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Boundedness of solutions for reversible systems
Applied Mathematics and Computation 152 (2004) 111–126 www.elsevier.com/locate/amc
Boundedness of solutions for reversible systems Xiaojing Yang Department of Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China
Abstract In this paper, we study the boundedness of all the solutions for the following second order nonlinear differential equation 0 d ¼ ðup ðx0 ÞÞ0 þ f ðxÞup ðx0 Þ þ gðxÞ ¼ eðtÞ dt where up ðuÞ ¼ jujp2 u, p > 1, f ; g 2 C 1 ðR n f0gÞ and odd, e 2 C 1 ðR=ZÞ is odd, g is subquasilinear: sgnðxÞ gðxÞ ! þ1, gðxÞ=up ðxÞ ! 0 as jxj ! 1. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Boundedness of solutions; Reversible systems; MoserÕs twist theorem; p-Laplacian
1. Introduction In this paper, we study the boundedness of all the solutions for the following second order nonlinear differential equation 0
ðup ðx0 ÞÞ þ f ðxÞup ðx0 Þ þ gðxÞ ¼ eðtÞ
ð1Þ p2
where f ; g 2 C 1 ðR n f0gÞ are odd, uðuÞ ¼ juj u, p > 1, e 2 C 1 ðR=ZÞ is odd, g is subquasilinear: sgnðxÞ gðxÞ ! þ1, gðxÞ=up ðxÞ ! 0 as jxj ! 1. In the early 60s, Littlewood [5] proposed to study the boundedness of all the solutions of the following second order nonlinear equation x00 þ gðxÞ ¼ eðtÞ
E-mail address: [email protected] (X. Yang). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00549-6
ð2Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
in the following two cases: (1) superlinear case: gðxÞ=x ! þ1 as jxj ! 1; (2) sublinear case: sgnðxÞ gðxÞ ! þ1 and gðxÞ=x ! 0 as jxj ! 1. Morris [9] first proved that all solutions of x00 þ 2x3 ¼ eðtÞ are bounded, where e 2 C 0 ðS 1 Þ, S 1 ¼ R=Z. Later, several authors have extended his results to the more general cases, see [1,3,4,6] and references therein. Recently, the boundedness of solutions for the following relatively simple sublinear equation x00 þ ua ðxÞ ¼ eðtÞ ð3Þ has been studied in [2,7], where 1 < a < 2. They proved that every solution of (3) is bounded if e 2 C 1 ðS 1 Þ, that is, if xðtÞ is a solution of (3), then it is defined in ð1; þ1Þ and satisfies supðjxðtÞj þ jx0 ðtÞjÞ < 1: t2R
For the more general problem x00 þ gðxÞ ¼ eðtÞ
ð4Þ
where g satisfies some reasonable sublinear assumptions, the boundedness of all solutions of (4) is proved in [8]. Inspired by the work of [2,6,8], we study the boundedness of all the solutions of (1) in subquasilinear case. In this paper, we denote by c > 1 and C > 1, respectively, two universal positive constants without regarding their values. The main result of this paper is Theorem 1. Assume f ; g 2 C 1 ðR n f0gÞ are odd, up ðuÞ ¼ juj C 1 ðS 1 Þ is odd and for x 6¼ 0 the following hold
p2
u, p > 1, e 2
i(i) xgðxÞ > 0 and there exist two positive constants a, b such that 1 GðxÞg0 ðxÞ 1 p < a6 > 1; and b 6 b < ; where q ¼ 2q g2 ðxÞ q p1 <
3aq2 þ 2aq q a2 q2 1 ; qðq 1 þ aqÞ
k (ii) xk d GðxÞ 6 CGðxÞ; k dx
kP0
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
113
(iii) k dk f ðxÞ x 6 Cjxjs GðxÞ; k P 0 dxk Rx where GðxÞ ¼ 0 gðsÞ ds, s is a positive constant such that 2
s>
q2 ½ð1 aÞ þ ðb aÞ þ 2q þ 1 : ð1 bÞqðq þ 1 aÞ
Then every solution of (1) is bounded. a2
Let p ¼ 2, gðxÞ ¼ jxj x, 4=3 < a < 2. Then GðxÞg0 ðxÞ=g2 ðxÞ ¼ ða 1Þ=a. In this case, 1=4 < a ¼ ða 1Þ=a ¼ b < 1=2. We have a2
Corollary 1. Let gðxÞ ¼ jxj x, 4=3 < a < 2, p ¼ 2. If there exists s > ð4 þ 5a2 Þ=ð2ð2a þ 1ÞÞ such that f 2 C 1 ðRÞ is odd and for x 6¼ 0, k P 0, k dk f ðxÞ 6 Cjxjas x dxk e 2 C 1 ðS 1 Þ is odd. Then every solution of x00 þ f ðxÞx0 þ jxj
a2
x ¼ eðtÞ
is bounded. Remark 1. The assumption of f ; g; e 2 C 1 can be relaxed to require that f ; g; e 2 C n0 with some n0 2 N sufficiently large and the assumption f ; g 2 C n0 ðR n f0gÞ can be relaxed to require that they hold for jxj P d0 1. The assumptions in (i) imply also D1 ¼ c1 c2 d1 > 0, D2 ¼ 2c1 c2 c2 d1 > 0 and D3 ¼ 2c1 1 c2 þ ac2 > 0, where c1 ¼ q=ðq þ 1 bqÞ 6 c2 ¼ q=ðq þ 1 aqÞ, d1 ¼ 2 þ ð1=qÞ a ð1 bÞs. Remark 2. From the assumptions on g, one can verify that the following hold 0 6 g0 ðxÞ 6 C;
g0 ðxÞ ! 0
sgnðxÞgðxÞ ! þ1;
as jxj ! 1
gðxÞ=up ðxÞ ! 0
ð5Þ as jxj ! 1
ð6Þ
ð1 bÞxgðxÞ 6 gðxÞ 6 ð1 aÞxgðxÞ
ð7Þ
cjxj1=ð1aÞ 6 GðxÞ 6 Cjxj1=ð1bÞ :
ð8Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
Example. Every solution of the equation x00 þ
xx0 2
ð1 þ x2 Þ
þ jxj
a2
x ¼ sin 2pt;
1
is bounded. The idea of proving the boundedness of solutions of (1) is as follows. By means of action and angle variables transformations, (1) is, outside of a large 2 disc Dr ¼ fðx; x0 Þ 2 R2 ; x2 þ ðx0 Þ 6 r2 g in the ðx; x0 Þ-plane, transformed into a perturbation of an integrable Hamiltonian system. The Poincare map of the transformed system is close to a so-called twist map in R2 =Dr . Then MoserÕs twist theorem for reversible system [10] guarantees the existence of arbitrary large invariant curves diffeomorphic to circles and surrounding the origin in the ðx; x0 Þ-plane. Every such curve is the base of a time-periodic and flow-invariant cylinder in the extended phase space ðx; x0 ; tÞ 2 R2 R, which confines the solutions in the interior and which leads to a bound of there solutions.
2. Reversible systems and action-angle variables In this section, some basic concepts of reversible systems will be given and then the action and angle variables will be introduced. Definition 1. Let X : X R ! Rn be continuous and 1-periodic in the last variables, where X Rn is an open set. The system x0 ¼ X ðx; tÞ
ð9Þ
is called reversible if there is an involution M : Rn ! Rn (i.e., M is a C 1 -diffeomorphism and M 2 ¼ IRn ) with MðXÞ ¼ X and such that DMðMxÞ X ðMx; tÞ ¼ X ðx; tÞ;
8ðx; tÞ 2 X R
where DM stands for the Jacobian matrix of M. Definition 2. Let A : Rn X ! Rn be a homeomorphism onto its image and let M : Rn Rn be a homeomorphism with M 2 ¼ IRn . A is called reversible with respect to M on a set B X \ AðXÞ, with MðBÞ ¼ B if A1 ¼ MAM;
8x 2 B
holds. The following definition and lemma will be useful when change of variables are involved.
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
115
Definition 3. Assume that T ð ; tÞ is an invertible transformation of X for any fixed t, and M is an involution of Rn with MðXÞ ¼ X. We say that T ðx; tÞ is Minvariant if equality M T ðx; tÞ ¼ T ðMx; tÞ holds. Lemma 1 [6]. Suppose that system (9) is reversible with respect to an involution M : Rn ! Rn . If a transformation T ð ; tÞ : X ! Rn is M-invariant and C 1 in x and t, then the transformed system of (9) under T is also reversible with respect to M. Suppose now the conditions of Theorem 1 hold, we will carry out the standard reduction to the action-angle variables by the method developed in [4]. If we introduce a new variable y by y ¼ up ðx0 Þ, then x0 ¼ uq ðyÞ and Eq. (1) is equivalent to the following system x0 ¼ uq ðyÞ;
y 0 ¼ f ðxÞy gðxÞ þ eðtÞ
ð10Þ
which is reversible with respect to the involution M : ðx; yÞ ! ðx; yÞ by the oddness of f , g and e, q ¼ p=ðp 1Þ. q Set H ðx; yÞ ¼ jyj =q þ GðxÞ. Then for large h > 0, H ðx; yÞ ¼ h denotes a simple closed curve in the ðx; yÞ-plane. It is well-known [4] that the auxiliary system x0 ¼ uq ðyÞ;
y 0 ¼ gðxÞ
ð11Þ
is an integrable Hamiltonian system with Hamiltonian function H ðx; yÞ and has action and angle variables ðr; hÞ. Denote the area surrounded by the closed integral curve Ch of (11) defined by H ðx; yÞ ¼ h by r. Then I r ¼: J ðhÞ ¼ y dx: ð12Þ H ðx;yÞ¼h 1 Denote by G1 þ and G the right and left inverse of G, respectively. Assume ðxþ ; 0Þ and ðx ; 0Þ are the intersection points of Ch with x-axis, i.e., 1 x ¼ G1 ðhÞ < 0 < Gþ ðhÞ ¼ xþ :
Then by the oddness of g, we have x ¼ xþ . It is easy to see that Z xþ J ðhÞ ¼ 4q1=q ðh GðxÞÞ1=q dx: 0
Now we define the generating function Sðx; rÞ as the area
ð13Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
Sðx; rÞ ¼
Z y dx C
where C is the part of the level curve Ch : H ðx; yÞ ¼ h connecting the y-axis with point ðx; yÞ, oriented clockwise and h ¼ J 1 ðrÞ ¼: IðrÞ. This defines S up to an H integer multiple of r ¼ Ch y dx, the area of the domain surrounded by the closed curve Ch . We define the map ðr; hÞ ! ðx; yÞ by y¼
oS ðx; rÞ; ox
h¼
oS ðx; rÞ or
ð14Þ
which is known to be symplectic since dx ^ dy ¼ dx ^ ðSxx dx þ Sxr drÞ ¼ Sxr dx ^ dr; dh ^ dr ¼ ðSrx dx þ Srr drÞ ^ dr ¼ Srx dx ^ dr: Lemma 2. Under the transformation ðr; hÞ ! ðx; yÞ, system (10) is changed into dr ¼ h1 ðr; hÞ þ h2 ðr; h; tÞ dt dh ¼ I 0 ðrÞ þ h3 ðr; hÞ þ h4 ðr; h; tÞ dt
ð15Þ
where h1 ðr; hÞ ¼ f ðxÞyxh h2 ðr; h; tÞ ¼ xh eðtÞ h3 ðr; hÞ ¼ f ðxÞyxr h4 ðr; h; tÞ ¼ xr eðtÞ
ð16Þ
IðrÞ ¼ J 1 ðrÞ and x ¼ xðr; hÞ, y ¼ yðr; hÞ are given implicitly by (14). Proof. Since the transformation (14) is symplectic, we have det D ¼ 1, where xr xh D¼ : yr yh It follows from (10) that dx dr dh ¼ xr þ xh ¼ uq ðyÞ; dt dt dt dy dr dh ¼ yr þ yh ¼ f ðxÞy gðxÞ þ eðtÞ: dt dt dt
ð17Þ
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
117
By using det D ¼ 1 and from (17), one obtains dr ¼ ½yh uq ðyÞ þ xh gðxÞ þ f ðxÞyxh xh eðtÞ; dt dh ¼ ½xr gðxÞ þ uq ðyÞyr f ðxÞyxr þ xr eðtÞ: dt
ð18Þ
From q
H ðx; yÞ ¼ jyj =q þ GðxÞ ¼ h ¼ IðrÞ we have jyj
q2
yyh þ gðxÞxh ¼ 0
jyj
q2
yyr þ gðxÞxr ¼ I 0 ðrÞ:
Substituting (19) into (18), we obtain (15) and (16).
ð19Þ
It is not difficult to verify that x is odd in h and y is even in h, hence we have h1 ðr; hÞ ¼ h1 ðr; hÞ h2 ðr; h; tÞ ¼ h2 ðr; h; tÞ h3 ðr; hÞ ¼ h3 ðr; hÞ
ð20Þ
h4 ðr; h; tÞ ¼ h4 ðr; h; tÞ which implies that system (15) is reversible with respect to the involution M : ðr; hÞ ! ðr; hÞ. Lemma 3. The following inequalities hold pffiffiffiffiffi pffiffiffiffiffi 2 2hxþ 6 J ðhÞ 6 4 2hxþ 8 < ch1þð1=qÞb 6 J ðhÞ 6 Ch1þð1=qÞa ch1 J ðhÞ 6 J 0 ðhÞ 6 Ch1 J ðhÞ : 1 0 ch J ðhÞ 6 J 00 ðhÞ 6 Ch1 J 0 ðhÞ:
ð21Þ ð22Þ
Moreover, J 0 ðhÞ ! þ1 as h ! þ1. Proof. (1) One can prove the inequality (21) by comparing the area p bounded by ffiffiffiffiffi Ch respectively with the area of the triangle or rectangle with sides 2h and xþ . (2) From the expression Z xþ 1=q ðh GðxÞÞ dx J ðhÞ ¼ 4q1=q 0
set GðxÞ ¼ sh, s 2 ½0; 1. Then h ds=dh ¼ GðsÞ=gðsÞ and gðxÞ dx ¼ h ds, moreover,
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
J ðhÞ ¼ 4q1=q
Z
1
h1þð1=qÞ 0
ð1 sÞ1=q ds gðxÞ
which implies that J 0 ðhÞ ¼
4q1=q h
Z
xþ 0
1 þ w0 ðxÞÞðh GðxÞ q
1=q dx:
Similarly, we can obtain
Z xþ w0 ðxÞ 1 dx p 4 J 00 ðhÞ ¼ 1=p : 1=p q h 0 ðh GðxÞÞ Since w0 ðxÞ ¼ 1 GðxÞg0 ðxÞ=g2 ðxÞ, we obtain from the assumptions in (i) of Theorem 1 1 1 1 þ b h1 J ðhÞ 6 J 0 ðhÞ 6 1 þ a h1 J ðhÞ q q which implies ch1þð1=qÞb 6 J ðhÞ 6 Ch1þð1=qÞa and chð1=qÞb 6 J 0 ðhÞ 6 Chð1=qÞa hence, J 0 ðhÞ ! þ1 as h ! þ1. Similarly, one can prove the last inequalities of (22). Lemma 4. For k P 0, we have k k d o 0 k k 0 dhk J ðhÞ 6 Ch J ðhÞ; ork J ðhÞ 6 Cr J ðhÞ
ð23Þ
and k o k ork xðr; hÞ 6 Cr jxj;
k o k ork yðr; hÞ 6 Cr jyj:
ð24Þ
The proof of Lemma 4 is similar to the proof of [4, Lemma A 4.1], so we omit it. It follows from IðrÞ ¼ J 1 ðrÞ and (22) that crc1 6 IðrÞ 6 Crc2
ð25Þ 1
with c1 ¼ ð1 þ ð1=qÞ aÞ ; c2 ¼ ð1 þ ð1=qÞ bÞ crc1 1 6 I 0 ðrÞ 6 Crc2 1 :
1
and ð26Þ
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
119
Lemma 5. Let x, y be defined by (14). Then ox q=p ðr; hÞ ¼ L0 ðhÞjyj sgnðyÞ; oh
oy ðr; hÞ ¼ J 0 ðhÞgðxðr; hÞÞ: oh
ð27Þ
Proof. For y > 0, we have Z x 1=q Sðx; yÞ ¼ q1=q ðhðrÞ GðsÞÞ ds 0
oS ðx; rÞ ¼ or
Z
x
qð1=qÞ1 ðhðrÞ GðsÞÞð1=qÞ1 ds h0 ðrÞ ¼ h:
0
Hence o2 S ox 1=q 0 ðx; rÞ ¼ q1=q ðhðrÞ GðxÞÞ h ðrÞ ¼ 1 or oh oh which yields ox 1=q 1 ðr; hÞ ¼ q1=q ðhðrÞ GðxÞÞ ðh0 ðrÞÞ ¼ J 0 ðhÞy q=p : oh Similarly, we have for y < 0 ox ¼ J 0 ðhÞjyjq=p : oh From above result and jyjq =q þ GðxÞ ¼ h, we obtain oy ox jyjq2 y þ gðxÞ ¼ 0 oh oh hence oy ¼ J 0 ðhÞgðxÞ: oh Lemma 6. For k P 0, m P 0, we have k o k d1 ork h1 ðr; hÞ 6 Cr I ðrÞ kþm o k 1a ork otm h2 ðr; h; tÞ 6 Cr I ðrÞ k o 6 Crk1 I d1 ðrÞ h ðr; hÞ 3 ork kþm o 6 Crk1 I 1a ðrÞ h ðr; h; tÞ ork otm 4 where d1 ¼ 2 þ ð1=qÞ a ð1 bÞs.
ð28Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
Proof. For k ¼ 0, by the expression of h1 ðr; hÞ h1 ðr; hÞ ¼ f ðxÞyxh we obtain by applying (iii) of Theorem 1, (21) and (27) s
jh1 ðr; hÞj 6 Cjxj GðxÞjyjjxh j s
6 Cjxj GðxÞh1=q J 0 ðhÞh1=p 6 Cjxjs GðxÞJ ðhÞ: From (8), we have jxj
s
6 GðxÞ
ðb1Þs
, therefore
jh1 ðr; hÞj 6 CGðxÞ1þðb1Þs J ðhÞ 6 Ch1þðb1Þs J ðhÞ 6 Ch2þð1=qÞaþðb1Þs ¼ CI d1 ðrÞ: For k P 1, we have k k o o h ðr; hÞ ¼ f ðxÞyx h ork 1 ork X i j l k o f ðxÞ o y o xh ¼ Cijl iþjþl¼k ori orj orl X oi f ðxÞ oj y ol xh 6C ori orj orl : iþjþl¼k Since oi f ðxÞ=ori is a sum of terms of X m6i
Cim
dm oi1 x oim x f ðxÞ
im ; ori1 or dxm
i1 þ ; im ¼ i;
we obtain by using (iii) and (24) i o f ðxÞ ms GðxÞri1 jxj rim jxj ori 6 Cjxj 6 Cri jxjs GðxÞ: From (24), j o y j j 1=q orj 6 Cr jyj 6 Cr h :
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
121
Finally, by (27) l l o xh o 0 q=p ¼ J ðhÞjyj orl orl s t Xo o 0 jyjq=p 6C J ðhÞ ors ort sþt¼l
6 Crs J 0 ðhÞrt jyj
q=p
6 Crl J 0 ðhÞh1=p : Combining above results, we obtain k o 6 Crk jxjs GðxÞh1þð1=qÞa 6 Crk I d1 ðrÞ: h ðr; hÞ ork 1 From the definition of h2 ðr; h; tÞ, by (22) and (27), we have for k ¼ 0, m P 0 m m o d ¼ jx h ðr; h; tÞ j eðtÞ h m otm 2 dt 6 Cjxh j 6 CJ 0 ðhÞh1=p 6 Ch1a ¼ CI 1a ðrÞ: For k P 1 k m kþm o d o k otm h2 ðr; h; tÞ ¼ ork xh ðr; hÞ dtm eðtÞ k o 6 C k xh ðr; hÞ or 6 Crk J 0 ðhÞh1=p 6 rk Ch1a ¼ Crk I 1a ðrÞ: Similarly, we can prove the rest inequalities in (28).
3. Further transformations In this section, we first introduce a series of M-invariant transformations such that in the transformed system, the growth speed of the term corresponding to h1 ðr; hÞ is the same as that of term of h2 ðr; h; tÞ if the action variable satisfies r 1. Then we change the role of angle and time variables.
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Lemma 7. There exists a M-invariant diffeomorphism T , having the form T : r ¼ q þ U ðq; hÞ;
h¼h
where U ðq; hÞ satisfies k o kþ1c1 d1 I ðqÞ; oqk U ðq; hÞ 6 Cq
k P 0:
Under this transformation, system (15) is transformed into the system dq ¼ h1 ðq; hÞ þ h2 ðq; h; tÞ dt dh ¼ I 0 ðqÞ þ h3 ðq; hÞ þ h4 ðq; h; tÞ dt
ð29Þ
h3 and h4 satisfy the same estimates as h2 , h3 and h4 in Lemma 6 where h2 , respectively and h1 satisfies k o kd2 d1 I ðqÞ ð30Þ oqk h1 ðq; hÞ 6 Cq where d2 ¼ c1 c2 d1 > 0. Moreover, system (29) is reversible with respect to the involution M : ðq; hÞ ! ðq; hÞ. Proof. Define a transformation S by S : q ¼ r þ V ðr; hÞ;
h¼h
ð31Þ
where V ðr; hÞ ¼
Z 0
h
h1 ðr; sÞ ds : I 0 ðrÞ
Since h1 ðr; hÞ ¼ h1 ðr; hÞ and jðok =ork Þh1 ðr; hÞj 6 Crk I d1 ðrÞ, jðo =orm ÞðI 0 ðrÞÞ1 j 6 Crmþ1c1 for k P 0, m P 0, one obtains
and
m
(i) V ðr; hÞ ¼ V ðr; hÞ; (ii) the transformation S is a diffeomorphism and satisfies k o 6 Crkþ1c1 I d1 ðrÞ: V ðr; hÞ ork
ð32Þ
Let T ¼ S1 : r ¼ q þ U ðq; hÞ, h ¼ h. Then system (15) is transformed into (29) with
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
oV oV h3 þ h1 h1 ðq; hÞ ¼ oh or oV oV h4 þ h2 þ h2 h2 ðq; h; tÞ ¼ oh or 0 0 h3 ðq; hÞ ¼ I ðqÞ I ðrÞ þ h3 h4 ðq; h; tÞ ¼ h4 ðr; h; tÞ
123
ð33Þ
with r ¼ q þ U ðq; hÞ. From the expression of h1 and (32), we obtain h1 ðq; hÞ 6 Crkc1 I 2d1 ðrÞ 6 Crkðc1 c2 d1 Þ I d1 ðrÞ 6 Crkd2 I d1 ðrÞ: Moreover, from the following lemma, system (29) is reversible with respect to involution M : ðq; hÞ ! ðq; hÞ. Lemma 8. For q 1, U satisfies i(i) jðok =oqk ÞU ðq; hÞj 6 Cqkþ1c1 I d1 ðrÞ; (ii) U ðq; hÞ ¼ U ðq; hÞ. The proof of Lemma 8 is similar to that of [6, Lemma 3] so we omit it. Now applying the transformation T n times with nd2 þ c2 d1 6 ð1 aÞc2 or equivalently n P ½c2 ðd1 1 þ aÞ=d2 where [A] denotes the integer part of A, (If d1 6 ð1 aÞ, we do not need the transformation T ) then the transformed term h1 satisfies k o k 1a ð34Þ oqk h1 ðq; hÞ 6 Cq I ðrÞ: 2 , h 3 and The terms of h h4 satisfy (28). tÞ ¼ tÞ, h~3 ðq; hÞ ¼ h3 ðq; hÞ, Set h ¼ h, h~1 ðq; hÞ, h~2 ðq; h; h2 ðq; h; tÞ ¼ h4 ðq; h; tÞ. Then system (29) is of the form h~4 ðq; h; dq ~ þ h~2 ðq; h; tÞ ¼ h1 ðq; hÞ dt ð35Þ dh 0 ~ ~ ¼ I ðqÞ þ h3 ðq; hÞ þ h4 ðq; h; tÞ: dt It is evident that h~i , i ¼ 1; 2; 3; 4, satisfies the same estimates as hi respectively. Moreover, system (35) is reversible with respect to the involution
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
! ðq; hÞ. For simplicity, we still denote h~ by h and h~i by hi , M : ðq; hÞ i ¼ 1; 2; 3; 4 in (35). Set f0 ðqÞ ¼ ðI 0 ðqÞÞ
1
h3 ðq; hÞ I 0 ðqÞðI 0 ðqÞ þ h3 ðq; hÞÞ h4 ðq; h; tÞ f2 ðq; h; tÞ ¼ 0 ðI ðqÞ þ h3 ðq; hÞÞðI 0 ðqÞ þ h3 ðq; hÞ þ h4 ðq; h; tÞÞ f3 ðq; h; tÞ ¼ ðh1 ðq; hÞ þ h2 ðq; h; tÞÞðf0 ðqÞ þ f1 ðq; hÞ þ f2 ðq; h; tÞÞ:
f1 ðq; hÞ ¼
ð36Þ
Then it is not difficult to verify that system (35) is equivalent to the following system dt ¼ f0 ðqÞ þ f1 ðq; hÞ þ f2 ðq; h; tÞ dh dq ¼ f3 ðq; h; tÞ: dh
ð37Þ
Moreover, one can verify by (36) that system (37) is reversible with respect to the involution M : ðq; tÞ ! ðq; tÞ. From the estimates (28), (34), and (36), one obtains the following estimates. For k þ m P 0, we have cq1c2 6 f0 ðqÞ 6 Cq1c1 k d f0 ðqÞ kþ1c2 6 Cqkþ1c1 cq 6 dqk k o kþ12c þc d 1 2 1 Cq f ðq; hÞ oqk 1 kþm o kþð1aÞc 1 2 Cq f ðq; h; tÞ oqk otm 2 kþm o kþð1aÞc þ1c 2 1 : oqk otm f3 ðq; h; tÞCq
ð38Þ
4. Proof of Theorem 1 Now we consider system (37) where fi , i ¼ 0; 1; 2; 3, satisfies (38). Set k ¼ f0 ðqÞ, h ¼ h and
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
125
g1 ðk; hÞ ¼ f1 ðqðkÞ; hÞ g2 ðk; h; tÞ ¼ f2 ðqðkÞ; h; tÞ g3 ðk; h; tÞ ¼ f3 ðqðkÞ; h; tÞf00 ðqðkÞÞ
ð39Þ
where q ¼ qðkÞ is the inverse of k ¼ f0 ðqÞ. It is evident that q 1 if and only if k 1 by the first two inequalities in (38) and 1 c2 > 0. The transformed system of (37) is dt ¼ k þ g1 ðk; hÞ þ g2 ðk; h; tÞ dh dk ¼ g3 ðk; h; tÞ: dh
ð40Þ
Lemma 9. The function gi , i ¼ 1; 2; 3, satisfies the following estimates k o 6 Ckkþ1e1 g ðk; hÞ okk 1 kþm o k1ððac2 Þ=ð1c2 ÞÞ okk otm g2 ðk; h; tÞ 6 Ck kþm o ke2 okk otm g3 ðq; h; tÞ 6 Ck
ð41Þ
for k þ m P 0, where e1 ¼
2c1 c2 c2 d1 > 0; 1 c2
e2 ¼
2c1 1 c2 þ ac2 > 0: 1 c1
ð42Þ
Moreover, system (40) is reversible with respect to the involution M : ðk; tÞ ! ðk; tÞ. Proof. Inequalities (41) are direct consequences of (38), inequalities (42) are consequences of the assumptions in (i), (iii) of Theorem 1 and Remark 1. The last statement is the result of Lemma 1. Proof of Theorem 1. Clearly, if kð0Þ ¼ k 1, the solutions of (40) do exist on 0 6 h 6 1 and are uniquely determined with respect to the initial value. By Lemma 9, the Poincare map of solutions of (40) is of the form ðk; tÞ ! ðk1 ; t1 Þ with P : k1 ¼ k þ P1 ðk; tÞ; t1 ¼ t þ sðkÞ þ P2 ðk; tÞ R1 where sðkÞ ¼ k þ 0 g1 ðk; hÞ dh. Moreover, there exists an e0 > 0 such that for 0 6 k þ m 6 4 and k 1,
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
kþm o e0 okk otm Pi ðk; tÞ 6 k ;
i ¼ 1; 2; for some 0 > 0
hold and the map P is reversible with respect to the involution M : ðk; tÞ ! ðk; tÞ. It follows from [10, Theorem 1.1] that the map P possesses a sequence of invariant circles tending to infinity. Hence, in the original system (10), there exists a corresponding sequence of invariant tori in phase space ðx; x0 ; tÞ 2 R2 S 1 . Since any solution of system (10) must stay within one of those tori, it is therefore bounded. Theorem 1 is thus proved.
References [1] R. Dieckekrhoff, E. Zehnder, Boundedness of solutions via the twist theorem, Ann. Scuola. Norm. Sup. Pisa. Cl. Sci. 14 (1) (1987) 79–95. [2] T. K€ upper, J. You, Existence of quasiperiodic solutions and LittlewoodÕs boundedness problem of Duffing equations with subquadratic potentials, Nonlinear Anal. 35 (1999) 549– 559. [3] S. Laederich, M. Levi, Invariant curves and time-dependent potentials, Ergod. Th. Dyn. Syst. 11 (1991) 365–378. [4] M. Levi, Quasi-periodic motions in superquadratic time-periodic potentials, Commun. Math. Phys. 143 (1) (1991) 43–83. [5] J. Littlewood, Some Problems in Real and Complex Analysis, Heath, Lexington, MA, 1968. [6] B. Liu, Boundedness of sollutions of nonlinear differential equations, J. Differen. Equat. 144 (1998) 66–98. [7] B. Liu, F. Zanolin, Boundedness of solutions for second order quasilinear ODEs, in press. [8] B. Liu, On LittlewoodÕs boundedness problem for sublinear Duffing equations, Trans. Am. Math. Soc. 352 (2001) 1567–1585. [9] G. Morris, A case of boundedness of LittlewoodÕs problem on oscillatory differential equations, Bull. Aust. Math. Soc. 14 (1976) 71–93. [10] M. Sevryuk, Reversible Systems, Lecture Notes in Mathematics, vol. 1211, Springer-Verlag, Berlin, 1986.
Boundedness of solutions for reversible systems Xiaojing Yang Department of Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China
Abstract In this paper, we study the boundedness of all the solutions for the following second order nonlinear differential equation 0 d ¼ ðup ðx0 ÞÞ0 þ f ðxÞup ðx0 Þ þ gðxÞ ¼ eðtÞ dt where up ðuÞ ¼ jujp2 u, p > 1, f ; g 2 C 1 ðR n f0gÞ and odd, e 2 C 1 ðR=ZÞ is odd, g is subquasilinear: sgnðxÞ gðxÞ ! þ1, gðxÞ=up ðxÞ ! 0 as jxj ! 1. Ó 2003 Elsevier Inc. All rights reserved. Keywords: Boundedness of solutions; Reversible systems; MoserÕs twist theorem; p-Laplacian
1. Introduction In this paper, we study the boundedness of all the solutions for the following second order nonlinear differential equation 0
ðup ðx0 ÞÞ þ f ðxÞup ðx0 Þ þ gðxÞ ¼ eðtÞ
ð1Þ p2
where f ; g 2 C 1 ðR n f0gÞ are odd, uðuÞ ¼ juj u, p > 1, e 2 C 1 ðR=ZÞ is odd, g is subquasilinear: sgnðxÞ gðxÞ ! þ1, gðxÞ=up ðxÞ ! 0 as jxj ! 1. In the early 60s, Littlewood [5] proposed to study the boundedness of all the solutions of the following second order nonlinear equation x00 þ gðxÞ ¼ eðtÞ
E-mail address: [email protected] (X. Yang). 0096-3003/$ - see front matter Ó 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(03)00549-6
ð2Þ
112
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
in the following two cases: (1) superlinear case: gðxÞ=x ! þ1 as jxj ! 1; (2) sublinear case: sgnðxÞ gðxÞ ! þ1 and gðxÞ=x ! 0 as jxj ! 1. Morris [9] first proved that all solutions of x00 þ 2x3 ¼ eðtÞ are bounded, where e 2 C 0 ðS 1 Þ, S 1 ¼ R=Z. Later, several authors have extended his results to the more general cases, see [1,3,4,6] and references therein. Recently, the boundedness of solutions for the following relatively simple sublinear equation x00 þ ua ðxÞ ¼ eðtÞ ð3Þ has been studied in [2,7], where 1 < a < 2. They proved that every solution of (3) is bounded if e 2 C 1 ðS 1 Þ, that is, if xðtÞ is a solution of (3), then it is defined in ð1; þ1Þ and satisfies supðjxðtÞj þ jx0 ðtÞjÞ < 1: t2R
For the more general problem x00 þ gðxÞ ¼ eðtÞ
ð4Þ
where g satisfies some reasonable sublinear assumptions, the boundedness of all solutions of (4) is proved in [8]. Inspired by the work of [2,6,8], we study the boundedness of all the solutions of (1) in subquasilinear case. In this paper, we denote by c > 1 and C > 1, respectively, two universal positive constants without regarding their values. The main result of this paper is Theorem 1. Assume f ; g 2 C 1 ðR n f0gÞ are odd, up ðuÞ ¼ juj C 1 ðS 1 Þ is odd and for x 6¼ 0 the following hold
p2
u, p > 1, e 2
i(i) xgðxÞ > 0 and there exist two positive constants a, b such that 1 GðxÞg0 ðxÞ 1 p < a6 > 1; and b 6 b < ; where q ¼ 2q g2 ðxÞ q p1 <
3aq2 þ 2aq q a2 q2 1 ; qðq 1 þ aqÞ
k (ii) xk d GðxÞ 6 CGðxÞ; k dx
kP0
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
113
(iii) k dk f ðxÞ x 6 Cjxjs GðxÞ; k P 0 dxk Rx where GðxÞ ¼ 0 gðsÞ ds, s is a positive constant such that 2
s>
q2 ½ð1 aÞ þ ðb aÞ þ 2q þ 1 : ð1 bÞqðq þ 1 aÞ
Then every solution of (1) is bounded. a2
Let p ¼ 2, gðxÞ ¼ jxj x, 4=3 < a < 2. Then GðxÞg0 ðxÞ=g2 ðxÞ ¼ ða 1Þ=a. In this case, 1=4 < a ¼ ða 1Þ=a ¼ b < 1=2. We have a2
Corollary 1. Let gðxÞ ¼ jxj x, 4=3 < a < 2, p ¼ 2. If there exists s > ð4 þ 5a2 Þ=ð2ð2a þ 1ÞÞ such that f 2 C 1 ðRÞ is odd and for x 6¼ 0, k P 0, k dk f ðxÞ 6 Cjxjas x dxk e 2 C 1 ðS 1 Þ is odd. Then every solution of x00 þ f ðxÞx0 þ jxj
a2
x ¼ eðtÞ
is bounded. Remark 1. The assumption of f ; g; e 2 C 1 can be relaxed to require that f ; g; e 2 C n0 with some n0 2 N sufficiently large and the assumption f ; g 2 C n0 ðR n f0gÞ can be relaxed to require that they hold for jxj P d0 1. The assumptions in (i) imply also D1 ¼ c1 c2 d1 > 0, D2 ¼ 2c1 c2 c2 d1 > 0 and D3 ¼ 2c1 1 c2 þ ac2 > 0, where c1 ¼ q=ðq þ 1 bqÞ 6 c2 ¼ q=ðq þ 1 aqÞ, d1 ¼ 2 þ ð1=qÞ a ð1 bÞs. Remark 2. From the assumptions on g, one can verify that the following hold 0 6 g0 ðxÞ 6 C;
g0 ðxÞ ! 0
sgnðxÞgðxÞ ! þ1;
as jxj ! 1
gðxÞ=up ðxÞ ! 0
ð5Þ as jxj ! 1
ð6Þ
ð1 bÞxgðxÞ 6 gðxÞ 6 ð1 aÞxgðxÞ
ð7Þ
cjxj1=ð1aÞ 6 GðxÞ 6 Cjxj1=ð1bÞ :
ð8Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
Example. Every solution of the equation x00 þ
xx0 2
ð1 þ x2 Þ
þ jxj
a2
x ¼ sin 2pt;
1
is bounded. The idea of proving the boundedness of solutions of (1) is as follows. By means of action and angle variables transformations, (1) is, outside of a large 2 disc Dr ¼ fðx; x0 Þ 2 R2 ; x2 þ ðx0 Þ 6 r2 g in the ðx; x0 Þ-plane, transformed into a perturbation of an integrable Hamiltonian system. The Poincare map of the transformed system is close to a so-called twist map in R2 =Dr . Then MoserÕs twist theorem for reversible system [10] guarantees the existence of arbitrary large invariant curves diffeomorphic to circles and surrounding the origin in the ðx; x0 Þ-plane. Every such curve is the base of a time-periodic and flow-invariant cylinder in the extended phase space ðx; x0 ; tÞ 2 R2 R, which confines the solutions in the interior and which leads to a bound of there solutions.
2. Reversible systems and action-angle variables In this section, some basic concepts of reversible systems will be given and then the action and angle variables will be introduced. Definition 1. Let X : X R ! Rn be continuous and 1-periodic in the last variables, where X Rn is an open set. The system x0 ¼ X ðx; tÞ
ð9Þ
is called reversible if there is an involution M : Rn ! Rn (i.e., M is a C 1 -diffeomorphism and M 2 ¼ IRn ) with MðXÞ ¼ X and such that DMðMxÞ X ðMx; tÞ ¼ X ðx; tÞ;
8ðx; tÞ 2 X R
where DM stands for the Jacobian matrix of M. Definition 2. Let A : Rn X ! Rn be a homeomorphism onto its image and let M : Rn Rn be a homeomorphism with M 2 ¼ IRn . A is called reversible with respect to M on a set B X \ AðXÞ, with MðBÞ ¼ B if A1 ¼ MAM;
8x 2 B
holds. The following definition and lemma will be useful when change of variables are involved.
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
115
Definition 3. Assume that T ð ; tÞ is an invertible transformation of X for any fixed t, and M is an involution of Rn with MðXÞ ¼ X. We say that T ðx; tÞ is Minvariant if equality M T ðx; tÞ ¼ T ðMx; tÞ holds. Lemma 1 [6]. Suppose that system (9) is reversible with respect to an involution M : Rn ! Rn . If a transformation T ð ; tÞ : X ! Rn is M-invariant and C 1 in x and t, then the transformed system of (9) under T is also reversible with respect to M. Suppose now the conditions of Theorem 1 hold, we will carry out the standard reduction to the action-angle variables by the method developed in [4]. If we introduce a new variable y by y ¼ up ðx0 Þ, then x0 ¼ uq ðyÞ and Eq. (1) is equivalent to the following system x0 ¼ uq ðyÞ;
y 0 ¼ f ðxÞy gðxÞ þ eðtÞ
ð10Þ
which is reversible with respect to the involution M : ðx; yÞ ! ðx; yÞ by the oddness of f , g and e, q ¼ p=ðp 1Þ. q Set H ðx; yÞ ¼ jyj =q þ GðxÞ. Then for large h > 0, H ðx; yÞ ¼ h denotes a simple closed curve in the ðx; yÞ-plane. It is well-known [4] that the auxiliary system x0 ¼ uq ðyÞ;
y 0 ¼ gðxÞ
ð11Þ
is an integrable Hamiltonian system with Hamiltonian function H ðx; yÞ and has action and angle variables ðr; hÞ. Denote the area surrounded by the closed integral curve Ch of (11) defined by H ðx; yÞ ¼ h by r. Then I r ¼: J ðhÞ ¼ y dx: ð12Þ H ðx;yÞ¼h 1 Denote by G1 þ and G the right and left inverse of G, respectively. Assume ðxþ ; 0Þ and ðx ; 0Þ are the intersection points of Ch with x-axis, i.e., 1 x ¼ G1 ðhÞ < 0 < Gþ ðhÞ ¼ xþ :
Then by the oddness of g, we have x ¼ xþ . It is easy to see that Z xþ J ðhÞ ¼ 4q1=q ðh GðxÞÞ1=q dx: 0
Now we define the generating function Sðx; rÞ as the area
ð13Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
Sðx; rÞ ¼
Z y dx C
where C is the part of the level curve Ch : H ðx; yÞ ¼ h connecting the y-axis with point ðx; yÞ, oriented clockwise and h ¼ J 1 ðrÞ ¼: IðrÞ. This defines S up to an H integer multiple of r ¼ Ch y dx, the area of the domain surrounded by the closed curve Ch . We define the map ðr; hÞ ! ðx; yÞ by y¼
oS ðx; rÞ; ox
h¼
oS ðx; rÞ or
ð14Þ
which is known to be symplectic since dx ^ dy ¼ dx ^ ðSxx dx þ Sxr drÞ ¼ Sxr dx ^ dr; dh ^ dr ¼ ðSrx dx þ Srr drÞ ^ dr ¼ Srx dx ^ dr: Lemma 2. Under the transformation ðr; hÞ ! ðx; yÞ, system (10) is changed into dr ¼ h1 ðr; hÞ þ h2 ðr; h; tÞ dt dh ¼ I 0 ðrÞ þ h3 ðr; hÞ þ h4 ðr; h; tÞ dt
ð15Þ
where h1 ðr; hÞ ¼ f ðxÞyxh h2 ðr; h; tÞ ¼ xh eðtÞ h3 ðr; hÞ ¼ f ðxÞyxr h4 ðr; h; tÞ ¼ xr eðtÞ
ð16Þ
IðrÞ ¼ J 1 ðrÞ and x ¼ xðr; hÞ, y ¼ yðr; hÞ are given implicitly by (14). Proof. Since the transformation (14) is symplectic, we have det D ¼ 1, where xr xh D¼ : yr yh It follows from (10) that dx dr dh ¼ xr þ xh ¼ uq ðyÞ; dt dt dt dy dr dh ¼ yr þ yh ¼ f ðxÞy gðxÞ þ eðtÞ: dt dt dt
ð17Þ
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
117
By using det D ¼ 1 and from (17), one obtains dr ¼ ½yh uq ðyÞ þ xh gðxÞ þ f ðxÞyxh xh eðtÞ; dt dh ¼ ½xr gðxÞ þ uq ðyÞyr f ðxÞyxr þ xr eðtÞ: dt
ð18Þ
From q
H ðx; yÞ ¼ jyj =q þ GðxÞ ¼ h ¼ IðrÞ we have jyj
q2
yyh þ gðxÞxh ¼ 0
jyj
q2
yyr þ gðxÞxr ¼ I 0 ðrÞ:
Substituting (19) into (18), we obtain (15) and (16).
ð19Þ
It is not difficult to verify that x is odd in h and y is even in h, hence we have h1 ðr; hÞ ¼ h1 ðr; hÞ h2 ðr; h; tÞ ¼ h2 ðr; h; tÞ h3 ðr; hÞ ¼ h3 ðr; hÞ
ð20Þ
h4 ðr; h; tÞ ¼ h4 ðr; h; tÞ which implies that system (15) is reversible with respect to the involution M : ðr; hÞ ! ðr; hÞ. Lemma 3. The following inequalities hold pffiffiffiffiffi pffiffiffiffiffi 2 2hxþ 6 J ðhÞ 6 4 2hxþ 8 < ch1þð1=qÞb 6 J ðhÞ 6 Ch1þð1=qÞa ch1 J ðhÞ 6 J 0 ðhÞ 6 Ch1 J ðhÞ : 1 0 ch J ðhÞ 6 J 00 ðhÞ 6 Ch1 J 0 ðhÞ:
ð21Þ ð22Þ
Moreover, J 0 ðhÞ ! þ1 as h ! þ1. Proof. (1) One can prove the inequality (21) by comparing the area p bounded by ffiffiffiffiffi Ch respectively with the area of the triangle or rectangle with sides 2h and xþ . (2) From the expression Z xþ 1=q ðh GðxÞÞ dx J ðhÞ ¼ 4q1=q 0
set GðxÞ ¼ sh, s 2 ½0; 1. Then h ds=dh ¼ GðsÞ=gðsÞ and gðxÞ dx ¼ h ds, moreover,
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
J ðhÞ ¼ 4q1=q
Z
1
h1þð1=qÞ 0
ð1 sÞ1=q ds gðxÞ
which implies that J 0 ðhÞ ¼
4q1=q h
Z
xþ 0
1 þ w0 ðxÞÞðh GðxÞ q
1=q dx:
Similarly, we can obtain
Z xþ w0 ðxÞ 1 dx p 4 J 00 ðhÞ ¼ 1=p : 1=p q h 0 ðh GðxÞÞ Since w0 ðxÞ ¼ 1 GðxÞg0 ðxÞ=g2 ðxÞ, we obtain from the assumptions in (i) of Theorem 1 1 1 1 þ b h1 J ðhÞ 6 J 0 ðhÞ 6 1 þ a h1 J ðhÞ q q which implies ch1þð1=qÞb 6 J ðhÞ 6 Ch1þð1=qÞa and chð1=qÞb 6 J 0 ðhÞ 6 Chð1=qÞa hence, J 0 ðhÞ ! þ1 as h ! þ1. Similarly, one can prove the last inequalities of (22). Lemma 4. For k P 0, we have k k d o 0 k k 0 dhk J ðhÞ 6 Ch J ðhÞ; ork J ðhÞ 6 Cr J ðhÞ
ð23Þ
and k o k ork xðr; hÞ 6 Cr jxj;
k o k ork yðr; hÞ 6 Cr jyj:
ð24Þ
The proof of Lemma 4 is similar to the proof of [4, Lemma A 4.1], so we omit it. It follows from IðrÞ ¼ J 1 ðrÞ and (22) that crc1 6 IðrÞ 6 Crc2
ð25Þ 1
with c1 ¼ ð1 þ ð1=qÞ aÞ ; c2 ¼ ð1 þ ð1=qÞ bÞ crc1 1 6 I 0 ðrÞ 6 Crc2 1 :
1
and ð26Þ
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
119
Lemma 5. Let x, y be defined by (14). Then ox q=p ðr; hÞ ¼ L0 ðhÞjyj sgnðyÞ; oh
oy ðr; hÞ ¼ J 0 ðhÞgðxðr; hÞÞ: oh
ð27Þ
Proof. For y > 0, we have Z x 1=q Sðx; yÞ ¼ q1=q ðhðrÞ GðsÞÞ ds 0
oS ðx; rÞ ¼ or
Z
x
qð1=qÞ1 ðhðrÞ GðsÞÞð1=qÞ1 ds h0 ðrÞ ¼ h:
0
Hence o2 S ox 1=q 0 ðx; rÞ ¼ q1=q ðhðrÞ GðxÞÞ h ðrÞ ¼ 1 or oh oh which yields ox 1=q 1 ðr; hÞ ¼ q1=q ðhðrÞ GðxÞÞ ðh0 ðrÞÞ ¼ J 0 ðhÞy q=p : oh Similarly, we have for y < 0 ox ¼ J 0 ðhÞjyjq=p : oh From above result and jyjq =q þ GðxÞ ¼ h, we obtain oy ox jyjq2 y þ gðxÞ ¼ 0 oh oh hence oy ¼ J 0 ðhÞgðxÞ: oh Lemma 6. For k P 0, m P 0, we have k o k d1 ork h1 ðr; hÞ 6 Cr I ðrÞ kþm o k 1a ork otm h2 ðr; h; tÞ 6 Cr I ðrÞ k o 6 Crk1 I d1 ðrÞ h ðr; hÞ 3 ork kþm o 6 Crk1 I 1a ðrÞ h ðr; h; tÞ ork otm 4 where d1 ¼ 2 þ ð1=qÞ a ð1 bÞs.
ð28Þ
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
Proof. For k ¼ 0, by the expression of h1 ðr; hÞ h1 ðr; hÞ ¼ f ðxÞyxh we obtain by applying (iii) of Theorem 1, (21) and (27) s
jh1 ðr; hÞj 6 Cjxj GðxÞjyjjxh j s
6 Cjxj GðxÞh1=q J 0 ðhÞh1=p 6 Cjxjs GðxÞJ ðhÞ: From (8), we have jxj
s
6 GðxÞ
ðb1Þs
, therefore
jh1 ðr; hÞj 6 CGðxÞ1þðb1Þs J ðhÞ 6 Ch1þðb1Þs J ðhÞ 6 Ch2þð1=qÞaþðb1Þs ¼ CI d1 ðrÞ: For k P 1, we have k k o o h ðr; hÞ ¼ f ðxÞyx h ork 1 ork X i j l k o f ðxÞ o y o xh ¼ Cijl iþjþl¼k ori orj orl X oi f ðxÞ oj y ol xh 6C ori orj orl : iþjþl¼k Since oi f ðxÞ=ori is a sum of terms of X m6i
Cim
dm oi1 x oim x f ðxÞ
im ; ori1 or dxm
i1 þ ; im ¼ i;
we obtain by using (iii) and (24) i o f ðxÞ ms GðxÞri1 jxj rim jxj ori 6 Cjxj 6 Cri jxjs GðxÞ: From (24), j o y j j 1=q orj 6 Cr jyj 6 Cr h :
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
121
Finally, by (27) l l o xh o 0 q=p ¼ J ðhÞjyj orl orl s t Xo o 0 jyjq=p 6C J ðhÞ ors ort sþt¼l
6 Crs J 0 ðhÞrt jyj
q=p
6 Crl J 0 ðhÞh1=p : Combining above results, we obtain k o 6 Crk jxjs GðxÞh1þð1=qÞa 6 Crk I d1 ðrÞ: h ðr; hÞ ork 1 From the definition of h2 ðr; h; tÞ, by (22) and (27), we have for k ¼ 0, m P 0 m m o d ¼ jx h ðr; h; tÞ j eðtÞ h m otm 2 dt 6 Cjxh j 6 CJ 0 ðhÞh1=p 6 Ch1a ¼ CI 1a ðrÞ: For k P 1 k m kþm o d o k otm h2 ðr; h; tÞ ¼ ork xh ðr; hÞ dtm eðtÞ k o 6 C k xh ðr; hÞ or 6 Crk J 0 ðhÞh1=p 6 rk Ch1a ¼ Crk I 1a ðrÞ: Similarly, we can prove the rest inequalities in (28).
3. Further transformations In this section, we first introduce a series of M-invariant transformations such that in the transformed system, the growth speed of the term corresponding to h1 ðr; hÞ is the same as that of term of h2 ðr; h; tÞ if the action variable satisfies r 1. Then we change the role of angle and time variables.
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Lemma 7. There exists a M-invariant diffeomorphism T , having the form T : r ¼ q þ U ðq; hÞ;
h¼h
where U ðq; hÞ satisfies k o kþ1c1 d1 I ðqÞ; oqk U ðq; hÞ 6 Cq
k P 0:
Under this transformation, system (15) is transformed into the system dq ¼ h1 ðq; hÞ þ h2 ðq; h; tÞ dt dh ¼ I 0 ðqÞ þ h3 ðq; hÞ þ h4 ðq; h; tÞ dt
ð29Þ
h3 and h4 satisfy the same estimates as h2 , h3 and h4 in Lemma 6 where h2 , respectively and h1 satisfies k o kd2 d1 I ðqÞ ð30Þ oqk h1 ðq; hÞ 6 Cq where d2 ¼ c1 c2 d1 > 0. Moreover, system (29) is reversible with respect to the involution M : ðq; hÞ ! ðq; hÞ. Proof. Define a transformation S by S : q ¼ r þ V ðr; hÞ;
h¼h
ð31Þ
where V ðr; hÞ ¼
Z 0
h
h1 ðr; sÞ ds : I 0 ðrÞ
Since h1 ðr; hÞ ¼ h1 ðr; hÞ and jðok =ork Þh1 ðr; hÞj 6 Crk I d1 ðrÞ, jðo =orm ÞðI 0 ðrÞÞ1 j 6 Crmþ1c1 for k P 0, m P 0, one obtains
and
m
(i) V ðr; hÞ ¼ V ðr; hÞ; (ii) the transformation S is a diffeomorphism and satisfies k o 6 Crkþ1c1 I d1 ðrÞ: V ðr; hÞ ork
ð32Þ
Let T ¼ S1 : r ¼ q þ U ðq; hÞ, h ¼ h. Then system (15) is transformed into (29) with
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
oV oV h3 þ h1 h1 ðq; hÞ ¼ oh or oV oV h4 þ h2 þ h2 h2 ðq; h; tÞ ¼ oh or 0 0 h3 ðq; hÞ ¼ I ðqÞ I ðrÞ þ h3 h4 ðq; h; tÞ ¼ h4 ðr; h; tÞ
123
ð33Þ
with r ¼ q þ U ðq; hÞ. From the expression of h1 and (32), we obtain h1 ðq; hÞ 6 Crkc1 I 2d1 ðrÞ 6 Crkðc1 c2 d1 Þ I d1 ðrÞ 6 Crkd2 I d1 ðrÞ: Moreover, from the following lemma, system (29) is reversible with respect to involution M : ðq; hÞ ! ðq; hÞ. Lemma 8. For q 1, U satisfies i(i) jðok =oqk ÞU ðq; hÞj 6 Cqkþ1c1 I d1 ðrÞ; (ii) U ðq; hÞ ¼ U ðq; hÞ. The proof of Lemma 8 is similar to that of [6, Lemma 3] so we omit it. Now applying the transformation T n times with nd2 þ c2 d1 6 ð1 aÞc2 or equivalently n P ½c2 ðd1 1 þ aÞ=d2 where [A] denotes the integer part of A, (If d1 6 ð1 aÞ, we do not need the transformation T ) then the transformed term h1 satisfies k o k 1a ð34Þ oqk h1 ðq; hÞ 6 Cq I ðrÞ: 2 , h 3 and The terms of h h4 satisfy (28). tÞ ¼ tÞ, h~3 ðq; hÞ ¼ h3 ðq; hÞ, Set h ¼ h, h~1 ðq; hÞ, h~2 ðq; h; h2 ðq; h; tÞ ¼ h4 ðq; h; tÞ. Then system (29) is of the form h~4 ðq; h; dq ~ þ h~2 ðq; h; tÞ ¼ h1 ðq; hÞ dt ð35Þ dh 0 ~ ~ ¼ I ðqÞ þ h3 ðq; hÞ þ h4 ðq; h; tÞ: dt It is evident that h~i , i ¼ 1; 2; 3; 4, satisfies the same estimates as hi respectively. Moreover, system (35) is reversible with respect to the involution
124
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
! ðq; hÞ. For simplicity, we still denote h~ by h and h~i by hi , M : ðq; hÞ i ¼ 1; 2; 3; 4 in (35). Set f0 ðqÞ ¼ ðI 0 ðqÞÞ
1
h3 ðq; hÞ I 0 ðqÞðI 0 ðqÞ þ h3 ðq; hÞÞ h4 ðq; h; tÞ f2 ðq; h; tÞ ¼ 0 ðI ðqÞ þ h3 ðq; hÞÞðI 0 ðqÞ þ h3 ðq; hÞ þ h4 ðq; h; tÞÞ f3 ðq; h; tÞ ¼ ðh1 ðq; hÞ þ h2 ðq; h; tÞÞðf0 ðqÞ þ f1 ðq; hÞ þ f2 ðq; h; tÞÞ:
f1 ðq; hÞ ¼
ð36Þ
Then it is not difficult to verify that system (35) is equivalent to the following system dt ¼ f0 ðqÞ þ f1 ðq; hÞ þ f2 ðq; h; tÞ dh dq ¼ f3 ðq; h; tÞ: dh
ð37Þ
Moreover, one can verify by (36) that system (37) is reversible with respect to the involution M : ðq; tÞ ! ðq; tÞ. From the estimates (28), (34), and (36), one obtains the following estimates. For k þ m P 0, we have cq1c2 6 f0 ðqÞ 6 Cq1c1 k d f0 ðqÞ kþ1c2 6 Cqkþ1c1 cq 6 dqk k o kþ12c þc d 1 2 1 Cq f ðq; hÞ oqk 1 kþm o kþð1aÞc 1 2 Cq f ðq; h; tÞ oqk otm 2 kþm o kþð1aÞc þ1c 2 1 : oqk otm f3 ðq; h; tÞCq
ð38Þ
4. Proof of Theorem 1 Now we consider system (37) where fi , i ¼ 0; 1; 2; 3, satisfies (38). Set k ¼ f0 ðqÞ, h ¼ h and
X. Yang / Appl. Math. Comput. 152 (2004) 111–126
125
g1 ðk; hÞ ¼ f1 ðqðkÞ; hÞ g2 ðk; h; tÞ ¼ f2 ðqðkÞ; h; tÞ g3 ðk; h; tÞ ¼ f3 ðqðkÞ; h; tÞf00 ðqðkÞÞ
ð39Þ
where q ¼ qðkÞ is the inverse of k ¼ f0 ðqÞ. It is evident that q 1 if and only if k 1 by the first two inequalities in (38) and 1 c2 > 0. The transformed system of (37) is dt ¼ k þ g1 ðk; hÞ þ g2 ðk; h; tÞ dh dk ¼ g3 ðk; h; tÞ: dh
ð40Þ
Lemma 9. The function gi , i ¼ 1; 2; 3, satisfies the following estimates k o 6 Ckkþ1e1 g ðk; hÞ okk 1 kþm o k1ððac2 Þ=ð1c2 ÞÞ okk otm g2 ðk; h; tÞ 6 Ck kþm o ke2 okk otm g3 ðq; h; tÞ 6 Ck
ð41Þ
for k þ m P 0, where e1 ¼
2c1 c2 c2 d1 > 0; 1 c2
e2 ¼
2c1 1 c2 þ ac2 > 0: 1 c1
ð42Þ
Moreover, system (40) is reversible with respect to the involution M : ðk; tÞ ! ðk; tÞ. Proof. Inequalities (41) are direct consequences of (38), inequalities (42) are consequences of the assumptions in (i), (iii) of Theorem 1 and Remark 1. The last statement is the result of Lemma 1. Proof of Theorem 1. Clearly, if kð0Þ ¼ k 1, the solutions of (40) do exist on 0 6 h 6 1 and are uniquely determined with respect to the initial value. By Lemma 9, the Poincare map of solutions of (40) is of the form ðk; tÞ ! ðk1 ; t1 Þ with P : k1 ¼ k þ P1 ðk; tÞ; t1 ¼ t þ sðkÞ þ P2 ðk; tÞ R1 where sðkÞ ¼ k þ 0 g1 ðk; hÞ dh. Moreover, there exists an e0 > 0 such that for 0 6 k þ m 6 4 and k 1,
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X. Yang / Appl. Math. Comput. 152 (2004) 111–126
kþm o e0 okk otm Pi ðk; tÞ 6 k ;
i ¼ 1; 2; for some 0 > 0
hold and the map P is reversible with respect to the involution M : ðk; tÞ ! ðk; tÞ. It follows from [10, Theorem 1.1] that the map P possesses a sequence of invariant circles tending to infinity. Hence, in the original system (10), there exists a corresponding sequence of invariant tori in phase space ðx; x0 ; tÞ 2 R2 S 1 . Since any solution of system (10) must stay within one of those tori, it is therefore bounded. Theorem 1 is thus proved.
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