtain results of existence of quasi periodic solution to the system 1.1 . Moreover, when we consider only a frequency 2π/T, our results will cover the results of periodic solutions to the system 1.1 .
The present paper is organized as follows. In Section 1 we review some notations and definitions of almost periodic functions. In Section 2, we state our main theorems. In Section 3, in order to prove our main results, we will state our basic lemmas. In Section 4, we prove our main results and give an example. Now we give some notations and definitions of almost periodic functions.
Definition 1. 1 Fink 9 . A function f t is said to be Bohr almost periodic, if for any > 0, there is a constant l > 0, such that in any interval of length l , there exists τ such that the AP 0 R N is the space of the Bohr almost periodic functions from R to R N , endowed with the norm x ∞ sup t∈R |x t | and it is a Banach space.
A fundamental property of almost periodic functions is that such functions have convergent means, that is, the following limit exists: The Fourier-Bohr coefficients of AP 0 R N are the complex vectors and Λ x {λ ∈ R | a x, λ / 0} and it is a countable set, when p ∈ Z , B P R N is the completion of AP 0 R N with respect to the norm When p 2, B 2 R N is a Hilbert spaces and its norm · 2 is associated to the inner product the elements of these spaces B P R N are called Besicovitch almost periodic functions. We use the generalized derivative ∇u ∈ B 2 R N of u ∈ B 2 R N defined by ∇u − 1/s u · s − u → 0 s → 0 , and we will identify the equivalence class u and its continuous representant 1.11 Its norm is associated to the inner product u, v u, v 2 ∇u, ∇v 2 , and B 1,2 R N is a Hilbert space.
For convenience, it is easily obtained that V is a linear subspace of B 1,2 R N and V is a Hilbert space.

Main Theorems
In this section, we state our main results. First, we give the following list of assumptions on F: Abstract and Applied Analysis 5 is continuously differentiable on V , and I x is defined by Then 1.1 has at least a quasi periodic solution by saddle point theorem.
Remark 2.5. When V only contains a frequency 2π/T, F t, x is periodic in t with periodic T , which means that f 3 is satisfied; our results cover some results in 11 .

Basic Lemmas
To apply critical point theory to study the quasi periodic solution of 1.1 , we will state our basic lemmas, which will be used in the proofs of our main results.

6
Abstract and Applied Analysis It is easily obtained that there exists a constant C > 0, such that 3.4 Proof. By Lemma 3.1, the injection of V into C R, R N , with its natural norm · ∞ , is con- By Banach-Steinhaus theorem, {x k } is bounded in V , and hence in C R, R N , we need to show that the sequence {x k } is equiuniformly continuous, for any x k t ∈ V , x k t x k1 t x k2 t · · · x kp t .

3.7
Denoting T min min T 1 , T 2 , . . . , T p , T max max T 1 , T 2 , . . . , T p , 3.8 By Arzela-Ascoli theorem, {x k } is relatively compact on any compact of R. By the uniqueness of the weak limit, every uniformly convergent subsequence of {x k } converges to x on any compact of R.
then there exists C > 0, such that Proof. Since, by Lemma 3.1, x has the Fourier expansion a k e iλ k t .

8 Abstract and Applied Analysis
The Cauchy-Schwarz inequality and Parseval equality imply that

3.13
Lemma 3.4 saddle point theorem . Let X be a real Banach space, X X 1 X 2 , where X 1 / {0} and is finite dimensional. Suppose that I ∈ C 1 X, R satisfies the PS condition and I 1 there exist constants σ, ρ > 0, such that I ∂B ρ X 1 ≤ σ; Then I possesses a critical value c ≥ ω and

The Proof of Main Results
In this section, we prove the main results stated in Section 2.
Proof of Theorem 2.1. First step: we show that I has at every point x a directional derivative I x ∈ V * given by 2.3 . It follows easily from Lemma 3.1 and f 3 , for any x ∈ V that we have

4.3
Abstract and Applied Analysis 9 For x, h are fixed in V , there exists M > 0, such that |x t | ≤ M, |h t | ≤ M. Since ∇F t, x is almost periodic in t uniformly for x ∈ R N , we have that ∇F t, x is uniformly Moreover, by Lemma 3.1,

4.5
So I has, at x, a Gâteaux derivative I x ∈ V * . Second step: we show that the mapping is continuous. For any > 0, x is fixed in V and x ≤ M, let y ∈ V with x − y ≤ δ 0 < /2, by Lemma 3.1, it is easily obtained |x t | ≤ C 1 M and |y t | ≤ C 1 M δ 0 . Since ∇F t, x is uniformly continuous on R × K 1 , K 1 {x ∈ R N | |x| ≤ C 1 M δ 0 } is compact subset in R N , then there exists δ 1 , such that |x t − y t | ≤ δ 1 and we have ∇F t, x t − ∇F t, y t ≤ 2 C 1 .

4.8
The above inequality holds, which implies the continuity of I so that I is Fréchet differentiable on V .
If x is a critical point of I in V , for all h ∈ V , we have Since AP 1 R N is dense in B 1,2 R N , we have DI x h 0, for all h ∈ B 1,2 R N ; therefore, I x 0, and then we obtain 2.4 by using Blot 12 . The proof of Theorem 2.1 is completed.
Proof of Theorem 2.3. By Theorem 2.1, I is continuously differentiable on V . Next we will prove that I is weakly lower semicontinuous on V .
By Lemma 3.2, if {x k } ⊂ V converges weakly to x, then {x k } converges uniformly to x on any compact of R.
Since x k t ∈ AP 0 R n , and F t, · is almost periodic in t uniformly for x ∈ R n , then F t, x k t is almost periodic, and F t, x k t converges uniformly to F t, x t on any compact of R. Let

4.11
Then it is easily obtained that a k0 −→ a 0 , 4.12 moreover, 4.14 Moreover, 1/2 |∇u t | 2 is convex and continuous, so I is weakly lower semi-continuous.

Abstract and Applied Analysis
11

4.16
As x → ∞ if and only if the above inequality and f 5 imply that Since V is a Hilbert space and I is weakly lower semi-continuous, the proof of Theorem 2.3 is completed.
is continuously differentiable on V .

Abstract and Applied Analysis
For any v ∈ V ,

4.20
So we see that T −T x k t dt, since ∇I x k → 0, there exists some k 0 such that | ∇I x k , h | ≤ h for all k ≥ k 0 and h ∈ V ; we obtain, for k > k 0 , and hence

2T
T −T ∇F t, x k t − ∇F t, x t , x k t − x t dt 4.27 holds, and Lemma 3.2 implies that ∇x k − ∇x 2 → 0 as k → ∞, so x k − x ≤ x k − x 2 ∇x k − ∇x 2 → 0, and the PS c condition holds, then the proof of Theorem 2.4 is completed by saddle point theorem.