Yaqun PENG Xinli ZHANG Daxiong PIAO

Abstract The authors study the Lagrangian stability for the sublinear Duffing equations+e(t)|x|α−1x = p(t) with 0 < α < 1,where e and p are real analytic quasi-periodic functions with frequency ω.It is proved that if the mean value of e is positive and the frequency ω satisfies Diophantine condition,then every solution of the equation is bounded.

Keywords Hamiltonian system,Sublinear Duffing equation,Boundedness,Quasiperiodic solution,Invariant curve

1 Introduction

In 1976,Morris[1],by using Moser’s twist theorem,proved that all solutions of the equation

are bounded whenpis periodic and piecewise continuous.Since then,KAM theory has been the most powerful tool to study Littlewood’s boundedness problem for Duffing type equations

whereψis periodic int.And fruitful achievements have been made by many authors (see for examples [2–6]and references therein).

In 1999,K¨upper-You [7]proved that all solutions of the equation

are bounded,where 0<α<1 andp∈C∞(T).

In 2001,Liu [8]investigated the sublinear equation in the more general form

and concluded that all solutions of the equation are bounded withp∈C5(T) andϕ∈C6(R)satisfying the sublinear condition:

In 2009,Wang [9]studied the sublinear equation

where 0< α <1,e,p∈C5(T),He proved that the necessary and sufficient

In the dynamical point of view,it is natural to study Littlewood’s boundedness problem for(1.1) withψquasi-periodic int.

In 2000,Zharnitsky[10]proved an invariant curve theorem for a quasi-periodic planar mapping and applied it to answering a question asked by Levi-Zehnder[11],that is the boundedness of solutions of the Fermi-Ulam model.

In 2005,Liu[12]established some invariant curve theorems for some planar reversible mappings with quasi-periodic perturbations.As an application,he proved the existence of quasiperiodic solutions and the boundedness of all solutions of an asymmetric oscillation

whenpis a real analytic,even and quasi-periodic function with the frequencyωsatisfying the Diophantine condition.

Recently Huang-Li-Liu [13–14]proved the existence of invariant curves for quasi-periodic smooth mappings and used the theory to get the existence of quasi-periodic solutions and the boundedness of all solutions of (1.3) whenpis a smooth quasi-periodic function with the frequency satisfying the Diophantine condition (see the results in Appendix).

Motivated by the above references,especially by Wang[9]and Huang-Li-Liu[13–14],we are going to investigate the boundedness problem of the special quasi-periodic subilinear Duffing equations

with 0< α <1,whereeandpare real analytic quasi-periodic functions and their frequencyω=(ω1,ω2,···,ωn) satisfies the Diophantine condition

It is well known that for any quasi-periodic functionf,its mean valuealways exists.Denote it by [f].

Our main result is the following theorem.condition that the equation posses the Lagrangian stability is

Theorem 1.1Assume thate,pare real analytic quasi-periodic functions with the frequencyω= (ω1,···,ωn) satisfying the Diophantine condition (1.5).If [e]>0,then (1.4) has quasiperiodic solutions and all the solutions of (1.4) are bounded,i.e.,every solutionx(t) of (1.4)exists fort∈R and

Remark 1.1The main idea of the proof of Theorem 1.1 is similar to the one in [9].But here,due to the quasi-periodicity ofeandp,we meet the so called “small divisor” problems,so we need much more regularity estimates and introducing new function class Fω(r0,l0) of quasi-periodic functions as a tool.To meet the requirements of the invariant curve theorem established by Huang-Li-Liu in [13],we must suppose thateandpare analytic quasi-periodic functions.It seems an interesting question to consider the smooth case.

The rest of our paper is organized as follows.In Section 2,we will give some definitions and proprieties and the integral proposition of quasi-periodic functions.In Section 3,we will introduce the action-angle variables and the new function class Fω(r0,l0) of quasi-periodic functions,then change action-angle variables.In Section 4,we will make further canonical transformations and obtain a new transformed Hamiltonian system.In Section 5,we will prove the existence of quasi-periodic solutions and the boundedness of all solutions for (1.4).Here we point out that though our proof appears a simple variant of [9],there is a huge difference between our quasi-periodic case and the periodic case in [9].In fact,in our proof we use the integral proposition of quasi-periodic functions in Section 2 and the proprieties of the new function class Fω(r0,l0) in Section 3.

2 Preliminaries

We first recall some basic knowledge on the analytic quasi-periodic functions.For further contents,one can refer to [15,Chapter 3].

Definition 2.1(see [15]) A functionf:R →R is called a real analytic quasi-periodic function with the frequencyω,if it can be represented by a Fourier series

wherek=(k1,k2,···,kn),〈k,ω〉=k1ω1+k2ω2+···+knωn0 ifk0,andfkexponentially decays with |k|,where |k|=|k1|+|k2|+···+|kn|.

The set of all such functions is denoted byQ(ω).

It is not difficult to seef0=[f].

For eachf∈Q(ω),there is a real analytic functionF(θ) =F(θ1,θ2,···,θn):Rn→R which is 1-periodic in each variableθj(1 ≤j≤n) and bounded in a complex neighborhoodof Rnfor somer >0 such that

ThenFhas a Fourier expansion

ThisFis called the shell function off.

LetQr(ω) ⊆Q(ω) be the set of real analytic quasi-periodic functionfsuch that the corresponding shell functionsFis bounded on the subset Πnr= {(θ1,θ2,···,θn) ∈Cn:|Imθj| ≤r,j=1,2,···,n} with the supremum norm

Define |f|r=|F|r.

It is well known that indefinite integral of a periodic function is still a periodic function if the mean value of the function is zero.It is easy to prove that this conclusion is not valid for a quasi-periodic function.However we have the following result for a real analytic quasi-periodic function.

Proposition 2.1Iff∈Q(ω) with the frequencyωsatisfing the Diophantine condition

(1.5),and

theng∈Q(ω).

ProofFrom Definition 2.1,

Suppose |fk|≤|f|re|k|for somer >0 and>0.Then from (1.5),we have

So

which implies that the functiongis well defined,whereCis a positive constant.Sinceg(t)=fork0,noting the fact thatfkdecay exponentially,we seeg∈Q(ω).

Lemma 2.1(see [13]) The setQ(ω) has the following properties:

(1) Iff,g∈Q(ω),thenf±g,g(·+f(·))∈Q(ω).

(2) Ifωsatisfies Diophantine condition,f∈Q(ω) andτ=βt+f(t) withβ+f′>0,then the inverse relation is given byt=β−1τ+g(τ) whereg∈.In particular,ifβ=1,theng∈Q(ω).

Lemma 2.2Iff,g∈Q(ω),thenf·g∈Q(ω).

ProofSincef,g∈Q(ω),we denotef,grespectively as

wherefk,gksatisfy |fk|≤M1e−|k|ρ1,|gk|≤M2e−|k|ρ2for positive constantsM1,M2,ρ1,ρ2.

Letζ(t)=f(t)g(t).Then

where

(i) Consider the caseρ2>ρ1>0.We have

therefore,ζ∈Q(ω).

(ii) Consider the caseρ1>ρ2>0.We have

therefore,ζ∈Q(ω).

(iii) Consider the caseρ1=ρ2and choose a constant 0<ρ<ρ1=ρ2.We have

therefore,ζ∈Q(ω).We complete the proof now.

3 Action-Angle Variables

We will first introduce the action-angle variables after two canonical transformations and then change action-angle variables in this section.Moreover,we give the definition and properties of a new function class Fω(r0,l0) of quasi-periodic functions in order to estimate the Hamiltonian.

3.1 A canonical transformation

(1.4) can be written as a Hamiltonian system

The corresponding Hamiltonian is

wheree1,qare as the following:

Sincee,p∈Q(ω),from Proposition 2.1,qis well defined andq∈Q(ω).

To make the Hamiltonian system simple,we introduce a transformation

whereG1(x,t) will be determined later.Under Φ1,Hamiltonian function (3.2) is transformed to

Let

then

Define

From Proposition 2.1,we seeE∈Q(ω).From Lemma 2.2,E2∈Q(ω).Therefore,G1(x,·) ∈Q(ω) for everyx∈R.Then the Hamiltonian function (3.2) becomes

and the corresponding Hamiltonian system is

3.2 Introducing action-angle variables

In order to introduce the action-angle variables,firstly consider the corresponding autonomous Hamiltonian system of (3.4),

Let (x0(t),z0(t)) be the periodic solution of (3.5) satisfying the initial value

andT0>0 be its minimal period.Introduce the functions C and S by

The functions C,S satisfy

The action and angle variables are introduced by the canonical transformation

Under Φ2,the Hamiltonian (3.3) is transformed into

We introduce the quasi-periodic function space Fω(r0,l0) as follows.

3.3 A function class Fω(r0,l0)

Givenr0∈R,l0≥0,denote Fω(r0,l0) the set of functions in (λ,θ,t) ∈R+×T×R:fisC∞inλ,Cl0inθ,f(λ,θ,·)∈Q(ω) for all (λ,θ)∈R+×T and satisfies

Lemma 3.1Fω(r0,l0) has the following properties:

(i) Ifr1

(ii) Iff∈Fω(r0,l0),then∈Fω(r0-j0,l0).

(iii) Iff1∈Fω(r1,l1) andf2∈Fω(r2,l2),thenf1·f2∈Fω(r1+r2,min{l1,l2}).

(iv) Iff∈Fω(r0,l0) satisfies |f(λ,·,·)|≥cλr0forλ>λ0,then∈Fω(-r0,l0).

Proof(i)f∈Fω(r1,l0),r1

(ii)f∈Fω(r0,l0),then

(iii)f1∈Fω(r1,l1) andf2∈Fω(r2,l2),from Lemma 2.2,f1·f2∈Q(ω).

(iv)f∈Fω(r0,l0),then

Forf∈Fω(r0,l0),denote the mean value overt-variables by [f]:

Define C1(θ)=(dbC(θ))α.It is obvious that C1∈C0for 0<α<1.Rewrite (3.6) as

Denote

From the definition of the function space Fω(r0,l0) and Lemma 3.1,we have

Define

Then the Hamiltonian (3.7) becomes

where

Since the Hamiltonian (3.9) is onlyC0onθ,we cannot guarantee that the Poincare map of (3.9) is smooth enough as required in the quasi-periodic invariant curve theorem obtained by Huang-Li-Liu in [13].To solve this probelm,we will exchange the role ofθandtin the following part.

3.4 Changing action-angle variables

Similarly,from(3.10),H0∈Fω(a,2),H1∈Fω(2a-1,0),for large enoughI >0,so there exists a function I(H,t,θ) such that

which can be rewritten as

From (3.11) and (3.13),it is obvious that

Consider the function

Let

It is easy to deduce that I1isC∞on C,C1and S respectively.From the definition ofH1,I0and Lemmas 2.1–2.2,I1∈Q(ω).

The Hamiltonian (3.9) becomes

withθ,t,Hbeing the new time variables,new angle variables and new action variables respectively.From the proprieties of the function space Fω(r0,l0) (see Lemma 3.1),we have

Furthermore,for a positive constantC0,

4 More Transformations

We will make more transformations since the Poincare mapping of the Hamiltonian system(3.17)is not a small perturbation of a stand quasi-periodic twist mapping.Notice that all these transformations are quasi-periodic in the time variable.We will discuss the quasi-periodicity after every transformation.

It should be noticed that C1∈C0.In order to make more transformations,we will improve the smoothness of C1by constructing a smooth approximation function C2of C1.DenoteFor the same reason,we also find a smooth approximation function S2of S1.The method can be found in[9],so we state the two conclusion in the following Lemma 4.1 without detail proof for simplicity.Here we point out that though our proof appears a simple variant of[9],there is a big difference between our quasi-periodic case and the periodic case in [9].In fact,in our proof we use the integral proposition of quasi-periodic functions in Section 2 and the proprieties of the new function class Fω(r0,l0) in Section 3.

Lemma 4.1(see [9]) For anyε>0,there existC1periodic functions C2,S2such that

and

where constantsD1,D2>0 are independent ofε.

From (3.17),

Let

Then (3.17) is rewritten as

Lemma 4.2For the initial action variableH0>0 large enough,there exists a canonical transformation such that the Hamiltonian (4.5) is transformed into

where

Moreover,for a positive constantC0,

Proofbe the parameter in Lemma 4.1 with the constant 0

From (3.18),(4.4) and Lemma 4.1,

whereD1is a constant independent ofH0denoted in Lemma 4.1,so

Introduce a canonical transformation

where the functionG2(λ,t,θ) will be determined later.Under Φ3,the Hamiltonian (4.5) is transformed into

where [I1](λ,C(θ),C2(θ),S(θ))

Let

then

I1isC∞in C,C2,S respectively and I1∈Q(ω)with the frequencyωsatisfying the Diophantine condition (1.5).According to Proposition 2.1,andC∞in C,C2,S respectively.ThusG2(λ,·,C(θ),C2(θ),S(θ)) ∈Q(ω) andC∞in C,C2,S respectively.It is obvious that

Denote

Then (4.12) can be rewritten as

Define

In the above denotation,t=t(λ,τ,C(θ),C2(θ),S(θ)).

It is obvious that J0(λ,θ,C2(θ)) isC∞on C2andC1onθ.From Lemma 2.2,J1isC∞in C,C2,S andC1onθrespectively and J1(λ,·,C(θ),C2(θ),S(θ),S1(θ))∈Q(ω).From Lemma 2.1,isC1onθand∈Q(ω).

Moreover,from (4.9),

From the definition ofG2,rewrite (4.11) as

Define

From (4.7),

Define

Rewrite (4.13) as

However,the Poincare map of the Hamiltonian system corresponding to(4.5)does not have the form of the Poincare mapping in [14],therefore we should make another transformation.In the above proof,C′2is onlyC0onθ.To satisfy the smoothness requirements,we establish aC1function C3which is an approximation of C′2similar as in Lemma 4.1.Then we can use the quasi-periodic twist theorem in [14]to the Poincare mapping.From [9],we have the following lemma.

Lemma 4.3(see [9]) For anyH0>0 and 0< ε0< c0,there exists aC1function C3(θ)such that

whereDis a constant independent ofH0.

From the above results,the Hamiltonian (4.6) is

where J0,J1,J2areC1onθ,C∞on C2and S2respectively and J1(λ,·,θ,C2(θ),S2(θ))∈Q(ω),J2(λ,·,θ,C2(θ))∈Q(ω),J3(λ,·,θ)∈Q(ω).

Lemma 4.4For the initial action variableH0>0 large enough (which implies that new initial action variableλ0large enough),there exists a canonical transformation which transforms the Hamiltonian (4.18) into

Moreover,we have

wherec0>0 andc1=are constants independent ofε0.

ProofIntroduce the canonical transformation

where the functionG3(ρ,τ,θ) will be determined later.Under Φ4,the Hamiltonian (4.18) is transformed into

where

Let

then

From the definition of J1,J2and Proposition 2.1,G3(ρ,·,θ)∈Q(ω).We can calculate that

Define

whereτ=τ(ρ,ς,θ,C3(θ)).Then,the Hamiltonian (4.18) is rewritten as

From Proportion 2.1,L1(ρ,·,θ),Letθ∗be the number such thatNote that C3(θ) = 0 forand there are similar results for C′3and C′2·C3.From the estimate on J2in (4.7) and C3in (4.3),

From (4.7) and for the reason thatis bounded,we can also prove other parts of(4.20).

5 Proof of Main Result

It is obvious that the solution (H(θ),t(θ)) of (3.17) with the initial conditionH(0) =H0,t(0)=t0satisfies

wherec,C >0 are two positive constants.As a consequence,for the solution (ρ(θ),ς(θ)) of(4.19) with the initial condition (ρ(0),ς(0))=(ρ(H0,t0,0),ς(H0,t0,0)),we have

Consider the Hamiltonian (4.19),

The corresponding Hamiltonian system is

where

Define

From (4.20),

Moreover,forρlarge enough,

Then the Poincare map is expressed as P1:{r|r >r∗,r∗≫1}×R →R2of the following form

where

whenε0in Lemma 4.1 is chosen small enough to make it smaller than the positive number which is dependent onc0,c1andaand make the Poincare map P1satisfy (6.1) in Theorem 6.1.

Define the quasi-periodic mapping M as

where the functionf(θ,·),g(θ,·)are quasi-periodic inθwith the frequencyω=(ω1,ω2,···,ωn).

Definition 5.1(see [13]) Let M be a mapping defined as (5.8).If M:R×[a0,b0]→R2is symplectic with respect to the usual symplectic structure dr∧dθand for every curve Γ:θ=ξ+ϕ(ξ),r=ψ(ξ),where the continuous functionsϕandψare quasi-periodic inξwith the frequencyω,there is

we say that M is an exact symplectic map.

Lemma 5.1(see [13]) If the mapping (5.8) is an exact symplectic map,then it has intersection property.

Proof of Theorem 1.1Since all the transformations are canonical,the Poincare map P1is an exact symplectic.For all the detail above,the Poincare map P1satisfies all the requirements in Theorem 6.1.For any rotation numberϖsatisfying (6.3),we can obtain a quasi-periodic invariant curve of P1with the form (6.4).Let (x,˙x) be the solution of (1.4) staying in the interior of some quasi-periodic invariant curves with appropriate rotation numberϖsatisfying(6.3).Since all the transformations are canonical,every solution starting from(x,˙x)is confined in the interior of the time quasi-periodic cylinder whose boundary is one of the quasi-periodic invariant curves and thus this solution is bounded.Notice that the initial value can be chosen large enough,thus,all the solutions are bounded.

6 Appendix

Assume thatf:R2→R is aCmsmooth function.Define

In 2017,Huang-Li-Liu [13]established the twist theorem of the following smooth quasiperiodic mapping.

Theorem 6.1(see [13]) The quasi-periodic mapping M given as (5.8),

is of class Cm(m >2τ+1>2n+1) and satisfies the intersection property.The functionsf(θ,r),g(θ,r) are quasi-periodic inθwith the frequencyω= (ω1,ω2,···,ωn) satisfying the following conditions:

where Γ is the Gamma function,γ,τare constants satisfying

ando1(m,Γ,γ,τ),o2(m,Γ,γ,τ) are sufficiently small functions ofm,Γ,γ,τanda0,b0>0 are two constants.

Then for any rotation numberϖsatisfying the inequalities

the quasi-periodic mapping M has an invariant curve Γ0with the form

whereφ,ψare quasi-periodic with the frequencyω= (ω1,···,ωn) and the invariant curve Γ0is continuous and quasi-periodic with the frequencyω.Moreover,the restriction of M onto Γ0is

Remark 6.1(see [13]) If all conditions of Theorem 6.1 hold,then the mapping M has many invariant curves Γ0,which can be labeled by the form

of the restriction of M onto Γ0.In fact,given anyϖsatisfying the inequalities(6.3),there exists an invariant curve Γ0of M which is quasi-periodic with the frequencyω,and the restriction of M onto Γ0has the form