# Boundedness of Solutions for a Class of Sublinear Reversible Oscillators with Periodic Forcing

Authors: Tingting Zhang; Jianguo Si
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study the boundedness of all solutions for the following differential equationx′′+f(x)x′+(B+εe(t))|x|α-1x=p(t), where f(x),p(t) are odd functions, e(t) is an even function, e(t),p(t) are smooth 1-periodic functions, B is a nonzero constant, and ε is a small parameter. A sufficient and necessary condition for the boundedness of all solutions of the above equation is established. Moreover, the existence of Aubry-Mather sets is obtained as well.


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## Body

## 1. Introduction

It is well known that the longtime behavior for periodically forced planar systems can be very intricate. For example, there are equations having unbounded solutions but with infinitely many zeros and with nearby unbounded solutions having randomly prescribed number of zeros and also periodic solutions; see [1]. In contrast to such unbounded phenomenon Littlewood [2] suggested to study the boundedness of all the solutions of the following differential equation:
(1)x¨+g(x)=h(t)
in the following two cases:(i)
superlinear case:g(x)/x→+∞ as x→±∞;(ii)
sublinear case:sgn(x)·g(x)→+∞ and g(x)/x→0 as x→±∞. Later, one calls this subject as Littlewood boundedness problem.The first result in superlinear case is obtained by Morris [3], who showed that all solutions of
(2)x¨+2x3=e(t)
are bounded, where e(t)∈C0. Later, a series results in superlinear case were obtained by several authors, see [4–13] and references therein. However, in general, it is harder to study the Lagrange stability of sublinear systems since smoothness of sublinear term is insufficient. There are only a few works in sublinear case so far. In 1999, Küpper and You [14] proved the first result in the study of the equation
(3)x¨+|x|α-1x=p(t),
where 0<α<1 and p(t)∈C∞(𝕋). Later, Liu [15] proved the same result for more general equation
(4)x¨+g(x)=e(t),
where g(x)∈C6 satisfying the sublinear condition (ii) and some inequalities, and e(t)∈C5(𝕋). In 2004, Ortega and Verzini [16] studied the boundedness of (4) in a special case with the variational method. In 2009, Wang [17] gave a sufficient and necessary condition for the boundedness of all solutions for sublinear equation
(5)x¨+e(t)|x|α-1x=p(t),
where e(t),p(t)∈C5(𝕋).As is widely known, there is a deep similarity between reversible and Hamiltonian dynamics. Many fundamental results of the Hamiltonian systems possess reversible counterparts. On boundedness problem for sublinear reversible systems, the first results were obtained by Li [18], later, Yang [19], in the study of a sublinear reversible systems
(6)x¨+f(x)x˙+|x|α-1x=e(t).
Recently, Wang [20] gave a sufficient and necessary condition for the boundedness of all solutions of the differential equation
(7)x¨+f(x)g(x˙)+γ|x|α-1x=p(t)
with 0<α<1, γ≠0.By the discussions about the sublinear Hamiltonian equation (1.3) in [17] motivations, we will study the boundedness of all solutions for a sublinear reversible system like
(8)x¨+f(x)x˙+(B+ɛe(t))|x|α-1x=p(t),
where B≠0 and 0<α<1. Furthermore, we also show that (8) has solutions of Mather type. The results obtained in [18–20] can be regarded as corollary of result of this paper.Remark 1.
Using the method of this paper we also can consider the more general equation(9)x¨+f(x)g(x˙)+(B+ɛe(t))|x|α-1x=p(t)
provided of adding suitable conditions for g(x). For convenience, we only consider the case g(x)≡x.Remark 2.
Adding the perturbation termɛe(t)|x|α-1x will lead to a new difficulty for estimating |S(θT0)|α-1C(θT0) appeared in (86). Fortunately, we can easily verify that ∫01‍|S(θT0)|α-1C(θT0)dθ is bounded by a constant (see in the proof of Lemma 12).Throughout this paper, we denote two universal positive constants without regarding their values byc<1 and C≥1, and suppose that the following conditions hold:(A1)
f(x)∈C4(ℝ),   p(t)∈C3(𝕋) and e(t)∈C3(𝕋), f(x) and p(t) are odd, e(t) is even, and e(t),   p(t) are both 1-periodic functions, 𝕋=ℝ/ℤ;(A2)
there is some positive constantμ such that the inequalities
(10)|xi+1f(i)(x)|≤C|x|α/2-βare satisfied for0≤i≤4 and all |x|≥μ, where 0<β<α/2.We decomposee(t) as e(t)=e¯+e^(t), where e¯ is the average of e(t) and e^(t) has zero mean value. That is e¯=∫01‍e(s)ds and ∫01‍e^(s)ds=0. If we write that A=B+ɛe¯, then it is easy to see that A and B have the same sign when 0<ɛ<ɛ* with 0<ɛ*<|B/e¯|.Now we state the main results of this paper.Theorem 3.
Assume thatB≠0 and (A1)-(A2) hold. Then there exists an 0<ɛ**<ɛ* such that for any 0<ɛ<ɛ**, every solution of (8) is bounded if and only if B>0.Theorem 4.
Under the conditions of Theorem3, there is an ɛ0>0 such that, for any ω∈(n,n+ɛ0), (8) has a solution (xω(t),xω′(t)) of Mather type with rotation number ω. More precisely:(i)
ifω=p/q is rational, the solutions (xω(t+i),xω′(t+i)), 1≤i≤q-1, are periodic solutions of period q; moreover, in this case
(11)limω→nmint∈ℝ(|xω(t)|+|xω′(t)|)=+∞;(ii)
ifω is irrational, the solution (xω(t),xω′(t)) is either a usual quasi-periodic solution or a generalized one.
We recall that a solution is called generalized quasi-periodic if the closed set(12){x(i),x′(i),i∈ℤ}¯
is a Denjoys minimal set.


## 2. Reversible Systems and Action-Angle Variables

In this section, we will assume thatB>0 and A>0. Firstly, we consider (8) which is equivalent to the following system:
(13)x˙=z+P(t),z˙=-A|x|α-1x-ɛe^(t)|x|α-1x-f(x)(z+P(t)),
where P(t)=∫0t‍p(s)ds. Then we can obtain that (13) is reversible with respect to the transformation (x,z)↦(-x,z) by (A1).Lemma 5.
There exists aG-invariant diffeomorphism (x,y)→(x,z) such that (13) is transformed into the following system:
(14)x˙=y+ɛE(t)|x|α-1x+P(t),y˙=-A|x|α-1x-[αɛE(t)|x|α-1+f(x)][y+ɛE(t)|x|α-1x+P(t)],
where E(t)=-∫0t‍e^(s)ds.Proof.
Introduce a transformationΨ:
(15)x=x,z=y+U(x,t),
where U(x,t) will be determined later. Under this transformation, the system (13) is transformed into a new system as follows:
(16)x˙=y+U(x,t)+P(t),y˙=-A|x|α-1x-ɛe^(t)|x|α-1x-(f(x)+∂U(x,t)∂x)[y+U(x,t)+P(t)]-∂U(x,t)∂t.
Now, we define the function U(x,t) by
(17)-ɛe^(t)|x|α-1x-∂U(x,t)∂t=0.
Since ∫01‍e^(t)dt=0, so we can obtain U(x,t)=ɛE(t)|x|α-1x. Then the new system can be expressed as in (14) by direct computation.
It is easy to know thatU(-x,-t)=U(x,t) by (A1), then we can obtain that the transformation Ψ is a G-invariant diffeomorphism.Let us consider the auxiliary system(18)x˙=y,y˙=-A|x|α-1x,
which is a time-independent Hamiltonian system with Hamiltonian
(19)H0(x,y)=y22+Aα+1|x|α+1.It is easy to see thatH0(x,y)>0,  (x,y)∈R2∖{0},   H0(0,0)=0. Note that each level line H0(x,y)=h>0 is a close orbit of system (18), hence, all the solutions of (18) are periodic with period tending to zero as h tends to infinity.Assume that(S(t),C(t)) is the solution of (18) with initial conditions (S(0),C(0))=(0,1), and let T0>0 be the minimal period. We can find that S(t) and C(t) satisfy(i)
S(t)∈C2(R), C(t)∈C1(R);(ii)
(S(-t),C(-t))=(-S(t),C(t)), (S(t+T0),C(t+T0))=(S(t),C(t));(iii)
S˙(t)=C(t), C˙(t)=-A|S(t)|α-1S(t);(iv)
(1/2)C2(t)+(A/(α+1))|S(t)|α+1=1/2;(v)
C(T0t)=0⇔t(mod(1/4))=0;(vi)
(S(T0(1/2-t)), C(T0(1/2-t)))=(S(T0t),-C(T0t));(vii)
S(T0t)=0⇔t(mod(1/2))=0.Then we introduce the transformation(20)Φ:ℝ+×𝕋⟶ℝ2∖{0},(ρ,φ)⟼(x,y)
which is
(21)x=ρbS(φT0),y=ρ1-bC(φT0),
where b=2/(3+α). It is easy to see that 1/2<b<2/3 by 0<α<1. Since (S(-t),C(-t))=(-S(t),C(t)), this transformation is invariant with respect to the involutions (ρ,φ)↦(ρ,-φ) and (x,y)↦(-x,y), and we can find that the mapping Φ is a generalized canonical transformation by (iv). In fact,
(22)|∂(x,y)∂(ρ,φ)|=|AbT0|S(φT0)|α+1+(1-b)T0C2(φT0)|=|(1-b)T0-α+12bT0C2(φT0)+(1-b)T0C2(φT0)|=(1-b)T0,(ρ˙φ˙)=(-dyφdxφdyρ-dxρ)(x˙y˙),
where d=((1-b)T0)-1.Under the transformationΦ, the system (18) is transformed into the simpler form
(23)ρ˙=-∂h0∂φ=0,φ˙=∂h0∂ρ=1T0·ρ1-2b,
where h0(ρ)=((2-2b)T0)-1·ρ2(1-b).The original system (13) is transformed into the system
(24)dρdt=l1(ρ,φ)+l2(ρ,φ,t)+ɛl3(ρ,φ,t)+αT0|S(φT0)|α-1C(φT0)ɛl4(ρ,φ,t),dφdt=h0′(ρ)+h1(ρ,φ)+h2(ρ,φ,t)+ɛh3(ρ,φ,t),
where
(25)l1(ρ,φ)=-dT0ρf(ρbS(φT0))C2(φT0)=:-dxφf(x)y,l2(ρ,φ,t)=-dT0ɛρ2-2bf(ρbS(φT0))×|S(φT0)|α-1S(φT0)C(φT0)E(t)+AdT0ρ1-b|S(φT0)|α-1S(φT0)P(t)-dT0ρbf(ρbS(φT0))C(φT0)P(t)=:-dɛxφ|x|α-1xf(x)E(t)-dyφP(t)-dxφf(x)P(t),l3(ρ,φ,t)=AdT0ρ3-4b|S(φT0)|2αE(t)=:-dyφ|x|α-1xE(t),l4(ρ,φ,t)=-dρ3-4bC(φT0)E(t)-dɛρ4-6b|S(φT0)|α-1S(φT0)E2(t)-dρ2-3bP(t)E(t),h1(ρ,φ)=dbf(ρbS(φT0))C(φT0)S(φT0)=:dxρf(x)y,h2(ρ,φ,t)=dbɛρ1-2bf(ρbS(φT0))|S(φT0)|α+1E(t)+αdbɛ2ρ3-6b|S(φT0)|2αE2(t)+d(1-b)ρ-bC(φT0)P(t)+dbρb-1f(ρbS(φT0))S(φT0)P(t)+αdbɛρ1-3b|S(φT0)|α-1S(φT0)E(t)P(t)=:dɛxρ|x|α-1xf(x)E(t)+αdɛ2xρ|x|2α-2xE2(t)+dyρP(t)+dxρf(x)P(t)+αdɛxρ|x|α-1E(t)P(t),h3(ρ,φ,t)=d(1-b+αb)ρ2-4b|S(φT0)|α-1S(φT0)C(φT0)E(t)=:dyρ|x|α-1xE(t)+αdxρ|x|α-1yE(t).Let(26)L2(ρ,φ,t)=l2(ρ,φ,t)+ɛl3(ρ,φ,t)+αT0|S(φT0)|α-1C(φT0)ɛl4(ρ,φ,t),H2(ρ,φ,t)=h2(ρ,φ,t)+ɛh3(ρ,φ,t).
Clearly, x is odd in φ and y is even in φ by the definitions of S(t) and C(t). Thus, by the evenness of P(t) and the oddness of f(x) and E(t) we have
(27)l1(ρ,-φ)=-l1(ρ,φ),L2(ρ,-φ,-t)=-L2(ρ,φ,t),h1(ρ,-φ)=h1(ρ,φ),H2(ρ,-φ,-t)=H2(ρ,φ,t).
This implies that system (24) is reversible with respect to the involutions (ρ,φ)↦(ρ,-φ).Lemma 6.
For0≤k+m≤4, the following inequalities hold:(1)
|(∂k/∂ρk)l1(ρ,φ)|≤Cρ-k+2-γ-(5/2)b,(2)
|(∂k+m/∂ρk∂tm)l2(ρ,φ,t)|≤Cρ-k+a,(3)
|(∂k+m/∂ρk∂tm)l3(ρ,φ,t)|≤Cρ-k+3-4b,(4)
|(∂k+m/∂ρk∂tm)l4(ρ,φ,t)|≤Cρ-k+3-4b,(5)
|(∂k/∂ρk)h1(ρ,φ)|≤Cρ-k+1-γ-(5/2)b,(6)
|(∂k+m/∂ρk∂tm)h2(ρ,φ,t)|≤Cρ-k+τ,(7)
|(∂k+m/∂ρk∂tm)h3(ρ,φ,t)|≤Cρ-k+2-4b,
whereγ=βb, a=max(3-(9/2)b-γ,1-b), and τ=max(3-6b,-b).Proof.
(1) It is easy to know that(∂k/∂ρk)l1(ρ,φ) is a sum of terms of the form
(28)d∂i1xφ∂ρi1·∂i2f(x)∂ρi2·∂i3y∂ρi3,i1+i2+i3=k,
where 0≤i1,i2,i3≤k. Meanwhile, ∂i2f(x)/∂ρi2 is a sum terms of the form
(29)f(s)(x)·∂l1x∂ρl1∂l2x∂ρl2⋯∂lsx∂ρls,0≤s≤i2,l1+⋯+ls=i2.
Hence, we obtain
(30)|∂k∂ρkl1(ρ,φ)|≤C|ρ-i1x·ρ-i2f(x)·ρ-i3y|≤Cρ-k·|x·f(x)·y|≤Cρ-k|x|α/2-β|y|≤Cρ-k+2-γ-(5/2)b
by the assumptions on f(x) and the definitions of x(ρ,φ) and y(ρ,φ).
(2) From the expression ofl2(ρ,φ,t), we have
(31)|∂k+m(-dɛxφ|x|α-1xf(x)E(t))∂ρk∂tm|≤C|∂k(-dɛxφ|x|α-1xf(x))∂ρk||E(m)(t)|≤Cɛ|ρ-i1x·ρ-i2f(x)·ραb-i3|≤Cɛρ-k+αb|x·f(x)|≤Cɛρ-k+3-γ-9b/2,|∂k+m(-dyφP(t))∂ρk∂tm|≤C|∂k(dyφ)∂ρk||dmP(t)dtm|≤Cρ-k+1-b,|∂k+m(-dxφf(x)P(t))∂ρk∂tm|≤C|∂k(-dxφf(x))∂ρk||dmP(t)dtm|≤Cρ-k+1-γ-3b/2.
We can find that(32)|∂k+m∂ρk∂tml2(ρ,φ,t)|≤Cρ-k+a,
where a=max(3-9b/2-γ,1-b).
(3) From the expression ofl3(ρ,φ,t), we have
(33)|∂k+m(dyφ|x|α-1xE(t))∂ρk∂tm|≤C|∂k(dyφ|x|α-1x)∂ρk||dm(E(t))dtm|≤Cρ-k+3-4b.
(4) From the expression ofl4(ρ,φ,t), we can obtain that
(34)|∂k+ml4(ρ,φ,t)∂ρk∂tm|≤C|∂k(-dρ3-4bC(φT0))∂ρk||dm(E(t))dtm|+C|∂k(-dρ2-3b)∂ρk||dm(P(t)E(t))dtm|+C|∂k(-dɛρ4-6b|S(φT0)|α-1S(φT0))∂ρk||dm(E2(t))dtm|≤Cρ-k+3-4b.
(5) From the definition ofh1(ρ,φ), we have
(35)|∂k∂ρkh1(ρ,φ)|≤C|ρ-i1-1x·ρ-i2f(x)·ρ-i3y|≤Cρ-k·|x·f(x)·y|≤Cρ-k-1|x|α/2-β|y|≤Cρ-k+1-γ-(5/2)b.
(6) From the definition ofh2(ρ,φ,t), we can obtain
(36)|∂k+m(dɛxρ|x|α-1xf(x)E(t))∂ρk∂tm|≤C|∂k(dxρ|x|α-1xf(x))∂ρk||E(m)(t)|≤Cɛ|ρ-i1-1x·ρ-i2f(x)·ραb-i3|≤Cɛρ-k+αb-1|x·f(x)|≤Cɛρ-k+2-γ-9b/2,|∂k+m(αdɛ2xρ|x|2α-2xE2(t))∂ρk∂tm|≤C|∂k(αdɛ2xρ|x|2α-2x)∂ρk||dm(E2(t))dtm|≤Cɛρ-k+3-6b,|∂k+m(-dyρP(t))∂ρk∂tm|≤C|∂k(dyρ)∂ρk||dmP(t)dtm|≤Cρ-k-b,|∂k+m(dxρf(x)P(t))∂ρk∂tm|≤C|∂k(-dxρf(x))∂ρk||dmP(t)dtm|≤Cρ-k-γ-3b/2,|∂k+m(αdɛxρ|x|α-1E(t)P(t))∂ρk∂tm|≤C|∂k(αdɛ2xρ|x|2α-2x)∂ρk||dm(E(t)P(t))dtm|≤Cɛρ-k+1-3b.
Hence, we can know that
(37)|∂k+m∂ρk∂tmh2(ρ,φ,t)|≤Cρ-k+τ,
where τ=max(3-6b,-b).
(7) From the expression ofh3(ρ,φ,t), we have(38)|∂k+m∂ρk∂tmh3(ρ,φ,t)|≤|∂k+m(d(1-b+αb)ρ2-4b|S(φT0)|α-1S(φT0)C(φT0)E(t))∂ρk∂tm|≤C|∂k(d(1-b+αb)ρ2-4b|S(φT0)|α-1S(φT0)C(φT0))∂ρk|×|dm(E(t))dtm|≤Cρ-k+2-4b.Forλ0>0, we define the domain
(39)𝒜λ0={(λ,φ,t):λ≥λ0,(φ,t)∈𝕋2}.Lemma 7.
There exists aG-invariant diffeomorphism Ψ1:
(40)ρ=I+U1(I,θ),φ=θ
such that 𝒜I+⊂Ψ1(𝒜I0)⊂𝒜I- for some I-<I0<I+. Under this transformation, (24) is transformed into the system
(41)dIdt=l~1(I,θ)+l~2(I,θ,t)+ɛl~3(I,θ,t)+αT0|S(θT0)|α-1C(θT0)ɛl~4(I,θ,t),dθdt=h0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t),
where
(42)l~1(I,θ)=∂V1(ρ,φ)∂ρ·l1(ρ,φ)+∂V1(ρ,φ)∂φ·h1(ρ,φ),l~2(I,θ,t)=l2(ρ,φ,t)+∂V1(ρ,φ)∂ρ·(l2(ρ,φ,t)+ɛl3(ρ,φ,t))+∂V1(ρ,φ)∂φ·(h2(ρ,φ,t)+ɛh3(ρ,φ,t))+ɛ(l3(ρ,φ,t)-l3(I,θ,t)),l~3(I,θ,t)=l3(I,θ,t),l~4(I,θ,t)=l4(ρ,φ,t)+∂V1(ρ,φ)∂ρ·l4(ρ,φ,t),h~1(I,θ)=h0′(ρ)-h0′(I)+h1(ρ,φ),h~2(I,θ,t)=h2(ρ,φ,t)+ɛ(h3(ρ,φ,t)-h3(I,θ,t)),h~3(I,θ,t)=h3(I,θ,t),
with
(43)V1(ρ,φ)=-∫0φ‍l1(ρ,s)h0′(ρ)ds.Proof.
Define a transformationΦ1 by
(44)Φ1:I=ρ+V1(ρ,φ),θ=φ.
By
(45)l1(ρ,-φ)=-l1(ρ,φ),|∂k∂ρkl1(ρ,φ)|≤Cρ-k+2-γ-(5/2)b,0≤k≤4,
we get
(46)V1(ρ,-φ)=V1(ρ,φ),(47)|∂k∂ρkV1(ρ,φ)|≤Cρ-k+1-γ-b/2.
Let Ψ1=Φ1-1:ρ=I+U1(I,θ), φ=θ. The system (24) is transformed into (41).Lemma 8.
ForI large enough, the following conclusions hold:(i)
|∂kU1(I,θ)/∂Ik|≤CI-k+1-γ-b/2,(ii)
U1(I,-θ)=U1(I,θ).Proof.
In view of(48)I=ρ+V1(ρ,φ),ρ=I+U1(I,θ),
we obtain
(49)U1(I,θ)=-V1(I+U1(I,θ),θ).
By |(∂k/∂ρk)V1(ρ,φ)|≤Cρ-k+1-γ-b/2, we have |(∂/∂ρ)V1(ρ,φ)|≤Cρ-γ-b/2≤1/2 for ρ large enough. Hence, U1 is uniquely determined by the contraction mapping principle. Moreover, U1(·,θ)∈C∞(𝒜I0), for some I0>0, as a consequence of the implicit function theorem and
(50)I-(1-γ-b/2)|U1(I,θ)|≤C.
Above all, ifk=1, from (47) and (49), we get
(51)|∂U1∂I|=|∂V1/∂ρ1+∂V1/∂ρ|≤∑n=0∞‍(Cρ-1+1-γ-b/2)n+1≤C·ρ-1+1-γ-b/2=C·I-1+1-γ-b/2(1+U1I)-1+1-γ-b/2≤C·I-1+1-γ-b/2.
We note that(52)∂kU1(I,θ)∂Ik=∂kV1(I+U1(I,θ),θ)∂Ik,
and the right side hand is sum of the term
(53)∂sV1∂ρs·∂k1(I+U1)∂Ik1⋯∂ks(I+U1)∂Iks,
where 1≤s≤k,   k1+⋯+ks=k,  ki≥1(for1≤i≤s). The highest order term in U1 is the one with s=1, namely, (∂V1/∂ρ)·(∂k(I+U1)/∂Ik). We move the part (∂V1/∂ρ)·(∂kU1/∂Ik) to the left hand side of (52). Since |(∂/∂ρ)V1(ρ,φ)|≤1/2 for ρ large enough, this also provides immediately a bound on ∂kU1(I,θ)/∂Ik. The rest part |(∂V1/∂ρ)·(∂kI/∂Ik)|≤CI-k+1-γ-b/2.
Now, we proceed inductively by assuming that forj≤k-1 the estimates
(54)|∂jU1(I,θ)∂Ij|≤CI-j+1-γ-b/2
hold and we wish to conclude that the same estimate holds for j=k.
Indeed, ifs≥2, we have
(55)|∂sV1∂ρs·∂k1(I+U1)∂Ik1⋯∂ks(I+U1)∂Iks|≤C·(I+U1)-s+1-γ-b/2·I-k1+1⋯I-ks+1≤C·I-k+1-γ-b/2
by
(56)|∂j(I+U1(I,θ))∂Ij|≤CI-j+1,1≤j≤k-1.
This proves (i) of  Lemma 8.
Now we check (ii). In fact, since(57)U1(I,θ)=-V1(I+U1(I,θ),θ),U1(I,-θ)=-V1(I+U1(I,-θ),θ),
we have
(58)|U1(I,θ)-U1(I,-θ)|≤supI≥I0|∂V1∂ρ||U1(I,θ)-U1(I,-θ)|.
From (47), we have |(∂/∂ρ)V1(ρ,φ)|≤1/2 for I≥I0 sufficiently large and therefore we obtain U1(I,θ)=U1(I,-θ).By the estimates in Lemma6, we can prove the following inequalities.Lemma 9.
For0≤k+m≤4, the following inequalities hold:(1)
|(∂k/∂Ik)l~1(I,θ)|≤CI-k+2-2γ-3b,(2)
|(∂k+m/∂Ik∂tm)l~2(I,θ,t)|≤CI-k+a,(3)
|(∂k+m/∂Ik∂tm)l~3(I,θ,t)|≤CI-k+3-4b,(4)
|(∂k+m/∂Ik∂tm)l~4(I,θ,t)|≤CI-k+3-4b,(5)
|(∂k/∂Ik)h~1(I,θ)|≤CI-k+1-γ-(5/2)b,(6)
|(∂k+m/∂Ik∂tm)h~2(I,θ,t)|≤CI-k+τ,(7)
|(∂k+m/∂Ik∂tm)h~3(I,θ,t)|≤CI-k+2-4b.Proof.
(1) From the estimates (1) and (5) of Lemmas6 and 8, it follows that(59)|∂k∂Ikl~1(I,θ)|≤|∂k∂Ik(∂V1(ρ,φ)∂ρ·l1(ρ,φ))|+|∂k∂Ik(∂V1(ρ,φ)∂φ·h1(ρ,φ))|≤C∑i1+i2=k‍|∂i1+1V1(ρ,φ)∂Ii1∂ρ||∂i2l1∂Ii2|+C∑i1+i2+i3=k‍∂i1ρ2b-1∂Ii1|∂i2l1∂Ii2||∂i3h1∂Ii3|≤C∑i1+i2=k‍(∑τ1+⋯+τs=i1‍|∂s+1V1(ρ,φ)∂ρs+1∂τ1(I+U1)∂Iτ1⋯∂τs(I+U1)∂Iτs|)×|∂i2l1∂Ii2|+C∑i1+i2+i3=k‍(∑τ1+⋯+τs=i1‍|∂sρ2b-1∂ρs∂τ1(I+U1)∂Iτ1⋯∂τs(I+U1)∂Iτs|)×|∂i2l1∂Ii2∂i3h1∂Ii3|≤Cρ-k+2-2γ-3b≤CI-k+2-2γ-3b. (2) Since
(60)l~2(I,θ,t)=l2(ρ,φ,t)+∂V1(ρ,φ)∂ρ·(l2(ρ,φ,t)+ɛl3(ρ,φ,t))+∂V1(ρ,φ)∂φ·(h2(ρ,φ,t)+ɛh3(ρ,φ,t))+ɛ(l3(ρ,φ,t)-l3(I,θ,t)),
we can prove that
(61)|∂k+m∂Ik∂tm(∂V1(ρ,φ)∂ρ·(l2(ρ,φ,t)+ɛl3(ρ,φ,t)))|≤CI-k+a,|∂k+m∂Ik∂tm(∂V1(ρ,φ)∂φ·(h2(ρ,φ,t)+ɛh3(ρ,φ,t)))|≤CI-k+a,|∂k+m∂Ik∂tml2(ρ,φ,t)|≤I-k+a.
Their proofs are similar to the proofs in (1).
Next, we check the last part ofl~2(I,θ,t). We get
(62)|∂k+m∂Ik∂tm(l3(ρ,φ,t)-l3(I,θ,t))|=|∂k+m∂Ik∂tm(∫01‍∂l3∂ρ(I+sU1(I,θ),θ,t)·U1(I,θ)ds)|≤∫01‍∑i1+i2=k‍|∂i1+m∂Ii1∂tm(∂l3(I+sU1,θ,t)∂ρ)||∂i2U1∂Ii2|ds≤C∫01‍∑i1+i2=k‍(I+U1)-i1+2-4bI-i2+1-γ-b/2ds≤CI-k+3-γ-(9/2)b≤CI-k+a,
by the estimate in Lemma 6 and the definition of a.
(3) It is clearly by (3) in Lemma6.
(4) It is clearly by (4) in Lemmas6 and 8.
(5) We have that(63)h~1(I,θ)=h0′(ρ)-h0′(I)+h1(ρ,φ),|∂k+m∂Ik∂tm(h0′(ρ)-h0′(I))|≤|∂k+m∂Ik∂tm(∫01‍d2h0(I+sU1)dρ2U1(I,θ)ds)|≤CI-k-2b+1-γ-b/2≤CI-k+1-γ-5b/2.
From the last inequalities and (5) in Lemma 6, we obtain
(64)|∂k∂Ikh~1(I,θ)|≤CI-k+1-γ-(5/2)b.
(6) Since
(65)h~2(I,θ,t)=h2(ρ,φ,t)+ɛ(h3(ρ,φ,t)-h3(I,θ,t)),|∂k+m∂Ik∂tmh2(ρ,φ,t)|≤CI-k+τ,
we just have to prove that
(66)|∂k+m∂Ik∂tm(h3(ρ,φ,t)-h3(I,θ,t))|≤CI-k+τ.
In fact,
(67)|∂k+m∂Ik∂tm(h3(ρ,φ,t)-h3(I,θ,t))|=|∂k+m∂Ik∂tm(∫01‍∂h3∂ρ(I+sU1(I,θ),θ,t)·U1(I,θ)ds)|≤∫01‍∑i1+i2=k‍|∂i1+m∂Ii1∂tm(∂h3(I+sU1,θ,t)∂ρ)||∂i2U1∂Ii2|ds≤C∫01‍∑i1+i2=k‍(I+U1)-i1+1-4bI-i2+1-γ-b/2ds≤CI-k+2-γ-(9/2)b≤CI-k+τ,
so we have proved (6).
(7) We have(68)|∂k+m∂Ik∂tmh~3(I,θ,t)|≤CI-k+2-4b,
by (7) in Lemma 6.


## 3. The Proof of Boundedness

In this section, all the solutions of (8) which are bounded will be proved via the KAM theory for reversible systems developed by Sevryuk [21] or Moser [22, 23] if B>0.We define the functionsη0,  η1,  η2,  η3, ξ1, ξ2, and ξ3 as
(69)η0(I)=1h0′(I),η1(I,θ)=-h~1(I,θ)h0′(I)(h0′(I)+h~1(I,θ)),η2(I,θ,t)=-(h~2(I,θ,t))×((h0′(I)+h~1(I,θ))×(h0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t)))-1,η3(I,θ,t)=-(h~3(I,θ,t))×((h0′(I)+h~1(I,θ))×(h0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t)))-1,ξ1(I,θ,t)=(l~1(I,θ)+l~2(I,θ,t))·(η0(I)+η1(I,θ)+η2(I,θ,t)+ɛη3(I,θ,t)),ξ2(I,θ,t)=l~3(I,θ,t)·(η0(I)+η1(I,θ)+η2(I,θ,t)+ɛη3(I,θ,t)),ξ3(I,θ,t)=l~4(I,θ,t)·(η0(I)+η1(I,θ)+η2(I,θ,t)+ɛη3(I,θ,t)).
Then system (41) is equivalent to the following system:
(70)dtdθ=η0(I)+η1(I,θ)+η2(I,θ,t)+ɛη3(I,θ,t),dIdθ=ξ1(I,θ,t)+ɛξ2(I,θ,t)+αT0|S(θT0)|α-1C(θT0)ɛξ3(I,θ,t).In addition, one can verify that system (70) is reversible with respect to involution G:(t,I)↦(-t,I).Then some estimates on the functionsηi(i=0,1,2,3) and ξi(i=1,2,3) are given.Lemma 10.
The following inequalities hold:(1)
cI2b-1≤|η0(I)|≤CI2b-1,(2)
|(∂k/∂Ik)η1(I,θ)|≤CI-k-1-γ+3b/2,(3)
|(∂k+m/∂Ik∂tm)η2(I,θ,t)|≤CI-k+τ+4b-2,(4)
|(∂k+m/∂Ik∂tm)η3(I,θ,t)|≤CI-k,(5)
|(∂k+m/∂Ik∂tm)ξ1(I,θ,t)|≤CI-k+a+2b-1,(6)
|(∂k+m/∂Ik∂tm)ξ2(I,θ,t)|≤CI-k+2-2b,(7)
|(∂k+m/∂Ik∂tm)ξ3(I,θ,t)|≤CI-k+2-2b, for 0≤k+m≤4.Proof.
(1) It is clear.
(2) Note that1-2b>1-γ-2b/5, and
(71)|h~1(I,θ)|≤CI1-γ-(5/2)b,
it follows that
(72)|h0′(I)+h~1(I,θ)|≥||h0′(I)|-|h~1(I,θ)||≥|h0′(I)|-|h~1(I,θ)|≥1T0I1-2b-CI1-γ-(5/2)b≥cI1-2b
as I≫1.
Moreover, we also have(73)|∂l∂Il(h0′(I)+h~1(I,θ))|≤|∂l∂Ilh0′(I)|+|∂l∂Ilh~1(I,θ)|≤CI-l+1-2b+CI-l+1-γ-(2/5)b≤CI-l+1-2b.
So
(74)|∂i∂Ii(1h0′(I)+h~1(I,θ))|≤C∑l1+⋯+ls=i‍|(-1)ss!(h0′(I)+h~1(I,θ))s+1|×|∂l1∂Il1(h0′(I)+h~1(I,θ))|⋯|∂ls∂Ils(h0′(I)+h~1(I,θ))|≤C∑l1+⋯+ls=i‍I(2b-1)(s+1)·I-i+(1-2b)s≤CI-i+2b-1.
From (72) and (74), it is easy to see that
(75)|∂k∂Ikη1(I,θ)|=|∂k∂Ik(h~1(I,θ)h0′(I)(h0′(I)+h~1(I,θ)))|≤C∑i1+i2+i3=k‍|∂i1∂Ii1h~1(I,θ)|·|∂i2∂Ii2(1h0′(I))|·|∂i3∂Ii3(1h0′(I)+h~1(I,θ))|≤C∑i1+i2+i3=k‍I-i1+1-γ-(2/5)bI-i2-1+2bI-i3-1+2b≤CI-k-1-γ+(3/2)b.
(3) We have
(76)|∂k+m∂Ik∂tmh~2(I,θ,t)|≤CI-k+τ,|∂k+m∂Ik∂tmh~3(I,θ,t)|≤CI-k+2-4b.
By (72), 1-2b>τ(τ=max(3-6b,-b)) and 1-2b>2-4b, we have
(77)|h0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t)|≥||h0′(I)+h~1(I,θ)|-|h~2(I,θ,t)+ɛh~3(I,θ,t)||≥|h0′(I)+h~1(I,θ)|-|h~2(I,θ,t)|-ɛ|h~3(I,θ,t)|≥cI1-2b-CIτ-CɛI2-4b≥cI1-2b,
for I≫1.
Leth0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t)=H(I,θ,t). We find that
(78)|∂l∂tl(1(h0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t))s+1)|≤C∑i1+⋯+ir=l‍|(-1)rr!(H(I,θ,t))s+1+r||∂i1∂ti1(H(I,θ,t))|⋯|∂ir∂tir(H(I,θ,t))|≤C∑i1+⋯+ir=l‍I(2b-1)(s+1+r)·ɛrI(2-4b)r≤CI(2b-1)(s+1),
so
(79)|∂k+l∂Ik∂tl(1h0′(I)+h~1(I,θ)+h~2(I,θ,t)+ɛh~3(I,θ,t))|≤C|∂l∂tl(∑i1+⋯+is=k‍(-1)ss!(H(I,θ,t))s+1·∂i1∂Ii1(H(I,θ,t))⋯∂is∂Iis(H(I,θ,t)))|≤C∑i1+⋯+is=k‍∑j0+j1+⋯+js=l‍|∂j0∂tj0(-1)ss!(H(I,θ,t))s+1|×|∂i1+j1∂Ii1∂tj1(H(I,θ,t))|⋯|∂is+js∂Iis∂tjs(H(I,θ,t))|≤C∑i1+⋯+is=k‍I(2b-1)(s+1)·I-(i1+⋯+is)-(1-2b)s≤CI-k+(2b-1).
Whenm=0, the proof of (3) is similar to the proof of (2).
Whenm>0, then
(80)|∂k+m∂Ik∂tmη2(I,θ,t)|=|∂k+m∂Ik∂tm(h~2(I,θ,t)(h0′(I)+h~1(I,θ))(h0′(I)+h~1+h~2+ɛh~3))|≤C|∂m∂tm(∑i1+i2+i3=k‍∂i1∂Ii1h~2·∂i2∂Ii2(1h0′(I)+h~1)·∂i3∂Ii3(1h0′(I)+h~1+h~2+ɛh~3))|≤C∑i1+i2+i3=k‍∑l1+l2=m‍|∂i1+l1h~2(I,θ,t)∂Ii1∂tl1|·|∂i2∂Ii2(1h0′(I)+h~1(I,θ))|·|∂i3+l2∂Ii3∂tl2(1h0′(I)+h~1+h~2+ɛh~3)|≤C∑i1+i2+i3=k‍I-i1+τI-i2+2b-1I-i3+2b-1≤CI-k+τ+4b-2.
(4) The proof of (4) is similar to the proof of (3).
(5) Letη0(I)+η1(I,θ)+η2(I,θ,t)+ɛη3(I,θ,t)=η(I,θ,t). By using the estimates on the functions l~i(i=1,2) and ηj(j=0,1,2,3), it follows that(81)|∂k+m∂Ik∂tmξ1(I,θ,t)|≤C|∂m∂tm(∑k1+k2=k‍∂k1(l~1(I,θ)+l~2(I,θ,t))∂Ik1·∂k2(η(I,θ,t))∂Ik2)|≤C∑k1+k2=k‍|∂k1l~1(I,θ)∂Ik1·∂k2+m(η2(I,θ,t)+ɛη3(I,θ,t))∂Ik2∂tm|+C∑lk+k2=k‍∑m1+m2=m‍|∂k1+m1l~2(I,θ,t)∂Ik1∂tm1·∂k2+m2(η(I,θ,t))∂Ik2∂tm2|≤CI-k1+2-γ-(5/2)b·(I-k2+τ+4b-2+ɛI-k2)+CI-k1+a·I-k2+2b-1≤CI-k+a+2b-1,when m≠0.
Whenm=0, then
(82)|∂k∂Ikξ1(I,θ,t)|≤C∑k1+k2=k‍|∂k1(l~1(I,θ)+l~2(I,θ,t))∂Ik1·∂k2(η(I,θ,t))∂Ik2|≤C(I-k1+2-γ-(5/2)b+I-k1+a)·I-k2+2b-1≤CI-k+a+2b-1,
by a>2-γ-5b/2.
(6) By using the estimates on the functionsl~3 and ηi(i=0,1,2,3), it follows that
(83)|∂k+m∂Ik∂tmξ2(I,θ,t)|≤C|∂m∂tm(∑k1+k2=k‍∂k1(l~3(I,θ,t))∂Ik1·∂k2(η(I,θ,t))∂Ik2)|≤C∑lk+k2=k‍∑m1+m2=m‍|∂k1+m1l~3(I,θ,t)∂Ik1∂tm1·∂k2+m2(η(I,θ,t))∂Ik2∂tm2|≤CI-k1+3-4b·I-k2+2b-1≤CI-k+2-2b.
(7) By using the estimates on the functionsl~4 and ηi(i=0,1,2,3), it follows that
(84)|∂k+m∂Ik∂tmξ3(I,θ,t)|≤C|∂m∂tm(∑k1+k2=k‍∂k1(l~4(I,θ,t))∂Ik1·∂k2(η(I,θ,t))∂Ik2)|≤C∑lk+k2=k‍∑m1+m2=m‍|∂k1+m1l~4(I,θ,t)∂Ik1∂tm1·∂k2+m2(η(I,θ,t))∂Ik2∂tm2|≤CI-k1+3-4b·I-k2+2b-1≤CI-k+2-2b.Lett=t,  θ=θ,  r=η0(I) and
(85)F0(r,θ)=η1(I(r),θ),F1(r,θ,t)=η2(I(r),θ,t)+ɛη3(I(r),θ,t),F2(r,θ,t)=η0′(I(r))·(ξ1(I(r),θ,t)+ɛξ2(I(r),θ,t)),F3(r,θ,t)=ɛη0′(I(r))·ξ3(I(r),θ,t),
where I(r) is the inverse function of r=η0(I).Then system (70) is transformed into the following form:
(86)dtdθ=r+F0(r,θ)+F1(r,θ,t),drdθ=F2(r,θ,t)+αT0|S(θT0)|α-1C(θT0)·F3(r,θ,t).
Moreover, one can verify that system (86) is reversible with respect to involution G:(t,r)↦(-t,r).It is easy to see thatI≫1 if and only if r≫1, and the solutions of system (86) do exist on 0≤θ≤1 when r(0)=r≫1.By using the estimates onηi and ξi(i=1,2,3) in Lemma 10, the following inequalities can be proved.Lemma 11.
For0≤k+m≤4 and r≫1, the following inequalities hold:(1)
|(∂k/∂rk)F0(r,θ)|≤Cr-k-(1+γ-3b/2)/(2b-1),(2)
|(∂k+m/∂rk∂tm)F1(r,θ,t)|≤C(r-k+(τ+4b-2)/(2b-1)+ɛr-k),(3)
|(∂k+m/∂rk∂tm)F2(r,θ,t)|≤C(r-k+(a+4b-3)/(2b-1)+ɛr-k),(4)
|(∂k+m/∂rk∂tm)F3(r,θ,t)|≤Cɛr-k.Proof.
Above all, we know thatr=η0(I)=T0I2b-1, so we can get I=((1/T0)r)1/(2b-1). Then we have
(87)|djIdrj|≤Cr-j+1/(2b-1),|djη0′(I(r))drj|≤C|dj(r1-1/(2b-1))drj|≤Cr-j+(2b-2)/(2b-1).
(1)  We have that(88)|∂kF0(r,θ)∂rk|≤∑k1+⋯+ks=k‍|∂sη1(I,θ)∂Is|·|dk1Idrk1|⋯|dksIdrks|≤CI-s-1-γ+(3/2)br-k+(1/(2b-1))s≤Cr-s(1/(2b-1))-(1+γ-(3/2)b)/(2b-1)r-k+(1/(2b-1))s≤Cr-k-(1+γ-(3/2)b)/(2b-1).
(2)  We have that(89)|∂k+mF1(r,θ,t)∂rk∂tm|≤C∑i1+⋯+is=k‍|∂s+mη2(I,θ,t)∂Is∂tmdi1Idri1⋯disIdris|+Cɛ∑j1+⋯+jν=k‍|∂ν+mη3(I,θ,t)∂Iν∂tmdj1Idrj1⋯djνIdrjν|≤C(r-k+((τ+4b-2)/(2b-1))+ɛr-k).
(3) We have that(90)|∂k+mF2(r,θ,t)∂rk∂tm|≤|∑k1+k2=k‍dk1η0′(I(r))drk1·(∂k2+mξ1(I(r),θ,t)∂rk2∂tm+ɛ∂k2+mξ2(I(r),θ,t)∂rk2∂tm)|≤C∑k1+k2=k‍r-k1+(2b-2)/(2b-1)×(∑i1+⋯+is=k2‍∂s+mξ1(I,θ,t)∂Is∂tmdi1Idri1⋯disIdris+ɛ∑j1+⋯+jν=k2‍∂ν+mξ2(I,θ,t)∂Iν∂tmdj1Idrj1⋯djνIdrjν)≤Cr-k1+(2b-2)/(2b-1)r(1/(2b-1))(-s+a+2b-1)r-k1+(1/(2b-1))s+Cɛr-k1+(2b-2)/(2b-1)r(1/(2b-1))(-ν+2-2b)r-k1+(1/(2b-1))ν≤Cr-k+(a+4b-3)/(2b-1)+Cɛr-k.
(4) We have that(91)|∂k+mF3(r,θ,t)∂rk∂tm|≤ɛ|∑k1+k2=k‍dk1η0′(I(r))drk1·(∂k2+mξ3(I(r),θ,t)∂rk2∂tm)|≤Cɛ∑k1+k2=k‍r-k1+(2b-2)/(2b-1)×(∑i1+⋯+is=k2‍∂s+mξ3(I,θ,t)∂Is∂tmdi1Idri1⋯disIdris)≤Cɛr-k.Lemma 12.
The time 1 mapΦ1 of the flow Φθ of the system (86) is of the form
(92)Φ1:r1=r+Q2(r,t),t1=t+ω^(r)+Q1(r,t),
where ω^(r)=r+∫01‍F0(r,θ)dθ. And there exists a μ0>0 such that, for 0≤k+m≤4, sufficiently large r and sufficiently small ɛ,
(93)|∂k+m∂rk∂tmQi(r,t)|≤Cr-μ0+ɛ,i=1,2
hold. Moreover, the map Φ1 is reversible with respect to the involution G:(t,r)↦(-t,r).Proof.
Since(94)∫01‍αT0|S(θT0)|α-1|C(θT0)|dθ=limϵ→0+∫ϵ1/4‍αT0|S(θT0)|α-1C(θT0)dθ-limϵ→0+∫1/41/2-ϵ‍αT0|S(θT0)|α-1C(θT0)dθ-limϵ→0+∫1/2+ϵ3/4‍αT0|S(θT0)|α-1C(θT0)dθ+limϵ→0+∫3/41-ϵ‍αT0|S(θT0)|α-1C(θT0)dθ=|S(T04)|α-1S(T04)-limϵ→0+|S(ϵT0)|α-1S(ϵT0)-[limϵ→0+|S((12-ϵ)T0)|α-1S((12-ϵ)T0)-|S(T04)|α-1S(T04)limϵ→0+]-[limϵ→0+|S(3T04)|α-1S(3T04)-limϵ→0+|S((12+ϵ)T0)|α-1S((12+ϵ)T0)]+[{3T04}limϵ→0+|S((1-ϵ)T0)|α-1S((1-ϵ)T0)limϵ→0+-|S(3T04)|α-1S(3T04)]=4|S(T04)|α-1S(T04)=4,
then we get ∫01‍αT0|S(θT0)|α-1|C(θT0)|dθ is bounded.
LetαT0|S(υT0)|α-1C(υT0)=S1(υ). Set (r(θ),t(θ))=Φθ(r,t) with Φ0=id for the flow:
(95)t(θ)=t+rθ+D1(r,t,θ),r(θ)=r+D2(r,t,θ).
Since
(96)Φθ=Φ0+∫0θ‍X·Φυdυ,
where X denotes the vector field of the system (86), we have
(97)t(θ)=t+∫0θ‍[r(υ)+F0(r(υ),υ)+F1(r(υ),υ,t(υ))]dυ=t+rθ+∫0θ‍[D2(r,t,υ)+F0(r+D2,υ)+F1(r+D2,υ,t+rυ+D1)]dυ=t+rθ+D1(r,t,θ),r(θ)=r+∫0θ‍[F2(r(υ),υ,t(υ))+S1(υ)F3(r(υ),υ,t(υ))]dυ=r+∫0θ‍[F2(r+D2,υ,t+rυ+D1)+S1(υ)F3(r+D2,υ,t+rυ+D1)]dυ=r+D2(r,t,θ),
which is equivalent to the following equations for D1 and D2:
(98)D1(r,t,θ)=∫0θ‍[D2(r,t,υ)+F0(r+D2,υ)+F1(r+D2,υ,t+rυ+D1)]dυ,D2(r,t,θ)=∫0θ‍[F2(r+D2,υ,t+rυ+D1)+S1(υ)F3(r+D2,υ,t+rυ+D1)]dυ.
LetD(r,t,θ)=(D1(r,t,θ),D2(r,t,θ)),  |D1(r,t,θ)|=sup(R+×𝕋×(0,1])(r,t,θ)∈|D1(r,t,θ)|. Define ∥D∥=:|D1|/3+2|D1|/3, and T(D)=:(T1(D),T2(D)), where
(99)T1(D)=∫0θ‍[D2(r,t,υ)+F0(r+D2,υ)+F1(r+D2,υ,t+rυ+D1)]dυ,T2(D)=∫0θ‍[F2(r+D2,υ,t+rυ+D1)+S1(υ)F3(r+D2,υ,t+rυ+D1)]dυ.
Next, we will prove that T is a contraction map. From the definition of T(D), we have
(100)|T1D-T1D~|=|∫0θ‍[D2-D~2+F0(r+D2,υ)-F0(r+D~2)+F1(r+D2,υ,t+rυ+D1)-F1(r+D~2,υ,t+rυ+D~1)]dυ{∫0θ‍}|≤|D2-D~2|+∫01‍|∂F0(r+s(D2-D~2),υ)∂r|·|D2-D~2|ds+∫01‍|∂F1(r+s(D2-D~2),υ,t+rυ+D1)∂r|·|D2-D~2|ds+∫01‍|∂F1(r+D~2,υ,t+rυ+s(D1-D~1))∂t|·|D1-D~1|ds≤65|D2-D~2|+14|D1-D~1|,|T2D-T2D~|=∫0θ‍[r+D~2,υ,t+rυ+D~1F2(r+D2,υ,t+rυ+D1)-F2(r+D~2,υ,t+rυ+D~1)+S1(υ)F3(r+D2,υ,t+rυ+D1)-S1(υ)F3(r+D~2,υ,t+rυ+D~1)]dυ≤∫01‍|∂F2(r+s(D2-D~2),υ,t+rυ+D1)∂r|·|D2-D~2|ds+∫01‍|∂F2(r+D~2,υ,t+rυ+s(D1-D~1))∂t|·|D1-D~1|ds+∫01‍|S1(υ)|dυ·∫01‍|∂F3(r+s(D2-D~2),υ,t+rυ+D1)∂r|·|D2-D~2|ds+∫01‍|S1(υ)|dυ·∫01‍|∂F3(r+D~2,υ,t+rυ+s(D1-D~1))∂t|·|D1-D~1|ds≤320|D2-D~2|+18|D1-D~1|,
by Lemma 11 and the boundedness of ∫01‍|S1(υ)|dυ. Then we have
(101)∥T(D)-T(D~)∥=13|T1(D)-T1(D~)|+23|T2(D)-T2(D~)|≤13×(65|D2-D~2|+14|D1-D~1|)+23×(320|D2-D~2|+18|D1-D~1|)=16|D1-D~1|+12|D2-D~2|≤34×(13|D1-D~1|+23|D2-D~2|)≤34∥D-D~∥,
by the definition of the norm ∥·∥.
Using the contraction principle, one verifies easily that forr≥r0, (98) has a unique solution in the space {|D1|≤1,|D2|≤1}. Moreover, D1 and D2 are smooth.
Next, we will estimateQ1(r,t) and Q2(r,t) as follows:
(102)Q1(r,t)=D1(r,t,1)-∫01‍F0(r,υ)dυ=∫01‍[D2(r,t,υ)∫01‍∂F0(r+sD2,υ)∂r+∫01‍∂F0(r+sD2,υ)∂r·D2ds+F1(r+D2,υ,t+rυ+D1)∂F0(r+sD2,υ)∂r]dυ,Q2(r,t)=D2(r,t,1)=∫01‍[F2(r+D2,υ,t+rυ+D1)+S1(υ)F3(r+D2,υ,t+rυ+D1)]dυ.
In order to prove (93), we just need to prove that
(103)|∂k+m∂rk∂tmDi(r,t,θ)|≤Cr-μ0+Cɛ,i=1,2
hold for k+m≤4.
(1) Whenk+m=0,
(104)|D2(r,t,θ)|≤∫0θ‍(|F2(r+D2,υ,t+rυ+D1)|+|S1(υ)||F3(r+D2,υ,t+rυ+D1)|)dυ≤C(r-μ0+ɛ)+∫01‍|S1(υ)|dυ·(Cɛ)≤C(r-μ0+ɛ),|D1(r,t,θ)|≤|D2(r,t,θ)|+∫0θ‍(|F0(r+D2,υ)|+|F1(r+D2,υ,t+rυ+D1)|)dυ,≤|D2(r,t,υ)|+∫0θ‍(Cr-μ0+Cr-μ0+Cɛ)dυ≤|D2(r,t,θ)|+C(r-μ0+ɛ)≤C(r-μ0+ɛ),
where μ0=min((1+γ-3b/2)/(2b-1),(2-4b-τ)/(2b-1),(3-4b-a)/(2b-1)).
(2) Whenm=0 and k≠0, we check the case when k=1 firstly
(105)|∂D2(r,t,θ)∂r|≤∫0θ‍|∂F2(r+D2,υ,t+rυ+D1)∂r|·(1+|∂D2(r,t,υ)∂r|)dυ+∫0θ‍|∂F2(r+D2,υ,t+rυ+D1)∂t|·(1+|∂D1(r,t,υ)∂r|)dυ+∫01‍|S1(υ)|dυ·∫0θ‍|∂F3(r+D2,υ,t+rυ+D1)∂r|+∫01‍|S1(υ)|dυ··(1+|∂D2(r,t,υ)∂r|)dυ+∫01‍|S1(υ)|dυ·∫0θ‍|∂F3(r+D2,υ,t+rυ+D1)∂t|+∫01‍|S1(υ)|dυ··(1+|∂D1(r,t,υ)∂r|)dυ≤Cr-1(r-μ0+ɛ)·(1+|∂D2(r,t,υ)∂r|)+C(r-μ0+ɛ)·(1+|∂D1(r,t,υ)∂r|),|∂D1(r,t,θ)∂r|≤|∂D2(r,t,θ)∂r|+∫0θ‍|∂F0(r+D2,υ)∂r|·(1+|∂D2(r,t,υ)∂r|)dυ+∫0θ‍|∂F1(r+D2,υ,t+rυ+D1)∂r|·(1+|∂D2(r,t,υ)∂r|)dυ+∫0θ‍|∂F1(r+D2,υ,t+rυ+D1)∂t|·(1+|∂D1(r,t,υ)∂r|)dυ≤|∂D2(r,t,θ)∂r|+Cr-1(r-μ0+ɛ)·(1+|∂D2(r,t,υ)∂r|)+C(r-μ0+ɛ)·(1+|∂D1(r,t,υ)∂r|).
Hence,
(106)|∂D1(r,t,θ)∂r|≤C(r-μ0+ɛ),|∂D2(r,t,θ)∂r|≤C(r-μ0+ɛ).
Now, we proceed inductively by assuming that for j<k-1 the estimates
(107)|∂jD1(r,t,θ)∂rj|≤C(r-μ0+ɛ),|∂jD2(r,t,θ)∂rj|≤C(r-μ0+ɛ),
hold and we wish to conclude that the same estimate holds for j=k(108)|∂kD2(r,t,θ)∂rk|≤∫0θ‍|∂F2(r+D2,υ,t+rυ+D1)∂r|·|∂kD2(r,t,θ)∂rk|dυ+∫0θ‍|∂F2(r+D2,υ,t+rυ+D1)∂t|·|∂kD1(r,t,υ)∂rk|dυ+C(r-μ0+ɛ)·∑k1+k2=k‍∑j1+⋯+jν=k2i1+⋯+is=k1‍|∂i1(r+D2)∂ri1|⋯|∂is(r+D2)∂ris|·∑k1+k2=k‍∑j1+⋯+jν=k2i1+⋯+is=k1‍×|∂j1(r+D1)∂rj1|⋯|∂jν(r+D1)∂rjν|+∫01‍|S1(υ)|dυ·∫0θ‍|∂F3(r+D2,υ,t+rυ+D1)∂r|·|∂kD2(r,t,υ)∂rk|dυ+∫01‍|S1(υ)|dυ·∫0θ‍|∂F3(r+D2,υ,t+rυ+D1)∂t|·|∂kD1(r,t,υ)∂rk|dυ+C(r-μ0+ɛ)·∑k1+k2=k‍∑j1+⋯+jν=k2i1+⋯+is=k1‍|∂i1(r+D2)∂ri1|⋯|∂is(r+D2)∂ris|·∑k1+k2=k‍∑j1+⋯+jν=k2i1+⋯+is=k1‍×|∂j1(r+D1)∂rj1|⋯|∂jν(r+D1)∂rjν|≤Cr-1(r-μ0+ɛ)·|∂kD2(r,t,υ)∂rk|+C(r-μ0+ɛ)·|∂kD1(r,t,υ)∂rk|+C(r-μ0+ɛ),|∂kD1(r,t,θ)∂rk|≤|∂kD2(r,t,θ)∂rk|+∫0θ‍|∂F0(r+D2,υ)∂r|·|∂kD2(r,t,υ)∂rk|dυ+C(r-μ0+ɛ)·∑i1+⋯+is=k‍|∂i1(r+D2)∂ri1|⋯|∂is(r+D2)∂ris|+∫0θ‍|∂F1(r+D2,υ,t+rυ+D1)∂r|·|∂kD2(r,t,υ)∂rk|dυ+∫0θ‍|∂F1(r+D2,υ,t+rυ+D1)∂t|·|∂kD1(r,t,υ)∂rk|dυ+C(r-μ0+ɛ)·∑k1+k2=k‍∑j1+⋯+jν=k2i1+⋯+is=k1‍|∂i1(r+D2)∂ri1|⋯|∂is(r+D2)∂ris|·∑k1+k2=k‍∑j1+⋯+jν=k2i1+⋯+is=k1‍×|∂j1(r+D1)∂rj1|⋯|∂jν(r+D1)∂rjν|≤|∂kD2(r,t,θ)∂rk|+Cr-1(r-μ0+ɛ)·|∂kD2(r,t,υ)∂rk|+C(r-μ0+ɛ)·|∂kD1(r,t,υ)∂rk|+C(r-μ0+ɛ),
where s+ν≤2. Hence,
(109)|∂kD1(r,t,θ)∂rk|≤C(r-μ0+ɛ),|∂kD2(r,t,θ)∂rk|≤C(r-μ0+ɛ).
(3) We can prove that(110)|∂mD1(r,t,θ)∂tm|≤C(r-μ0+ɛ),|∂mD2(r,t,θ)∂tm|≤C(r-μ0+ɛ)
similarly to (2) when m≠0.
(4) we have that(111)|∂2D2(r,t,θ)∂r∂t|≤∫0θ‍|∂F2(r+D2,υ,t+rυ+D1)∂r|·|∂2D2(r,t,θ)∂r∂t|dυ+∫0θ‍|∂F2(r+D2,υ,t+rυ+D1)∂t|·|∂2D1(r,t,θ)∂r∂t|dυ+C(r-μ0+ɛ)·(∂D2(r,t,θ)∂r+∂D1(r,t,θ)∂t+∂D1(r,t,θ)∂r+∂D2(r,t,θ)∂t+1)+∫01‍|S1(υ)|dυ·∫0θ‍|∂F3(r+D2,υ,t+rυ+D1)∂r|·|∂2D2(r,t,θ)∂r∂t|dυ+∫01‍|S1(υ)|dυ·∫0θ‍|∂F3(r+D2,υ,t+rυ+D1)∂t|·|∂2D1(r,t,θ)∂r∂t|dυ≤C(r-μ0+ɛ)·|∂2D2(r,t,θ)∂r∂t|+C(r-μ0+ɛ)·|∂2D1(r,t,θ)∂r∂t|+C(r-μ0+ɛ),|∂2D1(r,t,θ)∂r∂t|≤|∂2D2(r,t,θ)∂r∂t|+C(r-μ0+ɛ)·|∂2D2(r,t,θ)∂r∂t|+C(r-μ0+ɛ)·|∂2D1(r,t,θ)∂r∂t|+C(r-μ0+ɛ).
Hence,
(112)|∂2D1(r,t,θ)∂r∂t|≤C(r-μ0+ɛ),|∂2D2(r,t,θ)∂r∂t|≤C(r-μ0+ɛ).
(5) We can prove (103) similarly to (4) for the left k+m≤4.Proof of Boundedness.
From Theorem 1.1 in [21] we can see Φ1 possesses a sequence of invariant circles tending to infinity. So, in the original system (13), there exists a corresponding sequence of invariant tori in phase space (x,x˙,t)∈ℝ2×𝕋. Then any solution of system (13) is bounded because it must stay within one of those tori.


## 4. The Proof of Unboundedness

In this section, we will prove that all solutions of (8) are unbounded if B<0. In this case, A<0.Consider (8) which is equivalent to the following system:
(113)x˙=y,y˙=-A|x|α-1x-f(x)y-ɛe^(t)|x|α-1x+p(t).
Replacing (18) by an “auxiliary” system
(114)x˙=y,y˙=A|x|α-1x.Under the transformation (21), the system (113) is transformed into the form
(115)dρdt=-12-2bρ2(1-b)g^′(φ)+h^1(ρ,φ,t),dφdt=ρ1-2bg^(φ)+g^1(ρ,φ,t),
where
(116)g^(φ)=(1-b)d+2bdA|S(φT0)|α+1,h^1(ρ,φ,t)=-T0dρf(ρbS(φT0))S′(φT0)C(φT0)-T0dɛρ2-2b|S(φT0)|α-1S(φT0)S′(φT0)e^(t)+T0dρbS′(φT0)p(t),g^1(ρ,φ,t)=bdf(ρbS(φT0))S(φT0)C(φT0)+bdɛρ1-2b|S(φT0)|α+1e^(t)-bdρb-1S(φT0)p(t).Thus, the system (115) can be written in the form
(117)dρdt=-12(1-b)g^′(φ)ρ2(1-b)+O(ɛρ2(1-b)),dφdt=ρ1-2bg^(φ)+O(ɛρ1-2b).
From the equality
(118)12C2(t)+-Aα+1|S(t)|α+1=12,∀t∈ℝ,
it follows that
(119)0≤|S(φT0)|α+1≤-α+12A.
Hence, the function g^(φ) is C1, 1-periodic and change the sign. Since |S(T0-φT0)|=|S(φT0)| for any φ∈[0,1], there exists φ1∈(0,1/2) such that
(120)|S(T0-φ1T0)|α+1=|S(φ1T0)|α+1=-α+14A.
That is, g^(φ1)=g^(1-φ1)=0. In view of
(121)S(T0-φT0)=-S(φT0),C(T0-φT0)=C(φT0),
we find
(122)g^′(φ1)·g^′(1-φ1)=-(α+1)2(2bdAT0)2|S(φ1T0)|2(α-1)S2(φ1T0)C2(φ1T0)<0.
Hence, we obtain that g^′(φ1) or g^′(1-φ1) is negative. This proves that there exists a φ* such that g^(φ*)=0 and g^′(φ*)<0. Therefore, there are υ>0 and δ0>0 such that g^′(φ)<-δ0 for φ∈[φ*-υ,φ*+υ] and g^(φ)>0 for φ∈(φ*-υ,φ*),  g^(φ)<0 for φ∈(φ*,φ*+υ). Let
(123)𝒦J,υ={(ρ,φ)∈ℝ+×𝕋:ρ>J,φ∈[φ*-υ,φ*+υ]}.
Then, if J is sufficiently large, on the set 𝒦J,υ, we have
(124)-12(1-b)g^′(φ)ρ2(1-b)+O(ɛρ2(1-b))>δ02·ρ2(1-b),(125)ρ1-2bg^(φ)+O(ɛρ1-2b)>0,forρ≥J,φ∈[φ*-υ,φ*-υ2],ρ1-2bg^(φ)+O(ɛρ1-2b)<0,forρ≥J,φ∈[φ*+υ2,φ*+υ].From (117) and (124) we obtain, for t≥0,
(126)ρ(t,ρ0,φ0)=ρ0+∫0t‍(-12(1-b)g^′(φ)ρ2(1-b)+O(ɛρ2(1-b)))dt>ρ0+∫0t‍δ02·ρ2(1-b)dt≥ρ0>J.
Moreover, for ρ(t,ρ0,φ0)>J and φ(t,ρ0,φ0)∈[φ*-υ,φ*-υ/2]∪[φ*+υ/2,φ*+υ], we have
(127)ρ1-2bg^(φ)+O(ɛρ1-2b)=ρ1-2bg^′(φ¯)(φ-φ*)+O(ɛρ1-2b)<-δ02(φ-φ*)ρ1-2b.
From (126) and (127), it follows that any solution (ρ(t,ρ0,φ0),φ(t,ρ0,φ0)) of (115) with the initial condition (ρ(0,ρ0,φ0),φ(0,ρ0,φ0))=(ρ0,φ0)∈𝒦J,υ always stays in 𝒦J,υ and satisfies ρ(t,ρ0,φ0)>δt+ρ(0) with δ=δ03-2b/2, for all t≥0. The proof of Theorem 3 is completed.


## 5. The Proof of Theorem4

In this section, we will prove Theorem4 by using the abstract result on the existence of quasi-periodic solutions proved in [24] in the context Aubry-Mather theory for reversible systems. We only need to show that the Poincaré map (92) has the monotone property; that is,
(128)∂t1∂r(r,t)>0.We can get that(129)|∂F0(r,θ)∂r|≤Cr-1-(1+γ-3b/2)/(2b-1)
by Lemma 11, and
(130)|∂Q2(r,t)∂r|≤r-μ0+ɛ
by Lemma 12. Then we have
(131)∂t1∂r(r,t)=1+∫01‍∂F0∂rdθ+∂Q2∂r⟶c0,asr⟶+∞,
where c0≥1-ɛ. Therefore, we have
(132)∂t1∂r(r,t)>0
as r≫1 and ɛ≪1. This proves the validity of (128).



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*Source: 246343-2013-06-11.xml*