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

Abstract

We deal with the quasi-periodic solutions of the following second-order Hamiltonian systems 𝑥̈(𝑡)=∇𝐹(𝑡,𝑥(𝑡)), where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑁(𝑡)), and we present a new approach via variational methods and Minmax method to obtain the existence of quasi-periodic solutions to the above equation.

1. Introduction

In this paper, we consider the quasi-periodic solutions of the following second-order Hamiltonian system:𝑥̈(𝑡)=∇𝐹(𝑡,𝑥(𝑡)), (1.1) where 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑁(𝑡)), ∇𝐹(𝑡,𝑥)=(𝜕𝐹/𝜕𝑥₁,…,𝜕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(𝑡)=∇Φ(𝑥(𝑡))+𝑒(𝑡), (1.2) 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 formΦ(𝑥)=12⟨𝐴𝑥∣𝑥⟩+𝑈(𝑥), (1.3) where 𝐴 is a symmetric positive-definite matrix and 𝑈∈𝐶²(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 𝐶¹ on the ball 𝐵(𝑥₀,𝜌)(𝜌>0), where Φ reaches its minimum in this ball at 𝑥₀.

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 𝑥^′′(𝑡)=∇_𝑥Φ(𝑡,𝑥(𝑡)), (1.4) where Φ∈𝐶⁰(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 ∇_𝑥Φ is quasi-periodic in 𝑡, and ∇_𝑥Φ(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 𝐶¹ such that, for each 𝑥∈𝜕𝐾, the gradient ∇_𝑥Φ(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 𝐷₁𝐿𝑥(𝑡−𝑟),𝑥(𝑡−2𝑟),𝑥^′(𝑡−𝑟),𝑥^′(𝑡−2𝑟),𝑡−𝑟+𝐷₂𝐿𝑥(𝑡),𝑥(𝑡−𝑟),𝑥^′(𝑡),𝑥^′()=𝑑𝑡−𝑟,𝑡𝐷𝑑𝑡³𝐿𝑥(𝑡−𝑟),𝑥(𝑡−2𝑟),𝑥^′(𝑡−𝑟),𝑥^′(𝑡−2𝑟),𝑡−𝑟+𝐷₄𝐿𝑥(𝑡),𝑥(𝑡−𝑟),𝑥^′(𝑡),𝑥^′(𝑡−𝑟),𝑡, (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_𝑘𝐿 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 𝜖>0, there is a constant 𝑙_𝜖>0, such that in any interval of length 𝑙_𝜖, there exists 𝜏 such that the inequality |𝑓(𝑡+𝜏)−𝑓(𝑡)|<𝜖 is satisfied for all 𝑡∈𝑅.

Definition 1.2 (He [10]). 𝑓∈𝐶⁰(𝑅×𝑅^𝑚,𝑅^𝑁) is so called almost periodic in 𝑡 uniformly for 𝑥∈𝑅^𝑚 when, for each compact subset 𝐾 in 𝑅^𝑚, for each 𝜀>0, there exists 𝑙>0, and for each 𝛼∈𝑅, there exists 𝜏∈[𝛼,𝛼+𝑙] such that sup_𝑡∈𝑅sup_𝑥∈𝐾𝑓(𝑡+𝜏,𝑥)−𝑓(𝑡,𝑥)_𝑅𝑁<𝜀. (1.6)
𝐴𝑃⁰(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.
𝐴𝑃¹(𝑅^𝑁)={𝑥∈𝐴𝑃⁰(𝑅^𝑁)⋂𝐶¹(𝑅,𝑅^𝑁)∣𝑥^′(𝑡)∈𝐴𝑃⁰(𝑅^𝑁)}; 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→∞12𝑇^𝑇_−𝑇𝑥(𝑡)𝑑𝑡. (1.7)

The Fourier-Bohr coefficients of 𝐴𝑃⁰(𝑅^𝑁) are the complex vectors𝑎(𝑥,𝜆)=lim_𝑇→∞12𝑇^𝑇_−𝑇𝑒^{−𝑖𝜆𝑡}𝑥(𝑡)𝑑𝑡, (1.8)

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 𝐴𝑃⁰(𝑅^𝑁) with respect to the norm ‖𝑢‖_𝑝=lim_𝑇→∞12𝑇^𝑇_−𝑇|𝑢|^𝑝𝑑𝑡^1/𝑝. (1.9) When 𝑝=2,𝐵²(𝑅^𝑁) is a Hilbert spaces and its norm ‖⋅‖₂ is associated to the inner product ⟨𝑢,𝑣⟩₂=lim_𝑇→∞12𝑇^𝑇_−𝑇(𝑢,𝑣)𝑑𝑡, (1.10) the elements of these spaces 𝐵^𝑃(𝑅^𝑁) are called Besicovitch almost periodic functions.

We use the generalized derivative ∇𝑢∈𝐵²(𝑅^𝑁) of 𝑢∈𝐵²(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 𝑢(𝑡)=^𝑡₀∇𝑢(𝑡)𝑑𝑡+𝑐. (1.11) Then we define 𝐵^1,2(𝑅^𝑁)={𝑢∈𝐵²(𝑅^𝑁)∣∇𝑢∈𝐵²(𝑅^𝑁)}, endowed with the norm ‖𝑢‖=lim_𝑇→∞12𝑇^𝑇_−𝑇|𝑢(𝑡)|²+|∇𝑢(𝑡)|²𝑑𝑡^1/2. (1.12) 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, we denote Λ={(2𝑘𝜋/𝑇_𝑗)∣𝑘∈𝑍,𝑗=1,…,𝑝} (𝑇₁,…,𝑇_𝑝 is rationally independent), Λ(𝑥) is the set of all Fourier exponents {𝜆_𝑘} of 𝑥, which is called the spectrum of 𝑥;𝑉={𝑥∈𝐵^1,2(𝑅^𝑁)∣Λ(𝑥)⊆Λ}, it is easily obtained that 𝑉 is a linear subspace of 𝐵^1,2(𝑅^𝑁) 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^𝑁,𝑅), and 𝐹(𝑡,⋅) is almost periodic in 𝑡 uniformly for 𝑥∈𝑅^𝑁,(f₂)∇𝐹(𝑡,⋅) is almost periodic in 𝑡 uniformly for 𝑥∈𝑅^𝑁,(f₃) for any 𝜆∈𝑅⧵Λ, 𝑥∈𝑉,lim_𝑇→∞12𝑇^𝑇_−𝑇∇𝐹(𝑡,𝑥)𝑒^{−𝑖𝜆𝑡}𝑑𝑡=0. (2.1)

Theorem 2.1. Suppose 𝐹 satisfies (f₁)–(f₃), the functional 𝐼∶𝑉→𝑅, defined by 𝐼(𝑥)=lim_𝑇→∞12𝑇^𝑇_−𝑇12|∇𝑥|²+𝐹(𝑡,𝑥)𝑑𝑡 (2.2) is continuously differentiable on 𝑉, and 𝐼^′(𝑥) is defined by 𝐼^′(𝑥)ℎ=lim_𝑇→∞12𝑇^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∈𝐿¹_loc(𝑅), for a.e. 𝑡∈𝑅 and all 𝑥∈𝑅^𝑁, such that|∇𝐹(𝑡,𝑥)|≤𝑔(𝑡), (2.5)(f₅)There existslim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)𝑑𝑡⟶+∞as|𝑥|⟶∞. (2.6) Then (1.1) has at least a quasi periodic solution.

Theorem 2.4. Suppose that 𝐹 satisfies (f₁)–(f₄), and(f₆)One haslim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)𝑑𝑡⟶−∞as|𝑥|⟶∞. (2.7)
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 (f₃) is satisfied; our results cover some results in [11].

3. 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.

Lemma 3.1. If 𝑥∈𝑉, then 𝑥(𝑡)=^+∞_𝑘=−∞𝑎_𝑘𝑒^{𝑖𝜆𝑘𝑡}∈𝐴𝑃⁰𝑅^𝑁,‖𝑥‖_∞≤(𝐶+1)‖𝑥‖. (3.1)

Proof. For any 𝑥(𝑡)∈𝑉, 𝑥(𝑡)∼^+∞_𝑘=−∞𝑎_𝑘𝑒^{𝑖𝜆𝑘𝑡},∇𝑥(𝑡)∼^+∞_𝑘=−∞𝑖𝑎_𝑘𝜆_𝑘𝑒^{𝑖𝜆𝑘𝑡},𝜆_𝑘∈Λ. (3.2) It is easily obtained that there exists a constant 𝐶>0, such that ^𝑘=+∞_{𝑘=−∞,𝜆𝑘≠0}1𝜆²_𝑘≤𝐶²,𝑎₀+^𝑘=+∞_{𝑘=−∞,𝜆𝑘≠0}𝑎_𝑘≤‖𝑥‖₂+^𝑘=+∞_{𝑘=−∞,𝜆𝑘≠0}1𝜆²_𝑘^1/2𝑘=+∞_{𝑘=−∞,𝜆𝑘≠0}𝜆_𝑘𝑎_𝑘^21/2≤‖𝑥‖₂+𝐶‖∇𝑥‖₂≤(𝐶+1)‖𝑥‖. (3.3) So 𝑥(𝑡)=∑^+∞_𝑘=−∞𝑎_𝑘𝑒^{𝑖𝜆𝑘𝑡}∈𝐴𝑃⁰𝑅^𝑁,‖𝑥‖_∞≤(𝐶+1)‖𝑥‖. (3.4)

Lemma 3.2. For any {𝑥_𝑘}⊂𝑉, if the sequence {𝑥_𝑘} converges weakly to 𝑥, then {𝑥_𝑘} converges uniformly to 𝑥 on any compact subset of 𝑅.

Proof. By Lemma 3.1, the injection of 𝑉 into 𝐶(𝑅,𝑅^𝑁), with its natural norm ‖⋅‖_∞, is continuous. Since {𝑥_𝑘}⇀𝑥 in 𝑉, it follows that {𝑥_𝑘}⇀𝑥 in 𝐶(𝑅,𝑅^𝑁). 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)=^+∞_𝑚=−∞𝑎_𝑚𝑒^{𝑖𝜆𝑚𝑡}, (3.5) let 𝑥_𝑘𝑗(𝑡)=^+∞_{𝑚=−∞,𝜆𝑚∈2𝑛𝜋/𝑇𝑗∣𝑛∈𝑍}𝑎_𝑚𝑒^{𝑖𝜆𝑚𝑡}, (3.6) then 𝑥_𝑘(𝑡)=𝑥_𝑘1(𝑡)+𝑥_k2(𝑡)+⋯+𝑥_𝑘𝑝(𝑡). (3.7)
Denoting 𝑇_min=min𝑇₁,𝑇₂,…,𝑇_𝑝,𝑇_max=max𝑇₁,𝑇₂,…,𝑇_𝑝, (3.8) for 0≤𝑡−𝑠≤𝑇_min, we have 𝑥_𝑘(𝑡)−𝑥_𝑘(𝑠)≤^𝑡_𝑠𝑥^′_𝑘(𝜏)𝑑𝜏≤^𝑡_𝑠𝑥^′_𝑘1(𝜏)𝑑𝜏+^𝑡_𝑠𝑥^′_𝑘2(𝜏)𝑑𝜏+⋯+^t_𝑠𝑥^′_𝑘𝑛(𝜏≤())𝑑𝜏𝑡−𝑠^1/2𝑡_𝑠𝑥^′_𝑘1(𝜏)²𝑑𝜏^1/2+(𝑡−𝑠)^1/2𝑡_𝑠𝑥^′_𝑘2(𝜏)²𝑑𝜏^1/2+⋯+(𝑡−𝑠)^1/2𝑡_𝑠𝑥^′_𝑘𝑛(𝜏)²𝑑𝜏^1/2≤(𝑡−𝑠)^{1/2𝑠+𝑇1}_𝑠𝑥^′_𝑘1(𝜏)²𝑑𝜏^1/2+^𝑠+𝑇2_𝑠𝑥^′_𝑘2(𝜏)²𝑑𝜏^1/2+⋯+^{𝑠+𝑇𝑝}_𝑠𝑥^′_𝑘𝑛(𝜏)²𝑑𝜏^1/2≤(𝑡−𝑠)^1/2𝑇₁𝑥^′_𝑘2+𝑇₂𝑥^′_𝑘2+⋯+𝑇_𝑝𝑥^′_𝑘2≤(𝑡−𝑠)^1/2𝑝𝑇_max𝑥_𝑘≤𝐶₁(𝑡−𝑠)^1/2. (3.9)
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.

Lemma 3.3. If 𝑥∈𝑉 and lim_𝑇→∞12𝑇^𝑇_−𝑇𝑥(𝑡)=0, (3.10) then there exists 𝐶>0, such that ‖𝑥‖²_∞≤𝐶²‖∇𝑥‖²₂. (3.11)

Proof. Since, by Lemma 3.1, 𝑥 has the Fourier expansion 𝑥(𝑡)=^+∞_{𝑘=−∞,𝜆𝑘≠0}𝑎_𝑘𝑒^{𝑖𝜆𝑘𝑡}. (3.12) The Cauchy-Schwarz inequality and Parseval equality imply that |𝑥(𝑡)|²≤^+∞_{𝑘=−∞,𝜆𝑘≠0}𝑎_𝑘²≤^+∞_{𝑘=−∞,𝜆𝑘≠0}1𝜆²_𝑘^+∞_{𝑘=−∞,𝜆𝑘≠0}𝜆_𝑘𝑎_𝑘²≤𝐶²‖∇𝑥‖²₂. (3.13)

Lemma 3.4 (saddle point theorem). Let 𝑋 be a real Banach space, 𝑋=𝑋₁⨁𝑋₂, where 𝑋₁≠{0} and is finite dimensional. Suppose that 𝐼∈𝐶¹(𝑋,𝑅) satisfies the PS condition and(I₁) there exist constants 𝜎,𝜌>0, such that 𝐼_{𝜕𝐵𝜌⋂𝑋1}≤𝜎;(I₂) there exists a constant 𝜔>𝜎, such that 𝐼_𝑋2≥𝜔.
Then 𝐼 possesses a critical value 𝑐≥𝜔 and 𝑐=inf_ℎ∈Γmax_{𝑢∈𝐵𝜌⋂𝑋1}𝐼(ℎ(𝑢)), (3.14) where Γ={ℎ∈𝐶(𝐵_𝜌⋂𝑋₁,𝑋)|ℎ|_{𝜕𝐵𝜌⋂𝑋1}=𝑖𝑑}.

4. 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 (𝑓₃), for any 𝑥∈𝑉 that we have 𝐹(𝑡,𝑥(𝑡))∈𝐴𝑃⁰(𝑅),∇𝑥∈𝐵²𝑅^𝑁. (4.1)
So 𝐼 is everywhere finite on 𝑉. For 𝑥,ℎ fixed in 𝑉, 𝜆∈[−1,1], let us define 𝐺()=1𝜆,𝑡2|∇𝑥+𝜆∇ℎ|²(),𝜑(𝜆)+𝐹𝑡,𝑥+𝜆ℎ=lim_𝑇→∞12𝑇^𝑇_−𝑇𝐺(𝜆,𝑡)=𝐼(𝑥+𝜆ℎ). (4.2)
There exists 𝜃_𝑡∈[0,1], such that 1𝜆𝜑(𝜆)−𝜑(0)−lim_𝑇→∞12𝑇^𝑇_−𝑇[∇𝐹(𝑡,𝑥)ℎ+∇𝑥∇ℎ]𝑑𝑡=lim_𝑇→∞12𝑇^𝑇_−𝑇𝜆|∇ℎ|²+∇𝐹𝑡,𝑥(𝑡)+𝜃_𝑡𝜆ℎ(𝑡)ℎ−∇𝐹(𝑡,𝑥(𝑡))ℎ𝑑𝑡. (4.3)
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 continuous on 𝑅×𝐾, where 𝐾={𝑥∈𝑅^𝑁∣|𝑥|≤2𝑀} is compact subset in 𝑅^𝑁, so lim_𝜆→01𝜆𝜑(𝜆)−𝜑(0)=lim_𝑇→∞12𝑇^𝑇_−𝑇[∇𝐹(𝑡,𝑥)ℎ+∇𝑥∇ℎ]𝑑𝑡. (4.4)
Moreover, by Lemma 3.1, 𝐼′(𝑥),ℎ=lim_𝑇→∞12𝑇^𝑇_−𝑇[∇𝐹(𝑡,𝑥)ℎ+∇𝑥∇ℎ]𝑑𝑡≤𝑐₁‖ℎ‖_∞+𝑐₂‖∇ℎ‖₂≤𝑐₃‖ℎ‖. (4.5) So 𝐼 has, at 𝑥, a Gâteaux derivative 𝐼^′(𝑥)∈(𝑉)^∗.
Second step: we show that the mapping 𝐼^′∶𝑉⟶𝑉^∗,𝑥⟶𝐼^′(𝑥) (4.6) is continuous.
For any 𝜖>0, 𝑥 is fixed in 𝑉 and ‖𝑥‖≤𝑀, let 𝑦∈𝑉 with ‖𝑥−𝑦‖≤𝛿₀<𝜖/2, by Lemma 3.1, it is easily obtained |𝑥(𝑡)|≤(𝐶+1)𝑀 and |𝑦(𝑡)|≤(𝐶+1)(𝑀+𝛿₀). Since ∇𝐹(𝑡,𝑥) is uniformly continuous on 𝑅×𝐾₁, 𝐾₁={𝑥∈𝑅^𝑁∣|𝑥|≤(𝐶+1)(𝑀+𝛿₀)} is compact subset in 𝑅^𝑁, then there exists 𝛿₁, such that |𝑥(𝑡)−𝑦(𝑡)|≤𝛿₁ and we have ((𝑡(𝑡)≤𝜖∇𝐹𝑡,𝑥))−∇𝐹𝑡,𝑦2().𝐶+1 (4.7)
We denote 𝛿^∗=min{𝛿₀,𝛿₁/(𝐶+1)}, then, for all ℎ∈𝑉 and ‖ℎ‖≤1, such that ‖𝑥−𝑦‖≤𝛿^∗, we have 𝐼^′(𝑥)ℎ−𝐼^′𝑦ℎ≤lim_𝑇→∞12𝑇^𝑇_−𝑇()≤𝜖∇𝐹𝑡,𝑥−∇𝐹𝑡,𝑦ℎ+∇𝑥−∇𝑦∇ℎ𝑑𝑡2()‖ℎ‖𝐶+1^∞+lim_𝑇→∞12𝑇^𝑇_−𝑇∇𝑥−∇𝑦^21/2lim_𝑇→∞12𝑇^𝑇_−𝑇|∇ℎ|^21/2≤𝜖2+𝜖2=𝜖. (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 𝐼^′(𝑥)ℎ=lim_𝑇→∞12𝑇^𝑇_−𝑇[∇𝐹(𝑡,𝑥)ℎ+∇𝑥∇ℎ]𝑑𝑡=0, (4.9) by (𝑓₃), then for all ℎ∈𝐴𝑃¹(𝑅^𝑁), we have lim_𝑇→∞12𝑇^𝑇_−𝑇[∇𝐹(𝑡,𝑥)ℎ+∇𝑥∇ℎ]𝑑𝑡=0. (4.10)
Since 𝐴𝑃¹(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)∈𝐴𝑃⁰(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 𝐹𝑡,𝑥_𝑘(𝑡)=𝑎_𝑘0+^+∞_{𝑗=−∞,𝜆𝑗≠0}𝑎_𝑘𝑗𝑒^{𝑖𝜆𝑗𝑡},𝐹(𝑡,𝑥(𝑡))=𝑎₀+^+∞_{𝑗=−∞,𝜆𝑗≠0}𝑎_𝑗𝑒^{𝑖𝜆𝑗𝑡}. (4.11) Then it is easily obtained that 𝑎_𝑘0⟶𝑎₀, (4.12) moreover, lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹𝑡,𝑥_𝑘(𝑡)𝑑𝑡=𝑎_𝑘0, (4.13) so lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹𝑡,𝑥_𝑘(𝑡)𝑑𝑡⟶lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥(𝑡))𝑑𝑡. (4.14) Moreover, (1/2)|∇𝑢(𝑡)|² is convex and continuous, so 𝐼 is weakly lower semi-continuous.
For 𝑥∈𝑉, we have 𝑥=𝑥+𝑥, where 𝑥=lim_𝑇→∞12𝑇^𝑇_−𝑇𝑥(𝑡)𝑑𝑡, (4.15) then, lim_𝑇→∞12𝑇^𝑇_−𝑇(𝑡)𝐼(𝑥)𝑥=0,=lim_𝑇→∞12𝑇^𝑇_−𝑇12|∇𝑥|²𝑑𝑡+lim_𝑇→∞12𝑇^𝑇_−𝑇{𝐹(𝑡,𝑥)+[𝐹(𝑡,𝑥(𝑡))−𝐹(𝑡,𝑥=1)]}𝑑𝑡2‖∇𝑥‖²₂+lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)+¹₀(∇𝐹(𝑡,(𝑡(𝑡≥1𝑥+𝑠𝑥)),𝑥))𝑑𝑠𝑑𝑡2‖∇𝑥‖²₂−lim_𝑇→∞12𝑇^𝑇_−𝑇𝑔(𝑡)𝑑𝑡‖𝑥(𝑡)‖_∞+lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)≥1𝑑𝑡2‖∇𝑥‖²₂−𝑐‖∇𝑥‖₂+lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)𝑑𝑡. (4.16)
As ‖𝑥‖→∞ if and only if |𝑥|²+lim_𝑇→∞12𝑇^𝑇_−𝑇|∇𝑥(𝑡)|²𝑑𝑡^1/2⟶∞, (4.17) the above inequality and (f₅) imply that 𝐼(𝑥)⟶+∞as‖𝑥‖⟶∞. (4.18)
Since 𝑉 is a Hilbert space and 𝐼 is weakly lower semi-continuous, the proof of Theorem 2.3 is completed.

Proof of Theorem 2.4. Let 𝑉=𝑉⁺⨁𝑉⁻, 𝑉⁺ denote the subspace of functions with mean value zero in 𝑉, and 𝑉⁻ denote the subspace of constant functions in 𝑉. By Theorem 2.1, we know the functional 𝐼(𝑥)=lim_𝑇→∞12𝑇^𝑇_−𝑇12|∇𝑥|²+𝐹(𝑡,𝑥(𝑡))𝑑𝑡 (4.19) is continuously differentiable on 𝑉.
For any 𝑣∈𝑉⁺, 𝐼(𝑣)=lim_𝑇→∞12𝑇^𝑇_−𝑇12|∇𝑣|²𝑑𝑡+lim_𝑇→∞12𝑇^𝑇_−𝑇{𝐹()+[𝐹((𝑡(=1𝑡,0𝑡,𝑣))−𝐹𝑡,0)]}𝑑𝑡2‖∇𝑣‖²₂+lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,0)𝑑𝑡+lim_𝑇→∞12𝑇^𝑇_−𝑇¹₀(((𝑡(𝑡≥1∇𝐹𝑡,𝑠𝑣)),𝑣))𝑑𝑠𝑑𝑡2‖∇𝑣‖²₂−𝑐₁+‖𝑣‖_∞lim_𝑇→∞12𝑇^𝑇_−𝑇𝑔(𝑡)≥1𝑑𝑡2‖∇𝑣‖²₂−𝑐₁−𝐶‖∇𝑣‖. (4.20)
So we see that inf_𝑉+𝐼>−∞, (4.21) by (f₆), there exists 𝑅>0, such that sup_𝑆−𝑅𝐼<inf_𝑉+𝐼, (4.22) where 𝑆⁻_𝑅={𝑢∈𝑉⁻∣|𝑢|=𝑅}, so (I₁) and (I₂) of Lemma 3.4. are satisfied.
Finally, we show that (𝑃𝑆)_𝑐 condition holds, that is, each sequence {𝑥_𝑘} in 𝑉 such that 𝐼(𝑥_𝑘)→𝑐 and ∇𝐼(𝑥_𝑘)→0 contains a convergent subsequence.
Letting 𝑥_𝑘=𝑥_𝑘+𝑥_𝑘 with 𝑥_𝑘=lim_𝑇→∞(1/2𝑇)∫^𝑇_−𝑇𝑥_𝑘(𝑡)𝑑𝑡, since ∇𝐼(𝑥_𝑘)→0, there exists some 𝑘₀ such that |⟨∇𝐼(𝑥_𝑘),ℎ⟩|≤‖ℎ‖ for all 𝑘≥𝑘₀ and ℎ∈𝑉; we obtain, for 𝑘>𝑘₀, ⟨∇𝐼𝑥_𝑘,𝑥_𝑘⟩=lim_𝑇→∞12𝑇^𝑇_−𝑇∇𝑥_𝑘(𝑡)²+∇𝐹𝑡,𝑥_𝑘(𝑡),𝑥_𝑘(𝑡)𝑑𝑡≤𝑥_𝑘 (4.23) and hence 𝑥_𝑘≤𝐶₁,𝑘≥𝑘₀ (4.24) because of Lemma 3.3. Now 𝐼(𝑥_𝑘)→𝑐, hence there exists 𝐶₂, such that lim_𝑇→∞12𝑇^𝑇_−𝑇12∇𝑥_𝑘²𝑑𝑡+lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹𝑡,𝑥_𝑘+𝐹𝑡,𝑥_𝑘(𝑡)−𝐹𝑡,𝑥_𝑘𝑑𝑡≥𝐶₂, (4.25) by using (4.24), we obtain lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹𝑡,𝑥_𝑘𝑑𝑡≥𝐶₃, (4.26) and then |𝑥_𝑘|≤𝐶₄,𝑘∈𝑁. By (4.24), thus {𝑥_𝑘} is bounded in 𝑉 and hence contains a subsequence, relabeled {𝑥_𝑘} which weakly converges to some 𝑥∈𝑉. Now, the equality ∇𝐼𝑥_𝑘−∇𝐼(𝑥),𝑥_𝑘−𝑥=∇𝑥_𝑘−∇𝑥²₂+lim_𝑇→∞12𝑇^𝑇_−𝑇∇𝐹𝑡,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.
Example 4.1. Consider the scalar problem: 𝑥̈(𝑡)=𝑃sin𝑥−𝑏sgn𝑥+sin𝑏sgn𝑥+ℎ(𝑡), (4.28) where 0<𝑏<2𝜋 and 𝑏≠𝜋. 𝑃 is a projection operator from 𝐴P⁰(𝑅) to 𝑉₁, and 𝑉₁={𝑥∈𝐵^1,2(𝑅)∣Λ(𝑥)⊆Λ}. ℎ∈𝑉₁ and lim_𝑇→∞12𝑇^𝑇_−𝑇ℎ(𝑡)𝑑𝑡=0. (4.29)
In this case, 𝐹(𝑡,𝑥)=𝑃[(sin𝑏)|𝑥|−cos(|𝑥|−𝑏)]+ℎ(𝑡)𝑥, and hence, when 0<𝑏<𝜋, lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)𝑑𝑡=|𝑥|sin𝑏−cos(|𝑥|−𝑏)→+∞, (4.30) if |𝑥|→∞. So it is easy to check that the conditions (f₁)–(f₅) are satisfied, then (4.28) has at least a quasi periodic solution by using Theorem 2.3.
When 𝜋<𝑏<2𝜋, lim_𝑇→∞12𝑇^𝑇_−𝑇𝐹(𝑡,𝑥)𝑑𝑡=|𝑥|sin𝑏−cos(|𝑥|−𝑏)→−∞, (4.31) if |𝑥|→∞. So it is easy to check that the conditions (f₁)–(f₄) and (f₆) are satisfied, then (4.28) has at least a quasi periodic solution by using Theorem 2.4.

Acknowledgment

This project is supported by National Natural Science Foundation of China (No. 10871053 and No. 11031002).