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

Authors: Juhong Kuang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We deal with the quasi-periodic solutions of the following
second-order Hamiltonian systemsx¨(t)=∇F(t,x(t)), where x(t)=(x1(t),…,xN(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.


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## Body

## 1. Introduction

In this paper, we consider the quasi-periodic solutions of the following second-order Hamiltonian system:(1.1)ẍ(t)=∇F(t,x(t)),
where x(t)=(x1(t),…,xN(t)), ∇F(t,x)=(∂F/∂x1,…,∂F/∂xN) and ∂F/∂xk∈C(R×RN,R),k=1,…,N.A special class of the system (1.1) is the following autonomous second-order Hamilton system with convex potential Φ:(1.2)ẍ(t)=∇Φ(x(t))+e(t),
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(1.3)Φ(x)=12〈Ax∣x〉+U(x),
where A is a symmetric positive-definite matrix and U∈C2(RN,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.WhenF 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
(1.4)x′′(t)=∇xΦ(t,x(t)),
where Φ∈C0(R×K,R), K is a compact convex subset of RN 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
(1.5)D1L(x(t-r),x(t-2r),x′(t-r),x′(t-2r),t-r)+D2L(x(t),x(t-r),x′(t),x′(t-r),t)=ddt[D3L(x(t-r),x(t-2r),x′(t-r),x′(t-2r),t-r)+D4L(x(t),x(t-r),x′(t),x′(t-r),t)],
where L:(Rn)4×R→R is a differentiable function, Dj 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 DkL 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 Section1 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 functionf(t) is said to be Bohr almost periodic, if for any ϵ>0, there is a constant lϵ>0, such that in any interval of length lϵ, 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×Rm,RN) is so called almost periodic in t uniformly for x∈Rm when, for each compact subset K in Rm, for each ɛ>0, there exists l>0, and for each α∈R, there exists τ∈[α,α+l] such that
(1.6)sup⁡t∈Rsup⁡x∈K‖f(t+τ,x)-f(t,x)‖RN<ɛ.
AP0(RN) is the space of the Bohr almost periodic functions from R to RN, endowed with the norm ∥x∥∞=supt∈R|x(t)| and it is a Banach space.
AP1(RN)={x∈AP0(RN)⋂C1(R,RN)∣x′(t)∈AP0(RN)}; 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:(1.7)lim⁡T→∞12T∫-TTx(t)dt.The Fourier-Bohr coefficients ofAP0(RN) are the complex vectors(1.8)a(x,λ)=lim⁡T→∞12T∫-TTe-iλtx(t)dt,andΛ(x)={λ∈R∣a(x,λ)≠0} and it is a countable set,whenp∈Z+, BP(RN) is the completion of AP0(RN) with respect to the norm
(1.9)‖u‖p={lim⁡T→∞12T∫-TT|u|pdt}1/p.
When p=2,B2(RN) is a Hilbert spaces and its norm ∥·∥2 is associated to the inner product
(1.10)〈u,v〉2=lim⁡T→∞12T∫-TT(u,v)dt,
the elements of these spaces BP(RN) are called Besicovitch almost periodic functions.We use the generalized derivative∇u∈B2(RN) of u∈B2(RN) 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)u(t)=∫0t∇u(t)dt+c.
Then we define B1,2(RN)={u∈B2(RN)∣∇u∈B2(RN)}, endowed with the norm
(1.12)‖u‖={lim⁡T→∞12T∫-TT(|u(t)|2+|∇u(t)|2)dt}1/2.
Its norm is associated to the inner product 〈u,v〉=〈u,v〉2+〈∇u,∇v〉2, and B1,2(RN) 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(RN)∣Λ(x)⊆Λ}, it is easily obtained that V is a linear subspace of B1,2(RN) 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 onF:(f1)
F(t,·)∈C1(R×RN,R), and F(t,·) is almost periodic in t uniformly for x∈RN,(f2)
∇F(t,·) is almost periodic in t uniformly for x∈RN,(f3)
for anyλ∈R∖Λ, x∈V,(2.1)lim⁡T→∞12T∫-TT∇F(t,x)e-iλtdt=0.Theorem 2.1.
SupposeF satisfies (f1)–(f3), the functional I:V→R, defined by
(2.2)I(x)=lim⁡T→∞12T∫-TT[12|∇x|2+F(t,x)]dt
is continuously differentiable on V, and I′(x) is defined by
(2.3)I′(x)h=lim⁡T→∞12T∫-TT[∇x∇h+∇F(t,x)h]dt.
Moreover, ifx is a critical point of I in V, then
(2.4)∇F(t,x)=∇(∇x).Definition 2.2.
Whenx satisfies (2.4) in Theorem 2.1, we say that x is a weak solution of (1.1).Theorem 2.3.
Suppose thatF satisfies (f1)–(f3), and(f4)
there existsg∈Lloc1(R), for a.e. t∈R and all x∈RN, such that(2.5)|∇F(t,x)|≤g(t),(f5)
There exists(2.6)lim⁡T→∞12T∫-TTF(t,x)dt⟶+∞as|x|⟶∞.
Then (1.1) has at least a quasi periodic solution.Theorem 2.4.
Suppose thatF satisfies (f1)–(f4), and(f6)
One has(2.7)lim⁡T→∞12T∫-TTF(t,x)dt⟶-∞as|x|⟶∞.
Then (1.1) has at least a quasi periodic solution by saddle point theorem.Remark 2.5.
WhenV 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.
Ifx∈V, then
(3.1)x(t)=∑k=-∞+∞akeiλkt∈AP0(RN),‖x‖∞≤(C+1)‖x‖.Proof.
For anyx(t)∈V,
(3.2)x(t)∼∑k=-∞+∞akeiλkt,∇x(t)∼∑k=-∞+∞iakλkeiλkt,λk∈Λ.
It is easily obtained that there exists a constant C>0, such that
(3.3)∑k=-∞,λk≠0k=+∞1λk2≤C2,|a0|+∑k=-∞,λk≠0k=+∞|ak|≤‖x‖2+(∑k=-∞,λk≠0k=+∞1λk2)1/2(∑k=-∞,λk≠0k=+∞|λkak|2)1/2≤‖x‖2+C‖∇x‖2≤(C+1)‖x‖.
So
(3.4)x(t)=∑k=-∞+∞akeiλkt∈AP0(RN),‖x‖∞≤(C+1)‖x‖.Lemma 3.2.
For any{xk}⊂V, if the sequence {xk} converges weakly to x, then {xk} converges uniformly to x on any compact subset of R.Proof.
By Lemma3.1, the injection of V into C(R,RN), with its natural norm ∥·∥∞, is continuous. Since {xk}⇀x in V, it follows that {xk}⇀x in C(R,RN). By Banach-Steinhaus theorem, {xk} is bounded in V, and hence in C(R,RN), we need to show that the sequence {xk} is equiuniformly continuous, for any xk(t)∈V,
(3.5)xk(t)=∑m=-∞+∞ameiλmt,
let
(3.6)xkj(t)=∑m=-∞,λm∈{(2nπ/Tj)∣n∈Z}+∞ameiλmt,
then
(3.7)xk(t)=xk1(t)+xk2(t)+⋯+xkp(t).
Denoting(3.8)Tmin⁡=min⁡{T1,T2,…,Tp},Tmax⁡=max⁡{T1,T2,…,Tp},
for 0≤t-s≤Tmin, we have
(3.9)|xk(t)-xk(s)|≤∫st|xk′(τ)|dτ≤∫st|xk1′(τ)|dτ+∫st|xk2′(τ)|dτ+⋯+∫st|xkn′(τ)|dτ≤(t-s)1/2{∫st|xk1′(τ)|2dτ}1/2+(t-s)1/2{∫st|xk2′(τ)|2dτ}1/2+⋯+(t-s)1/2{∫st|xkn′(τ)|2dτ}1/2≤(t-s)1/2{[∫ss+T1|xk1′(τ)|2dτ]1/2+[∫ss+T2|xk2′(τ)|2dτ]1/2+⋯+[∫ss+Tp|xkn′(τ)|2dτ]1/2}≤(t-s)1/2{T1‖xk′‖2+T2‖xk′‖2+⋯+Tp‖xk′‖2}≤(t-s)1/2pTmax⁡‖xk‖≤C1(t-s)1/2.
By Arzela-Ascoli theorem,{xk} is relatively compact on any compact of R. By the uniqueness of the weak limit, every uniformly convergent subsequence of {xk} converges to x on any compact of R.Lemma 3.3.
Ifx∈V and
(3.10)lim⁡T→∞12T∫-TTx(t)=0,
then there exists C>0, such that
(3.11)‖x‖∞2≤C2‖∇x‖22.Proof.
Since, by Lemma3.1, x has the Fourier expansion
(3.12)x(t)=∑k=-∞,λk≠0+∞akeiλkt.
The Cauchy-Schwarz inequality and Parseval equality imply that
(3.13)|x(t)|2≤(∑k=-∞,λk≠0+∞|ak|)2≤(∑k=-∞,λk≠0+∞1λk2)(∑k=-∞,λk≠0+∞|λkak|2)≤C2‖∇x‖22.Lemma 3.4 (saddle point theorem).
LetX 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 constantsσ,ρ>0, such that I∂Bρ⋂X1≤σ;(I2)
there exists a constantω>σ, such that IX2≥ω.
ThenI possesses a critical value c≥ω and
(3.14)c=inf⁡h∈Γmax⁡u∈Bρ⋂X1I(h(u)),
where Γ={h∈C(Bρ¯⋂X1,X)|h|∂Bρ⋂X1=id}.


## 4. The Proof of Main Results

In this section, we prove the main results stated in Section2.Proof of Theorem2.1.
First step: we show thatI has at every point x a directional derivative I′(x)∈V* given by (2.3).
It follows easily from Lemma3.1 and (f3), for any x∈V that we have
(4.1)F(t,x(t))∈AP0(R),∇x∈B2(RN).
SoI is everywhere finite on V. For x,h fixed in V, λ∈[-1,1], let us define
(4.2)G(λ,t)=12|∇x+λ∇h|2+F(t,x+λh),φ(λ)=lim⁡T→∞12T∫-TTG(λ,t)=I(x+λh).
There existsθt∈[0,1], such that
(4.3)1λ[φ(λ)-φ(0)]-lim⁡T→∞12T∫-TT[∇F(t,x)h+∇x∇h]dt=lim⁡T→∞12T∫-TT[λ|∇h|2+(∇F(t,x(t)+θtλh(t))h-∇F(t,x(t))h)]dt.
Forx,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∈RN, we have that ∇F(t,x) is uniformly continuous on R×K, where K={x∈RN∣|x|≤2M} is compact subset in RN, so
(4.4)lim⁡λ→01λ[φ(λ)-φ(0)]=lim⁡T→∞12T∫-TT[∇F(t,x)h+∇x∇h]dt.
Moreover, by Lemma3.1,
(4.5)(I′(x),h)=lim⁡T→∞12T∫-TT[∇F(t,x)h+∇x∇h]dt≤c1‖h‖∞+c2‖∇h‖2≤c3‖h‖.
So I has, at x, a Gâteaux derivative I′(x)∈(V)*.
Second step: we show that the mapping(4.6)I′:V⟶V*,x⟶I′(x)
is continuous.
For anyϵ>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∈RN∣|x|≤(C+1)(M+δ0)} is compact subset in RN, then there exists δ1, such that |x(t)-y(t)|≤δ1 and we have
(4.7)∇F(t,x(t))-∇F(t,y(t))≤ϵ2(C+1).
We denoteδ*=min{δ0,δ1/(C+1)}, then, for all h∈V and ∥h∥≤1, such that ∥x-y∥≤δ*, we have
(4.8)|I′(x)h-I′(y)h|≤|lim⁡T→∞12T∫-TT[(∇F(t,x)-∇F(t,y))h+(∇x-∇y)∇h]dt|≤ϵ2(C+1)‖h‖∞+{lim⁡T→∞12T∫-TT|∇x-∇y|2}1/2{lim⁡T→∞12T∫-TT|∇h|2}1/2≤ϵ2+ϵ2=ϵ.
The above inequality holds, which implies the continuity ofI′ so that I is Fréchet differentiable on V.
Ifx is a critical point of I in V, for all h∈V, we have
(4.9)I′(x)h=lim⁡T→∞12T∫-TT[∇F(t,x)h+∇x∇h]dt=0,
by (f3), then for all h∈AP1(RN), we have
(4.10)lim⁡T→∞12T∫-TT[∇F(t,x)h+∇x∇h]dt=0.
SinceAP1(RN) is dense in B1,2(RN), we have DI(x)h=0, for all h∈B1,2(RN); therefore, I′(x)=0, and then we obtain (2.4) by using Blot [12]. The proof of Theorem 2.1 is completed.Proof of Theorem2.3.
By Theorem2.1, I is continuously differentiable on V. Next we will prove that I is weakly lower semicontinuous on V.
By Lemma3.2, if {xk}⊂V converges weakly to x, then {xk} converges uniformly to x on any compact of R.
Sincexk(t)∈AP0(Rn), and F(t,·) is almost periodic in t uniformly for x∈Rn, then F(t,xk(t)) is almost periodic, and F(t,xk(t)) converges uniformly to F(t,x(t)) on any compact of R.
Let(4.11)F(t,xk(t))=ak0+∑j=-∞,λj≠0+∞akjeiλjt,F(t,x(t))=a0+∑j=-∞,λj≠0+∞ajeiλjt.
Then it is easily obtained that
(4.12)ak0⟶a0,
moreover,
(4.13)lim⁡T→∞12T∫-TTF(t,xk(t))dt=ak0,
so
(4.14)lim⁡T→∞12T∫-TTF(t,xk(t))dt⟶lim⁡T→∞12T∫-TTF(t,x(t))dt.
Moreover, (1/2)|∇u(t)|2 is convex and continuous, so I is weakly lower semi-continuous.
Forx∈V, we have x=x¯+x̃, where
(4.15)x¯=lim⁡T→∞12T∫-TTx(t)dt,
then,
(4.16)lim⁡T→∞12T∫-TTx̃(t)=0,I(x)=lim⁡T→∞12T∫-TT12|∇x|2dt+lim⁡T→∞12T∫-TT{F(t,x¯)+[F(t,x(t))-F(t,x¯)]}dt=12‖∇x‖22+lim⁡T→∞12T∫-TT{F(t,x¯)+∫01(∇F(t,x¯+sx̃(t)),x̃(t))ds}dt≥12‖∇x‖22-lim⁡T→∞12T∫-TTg(t)dt‖x̃(t)‖∞+lim⁡T→∞12T∫-TTF(t,x¯)dt≥12‖∇x‖22-c‖∇x‖2+lim⁡T→∞12T∫-TTF(t,x¯)dt.
As∥x∥→∞ if and only if
(4.17)(|x¯|2+lim⁡T→∞12T∫-TT|∇x(t)|2dt)1/2⟶∞,
the above inequality and (f5) imply that
(4.18)I(x)⟶+∞as‖x‖⟶∞.
SinceV is a Hilbert space and I is weakly lower semi-continuous, the proof of Theorem 2.3 is completed.Proof of Theorem2.4.
LetV=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
(4.19)I(x)=lim⁡T→∞12T∫-TT{12|∇x|2+F(t,x(t))}dt
is continuously differentiable on V.
For anyv∈V+,
(4.20)I(v)=lim⁡T→∞12T∫-TT12|∇v|2dt+lim⁡T→∞12T∫-TT{F(t,0)+[F(t,v(t))-F(t,0)]}dt=12‖∇v‖22+lim⁡T→∞12T∫-TTF(t,0)dt+lim⁡T→∞12T∫-TT∫01(∇F(t,sv(t)),v(t))dsdt≥12‖∇v‖22-c1+‖v‖∞lim⁡T→∞12T∫-TTg(t)dt≥12‖∇v‖22-c1-C‖∇v‖.
So we see that(4.21)inf⁡V+I>-∞,
by (f6), there exists R>0, such that
(4.22)sup⁡SR-I<inf⁡V+I,
where SR-={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 {xk} in V such that I(xk)→c and ∇I(xk)→0 contains a convergent subsequence.
Lettingxk=x¯k+x̃k with x¯k=limT→∞(1/2T)∫-TTxk(t)dt, since ∇I(xk)→0, there exists some k0 such that |〈∇I(xk),h〉|≤∥h∥ for all k≥k0 and h∈V; we obtain, for k>k0,
(4.23)|〈∇I(xk),x̃k〉|=|lim⁡T→∞12T∫-TT(|∇xk(t)|2+(∇F(t,xk(t)),x̃k(t)))dt|≤‖x̃k‖
and hence
(4.24)‖x̃k‖≤C1,k≥k0
because of Lemma 3.3. Now I(xk)→c, hence there exists C2, such that
(4.25)lim⁡T→∞12T∫-TT12|∇xk|2dt+lim⁡T→∞12T∫-TT{F(t,x¯k)+[F(t,xk(t))-F(t,x¯k)]}dt≥C2,
by using (4.24), we obtain
(4.26)lim⁡T→∞12T∫-TTF(t,x¯k)dt≥C3,
and then |x¯k|≤C4,k∈N. By (4.24), thus {xk} is bounded in V and hence contains a subsequence, relabeled {xk} which weakly converges to some x∈V. Now, the equality
(4.27)〈∇I(xk)-∇I(x),xk-x〉=‖∇xk-∇x‖22+lim⁡T→∞12T∫-TT(∇F(t,xk(t))-∇F(t,x(t)),xk(t)-x(t))dt
holds, and Lemma 3.2 implies that ∥∇xk-∇x∥2→0 as k→∞, so ∥xk-x∥≤∥xk-x∥2+∥∇xk-∇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:(4.28)ẍ(t)=P[sin⁡(x-bsgn⁡x)+sin⁡(bsgn⁡x)]+h(t),
where 0<b<2π and b≠π. P is a projection operator from AP0(R) to V1, and V1={x∈B1,2(R)∣Λ(x)⊆Λ}. h∈V1 and
(4.29)lim⁡T→∞12T∫-TTh(t)dt=0.
In this case,F(t,x)=P[(sinb)|x|-cos(|x|-b)]+h(t)x, and hence, when 0<b<π,
(4.30)lim⁡T→∞12T∫-TTF(t,x)dt=|x|sin⁡b-cos⁡(|x|-b)→+∞,
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.
Whenπ<b<2π,
(4.31)lim⁡T→∞12T∫-TTF(t,x)dt=|x|sin⁡b-cos⁡(|x|-b)→-∞,
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.



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*Source: 271616-2012-02-20.xml*