Existence of invariant tori for differentiable Hamiltonian vector fields without action-angle variables

Chaos, Solitons & Fractals 68 (2014) 114–122

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Chaos, Solitons & Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Existence of invariant tori for differentiable Hamiltonian vector fields without action-angle variables Wu Hwan Jong ⇑, Jin Chol Paek Faculty of Mathematics, KIM IL SUNG University, Pyongyang, Democratic People’s Republic of Korea

a r t i c l e

i n f o

Article history: Received 11 March 2013 Accepted 4 August 2014 Available online 7 September 2014

a b s t r a c t We proved the theorem for existence of invariant tori in differentiable Hamiltonian vector fields without action-angle variables. It is a generalization of the result of de la Llave et al. [4] that deals with analytic vector fields. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction In this paper, we provide a KAM theorem on existence of invariant tori with a Diophantine vector for differentiable Hamiltonian vector fields. We studied differentiable Hamiltonian vector fields which may not be perturbations of integrable vector fields or may not be written in action-angle variables. The existence problem of invariant tori for Hamiltonian system is often appeared in the study of stability problem of system of planets in celestial mechanics [3,6,13], the calculation of long time stability region in particle accelerator [10], the analysis of relaxation times in analysis of relaxation phenomena in molecular dynamics [2] and the analysis of the complicated orbit structure exhibited by several classes of population models [7] and so on. Kolmogorov [12] has proposed the procedure which clarifies the existence of invariant tori for perturbed analytical Hamiltonian vector field with action-angle variables at first and Arnold [1] has given the rigorous proof based on Kolmogorov’s procedure. Moser [14,15] has proved the existence of invariant tori for analytical areapreserving twist mappings on 2-dimensional annulus with action-angle variables and moreover he has relaxed the assumption of analyticity of the map to C 333 differentiability. After that, Rüssmann [17] has relaxed the

⇑ Corresponding author. E-mail address: [email protected] (W.H. Jong). http://dx.doi.org/10.1016/j.chaos.2014.08.002 0960-0779/Ó 2014 Elsevier Ltd. All rights reserved.

differentiability condition with the existence of invariant tori in Hamiltonian system to C 5 class and Takens [18] has clarified that it is not enough for C 1 class. Finally Herman [11] has clarified that it is enough for C 3 class but not to C 2 -mappings whose second derivatives belong to the Hölder class C 1d where d > 0 is small. However these are all the results for the Hamiltonian systems which are perturbations of some integrable systems and written in action-angle variables. Readers can refer to good expository article [5] for KAM theory. On the other hands, action-angle variables have singularity at elliptic fixed points or in neighborhood of separatrix and the use of action-angle variables are too restrictive in the case of numerical analysis. Moreover, in many practical applications, we have to consider the system which is not near to integrable one but has approximate invariant tori with sufficiently small error [4]. de la Llave et al. [4] neither proved the existence of invariant tori for analytic Hamiltonian systems which are neither perturbed integrable systems nor written in action-angle variables. Haro and de la Llave [9] applied this result in the numerical computation of invariant tori. Gonzalez-Enriquez and de la Llave [8] considered finitely differentiable symplectic maps without actionangle variables. In this paper, we prove that there exists true invariant tori nearby approximately invariant tori for differentiable Hamiltonian vector fields which are neither perturbed integrable vector fields nor written in action-angle variables.

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2. Some definitions and the existence of invariant tori in the case of analytic Hamiltonian vector field Definition 1. Given c > 0 and r > n  1, we define Dn ðc; rÞ as the set of frequency vectors x ¼ ðx1 ; . . . ; xn Þ 2 Rn satisfying the Diophantine condition:

jk  xj P cjkjr ;

8k 2 Zn  f0g

where jkj ¼ jk1 j þ . . . þ jkn j. Let U q denote the complex closed strip of width q > 0, that is, U q ¼ fh 2 Cn ; jImhj 6 qg. Let B is a compact subset of Rn which is included in closure of its interior. Given a function g 2 C m ðBÞ, for m 2 Zþ ¼ N [ f0g we will denote the C m -norm of g on B by jgjC m ;B . Given a 1-periodic function n

n

n

K, continuous on T ¼ R =Z , we denote the average of K on Tn by

hKi ¼

Z Tn

@ x KðhÞ ¼ J rHðKðhÞÞ;

def

jf jC 0 ðUÞ ¼ supjf ðxÞj < 1: x2U

For l 2 N; C l ðUÞ denotes the space of functions f : U ! R with continuous derivatives up to order l such that def

jf jC l ðUÞ ¼ supfjDk f ðxÞjg < 1:

n X i¼1



xi

Let l ¼ p þ a with p 2 Zþ and 0 < a < 1. Define the Hölder space C l ðUÞ to be the set of all functions f : U ! R with continuous derivatives up to order p for which def

jf jC l ðUÞ ¼ jf jC p

( ) jDk f ðxÞ  Dk f ðyÞj < 1: þ sup x;y2U;x–y jx  yja jkj¼p

ð1Þ

 0 In ; In is n-dimensional unit matrix and In 0 is the derivative in direction x:

where J ¼

@xK ¼

Definition 2. Let Zþ denote the set of positive integers. Given U  Cd an open set, C 0 ðUÞ denotes the space of continuous functions f : U ! R such that

x2U jkj6l

KðhÞdh:

If X is an open subset of topological space of M then we  denote this fact by X  M. We will assume that U is either   n n T  U with U  R or U  R2n . The results for a Hamiltonian vector field with differential Hamiltonian function H : U ! R are based on the study of the equation

@x

@ x KðhÞ is a tangent vector of M ¼ KðTn Þ at z ¼ KðhÞ. And J rHðzÞ ¼ X H ðzÞ implies that J rHðzÞ is a vector of Hamilton vector field X H at z ¼ KðhÞ. Hence if (1) satisfies, then X H ðzÞ 2 T z M. Therefore KðTn Þ is a invariant under the Hamiltonian vector field with Hamiltonian function H; X H ¼ J rH. h

@K : @hi

In the above K : Tn ! U is the function to be determined and x 2 Dðc; rÞ  Rn . Proposition 1. Note that if the solution of (1) is embedding, then the KðTn Þ is invariant under the Hamiltonian vector field with Hamiltonian H; X H ¼ J rH. Proof. Consider the submanifold M ¼ KðTn Þ  U. We are enough to prove that X H ðzÞ 2 T z M; ð8z 2 MÞ. Take z 2 M arbitrarily and let z ¼ KðhÞ for some h 2 Tn . The tangent bundle of M ¼ KðTn Þ at z ¼ KðhÞ is given as T z M ¼ T h KðRn Þ, that is, for any tangent vector v 2 T z M, there exists a x 2 T h Tn ¼ Rn such that v 2 T h KðxÞ. Then from the following equality

1 DK 1 ðhÞx C B .. C ¼ T h KðxÞ ¼ DKðhÞx ¼ B . A @ DK 2n ðhÞx 1 0X n @K 1 ðhÞ x i C B @h C B i¼1 i n C X B @K C B . ¼B ðhÞxi ¼ @ x KðhÞ; C¼ .. C i¼1 @hi B C BX n A @ @K 2n ðhÞxi @hi

Definition 3. Let l P 0. Given U # Rd open, define the Banach space to be the set of all holomorphic functions f : U þ q ! C which are real-valued on U and such that

jf jC l ðUþqÞ < 1: For a matrix or vector-valued function G with components Gi; j in either C l ðUÞ or in AðU þ q; C l Þ we use the norm, respectively, def

def

jGjC l ðUÞ ¼ max jGi;j jC l ðUÞ or jGjC l ðUþqÞ ¼ max jGi;j jC l ðUþqÞ . i;j

i;j

The space of all functions g ¼ ðg 1 ; . . . ; g d Þ : U # Cn ! V # Cd such that g i 2 C l ðUÞ, for i ¼ 1; . . . ; d is denoted by C l ðU; VÞ. Definition 4. Given U # Cn , for x; y 2 U denote by dU ðx; yÞ the minimum length of arcs inside U joining x and y. We say that U is compensated if there exists a constant cU such that dU ðx; yÞ 6 C U jx  yj, for all x; y 2 U.

0

v

i¼1

Definition 5. Let # : Rd ! R be a function with continuous derivatives up to order m, for some m 2 N, and assume that grad#ðxÞ – 0 for all x 2 fx : #ðxÞ ¼ 0g. The set U ¼ fx 2 Rd : #ðxÞ 6 0g is called a closed domain with C m -boundary. An open domain is defined by fx 2 Rd : #ðxÞ < 0g. Definition 6. Given Hamiltonian H 2 C 1 ðUÞ and frequency vector x 2 Dn ðc; rÞ; K 2 C 1 ðTn ; UÞ is said to be non-degenerate if it satisfies the following two conditions:

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(1) jðSt  IdÞ½f jC r ðUÞ 6 jjf jC l ðUÞ tlþr ; 0 6 r < l.

(1) There exists a n  n matrix-valued function NðhÞ satisfying

(2) jSt ½f jC 0 ðUþCt1 Þ 6 jjf jC 0 ðUÞ . (3) jðSs  St Þ½f jC 0 ðUþs1 Þ 6 jjf jC l ðUÞ tl ;

NðhÞDKðhÞT DKðhÞ ¼ In :

s P t.

(4) jImðSt ½f ÞjC 0 ðUþCt1 Þ 6 jCt 1 jf jC l ðUÞ .

(2) The average of S; hSi is invertible with

SðhÞ ¼ NðhÞDKðhÞT ½AðhÞJ  JAðhÞ  DKðhÞNðhÞ Proof. See Proposition 23 in [8].

where

AðhÞ ¼ DX H ðKðhÞÞ ¼



Dx ry HðKðhÞÞ Dy ry HðKðhÞÞ Dx rx HðKðhÞÞ  Dy rx HðKðhÞÞ

 :

Proposition 3. Linear smoothing operator St have the following properties.

Theorem 1 (Theorem 5 in [4]). Let x 2 Dðc; rÞ. Assume that K 0 2 AðTn þ q; C 0 Þ is 1-periodic in Tn þ q and non-degenerate. Assume that H is real analytic and that is can be holomorphically extended to some neighborhood of the image of U q under K 0 :

 Br ¼ Br ðK 0 Þ :¼

(1) St commutes with constant coefficient differential operators. (2) St acts as the identity on polynomials. (3) St takes periodic functions into periodic functions. (4) St ½f ðxÞ 2 R for all x 2 Rd .

 z 2 Rn ; inf jz  K 0 ðhÞj < r : h2U q

h

Proof. See Remark 7 in [8].

h

Define the error function by

e0 ðhÞ ¼ J rHðK 0 ðhÞÞ  @ x K 0 ðhÞ: There exists a constant c > 0, depending on r; n; r; q; jHjC 3 ðBr Þ ; jDK 0 jC 0 ðTn þqÞ ; jN 0 jC 0 ðTn þqÞ ; jhE0 i1 j such that if

cc4 q4r je0 jC 0 ðTn þqÞ < 1 and

Lemma 2. Let 1 < l < m; m 2 N and U be either a compensated bounded open domain in Rd with C m -boundary, or Tn  U, with U  Rdn , a compensated bounded open domain with C m -boundary (n < dÞ. Then for each C P 0; 0 6 l < l and r 2 ð0; 1Þ, with 0 < r þ l < l, there exists a constant j0 ¼ j0 ðd; l; C; l; rÞ, such that for all t P e1=r satisfying

t1 ðC þ r logðtÞÞ 6 1;

cc2 q2r je0 jC 0 ðTn þqÞ < r; then there exists a solution for (1), K 1 2 AðTn þ q=2; C 0 Þ, satisfies the non-degenerate conditions. Moreover

jK 1  K 0 jC 0 ðTn þqÞ 6 cc2 q2r je0 jC0 ðTn þqÞ : 2

the following inequality holds:

jSt ½f jC l ðUþCt1 Þ 6 j0 jf jC l ðUÞ ;

f 2 C l ðUÞ:

Proof. See Lemma 33 in [8].

h

Proposition 2. The dependence of constant c on jHjC 3 ðBr Þ ;

jDK 0 jC 0 ðTn þqÞ ; jN 0 jC 0 ðTn þqÞ ; jhE0 i1 j is polynomial. That is,

Remark 1. For convenience, we will denote maxfj; j0 g by

there exists a polynomial, aðy1 ; y2 ; y3 ; y4 Þ with positive coefficients depending on r; n and such that

j.

c ¼ aðjHjC3 ;Br ; jDK 0 jC0 ðTn þqÞ ; jN0 jC0 ðTn þqÞ ; jhE0 i1 jÞ:

Lemma 3. Let 1 < l < m; m 2 N and U be either a compensated bounded open domain in Rd with C m -boundary, or Tn  U, with U  Rdn , a compensated bounded open domain with C m -boundary(n < dÞ. And let V be either a compensated bounded open domain in Rp with C m -boundary, or Ts  V, with V  Rps , a compensated bounded open domain with C m - boundary. Given the real numbers C P 0; b > 0 and 0 < l < l, there exist two constants j00 ¼ j00 ðd; p; l; C; l; bÞ and t0 ¼ t0 ðp; l; V; C; b; lÞ such that for each h 2 C l ðVÞ and f 2 C l ðU; VÞ, satisfying (i) the closure of f ðUÞ is contained in V and (ii) jf jC l ðUÞ 6 b, the following holds for all t P t1 :

Proof. See Remark 15 in [4].

h

3. The existence of invariant tori in the case of differentiable Hamiltonian vector field To prove the existence of invariant tori in the case of differentiable Hamiltonian, we use some approximation theorems. Lemma 1. Let 1 < l < m; m 2 N and U be either a compensated bounded open domain in Rd with C m -boundary, or Tn  U, with U  Rdn , a compensated bounded open domain with C m -boundary(n < dÞ. Then there exists a linear smoothing operator St : C l ðUÞ ! AðU þ t 1 ; C 0 Þ, for any C P 0, there exists a constant j ¼ jðd; l; CÞ such that for all t P 1 and f 2 C l ðUÞ the following holds:

^ l ; jSt ½h  St ½f   St ½h  f jC 0 ðUþCt1 Þ 6 j00 Mt

t P t0

where

^ ¼ jhj l ð1 þ jf js l Þ þ jhj l jf j l M C ðVÞ C ðVÞ C ðUÞ C ðUÞ and s is any number in ðl; 1Þ, if 0 < l < 1 < l and s ¼ l, if 1 6 l.

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Proof. See Lemma 37 in [8].

From Lemma 3, there exists a sufficiently large t2 ðt 2 > t1 Þ, such that for all t > t 2 ,

h

Lemma 4. Let l 2 N. If a sequence of functions ff k ðxÞg analytic on U þ q=2k1 satisfies the following inequality l

jf kþ1 ðxÞ  f k ðxÞjC0 ðTn þq=2k Þ 6 Að2k Þ ;

jDSt ½KjC 0 ðTn þt1 Þ 6 jjKjCl ðTn Þ ¼ r

ð4Þ

holds and from (4) of Lemma 1, we have

jImðSt ½KÞjC 0 ðTn þt1 Þ 6 jt 1 jKjC l ðTn Þ 6 t 1 r:

where A P 0 is a suitable constant, then f k ðxÞ converges to certain function f 2 C 1 ðTn Þ.

Hence

St ½K : Tn þ t 1 ! Ut þ t 1 r: Proof. See Lemma 1 of Chapter 3, Section 7 in [16].

Theorem 2. Let x 2 Dðc; rÞðr > n  1Þ; 4r þ 3 < l < m0 and U be either a compensated bounded open domain in R2n with C m -boundary, or Tn  U, with U  Rn , a compensated bounded open domain with C m -boundary. And let H 2 C l ðUÞ and K 2 C l ðTn ; UÞ is non-degenerate and g ¼ distðKðTn Þ; @UÞ=2 > 0. Define

Then there exist two constants c > 0 and q < 1 depending on r; n; l; jHjC l ðTn Þ ; jKjC l ðTn Þ ; jNjC l ðTn Þ ; jhE0 i1 j such that given 0 < q1 < q if ð4rþ1Þ

je0 jC0 ðTn Þ < minf1; r; gg;

ð2Þ

then there exists a solution of (1), K 1 2 C lð2rþ1Þ ðTn ; UÞ such that ð2rþmÞ

jK  K 1 jCm ðTn Þ 6 ~cc2 q1

jDSt ½KðhÞT DSt ½KðhÞ  NðhÞ1 jC0 ðTn Þ ¼ jDSt ½KðhÞT DSt ½KðhÞ  DKðhÞT DKðhÞjC 0 ðTn Þ 6 jDSt ½KðhÞT DSt ½KðhÞ  DSt ½KðhÞT DKðhÞjC 0 ðTn Þ þ jDSt ½KðhÞT DKðhÞ  DKðhÞT DKðhÞjC0 ðTn Þ 6 jDSt ½KðhÞT jC 0 ðTn Þ  jDSt ½KðhÞ  DKðhÞjC 0 ðTn Þ þ jDSt ½KðhÞT

e0 ðhÞ ¼ J rHðKðhÞÞ  @ x KðhÞðh 2 Tn Þ:

cc4 q1

Now we will prove the non-degeneracy of St ½K. Performing some computation, we have

h



ql1 1 þ je0 jC 0 ðTn Þ



 DKðhÞT jC 0 ðTn Þ  jDKðhÞjC 0 ðTn Þ 6 jSt ½DKðhÞT jC 0 ðTn Þ  jSt ½KðhÞ  KðhÞjC 1 ðTn Þ þ jSt ½KðhÞT  KðhÞT jC 1 ðTn Þ  jDKðhÞjC 0 ðTn Þ 6 jjDKðhÞjC0 ðTn Þ jjKðhÞjCl ðTn Þ t lþ1 þ jjKðhÞjC l ðTn Þ tlþ1 jDKðhÞjC0 ðTn Þ 6 j2 jKðhÞj2Cl ðTn Þ t lþ1 : Hence if we take sufficiently large t 3 > t2 such that

for all 0 6 m < l  ð2r þ 1Þ. Here r ¼ jjKjC 0 ðTn Þ and j is a constant of Lemma 1.

j jKðhÞj2Cl ðTn Þ jNjC0 ðTn Þ tl 3 6 1=2; 2

for all t > t3 , the following inequality holds: Proof. First Step: We obtain a couple of analytic approximate function of H and analytic approximate non-degenerate function of K which will satisfy the assumption of Theorem 1. Consider St ½H and St ½K, the approximate analytic functions of H; K. This couple will satisfy our demand for sufficiently large t. From the property (1) of St , we have

jDSt ½KðhÞT DSt ½KðhÞ  NðhÞ1 jC0 ðTn Þ  jNjC 0 ðTn Þ 6 1=2: The Neumann’s series theorem implies that DSt ½KðhÞT DSt ½KðhÞ is invertible for all h 2 Rd and its inverse, denoted by N t , satisfies

jNt  NjC 0 ðTn Þ 6 2jjKj2Cl ðTn Þ jNj2C0 ðTn Þ t lþ1 6 jNjC0 ðTn Þ : n

Now, let h 2 R þ t , Lemma 2 implies, for all t P t3

jSt ½K  KjC 0 ðTn Þ 6 jjKjC 0 ðTn Þ tl :

jDSt ½KðhÞ  DSt ½KðReðhÞÞj 6 jD2 St ½KjC 0 ðTn þt1 Þ jImðhÞj

Hence if t 1 is sufficiently large such that

6 jjKjCl ðTn Þt1 :

1 maxfjjKjC 0 ðTn Þ tl 1 ; t 1 g 6 g=2;

So we obtain

for all t > t1 , the following inequality holds:

jSt ½K  KjC 0 ðTn Þ 6

g 2

2 1 jDSt ½KT ðhÞDSt ½KðhÞ  N1 t ðReðhÞÞj 6 jjKjC l ðTn Þ t :

ð3Þ

:

2j jKj2C l ðTn Þ jNjC 0 ðTn Þ t1 6 1=2;

St ½KðTn Þ  Ut ;

for all t > t4 , (4) implies

where

Ut ¼

If we take sufficiently large t4 > t 3 such that 0

And therefore

(

ð5Þ

1

j0 jKj2Cl ðTn Þ jNt jC0 ðTn Þ t1 6 1=2: 2n

 t 1 Þ  Ug; U  R ; fx 2 U : Bðx; n  t1 Þ  Ug; U ¼ T n  UðU  Rn Þ: T  fx 2 U : Bðx;

Hence, from the Neumann’s series theorem, DSt ½KðhÞT DSt ½KðhÞ is invertible for all h 2 Rn þ t 1 and it satisfies

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þ jNðhÞDKðhÞT ½AðhÞJ  JAðhÞðDSt ½KðhÞN t ðhÞ

jNt jC 0 ðTn þt1 Þ 6 jNt jC 0 ðTn Þ þ 2jNt j2C0 ðTn Þ jjKj2Cl ðTn Þ t 1 :

 DKðhÞNðhÞÞjC 0 ðTn Þ 6 jMK tlþ1 2jjHjC l ðUÞ jjKjC l ðTn Þ 2jNjC 0 ðTn Þ

And from (5), we have

~ K 2jNj 0 n t 1 Þ; jNt jC 0 ðTn þt1 Þ 6 2jNjC 0 ðTn Þ ð1 þ jM C ðT Þ

ð6Þ

~ K ¼ 4jNj 0 n jKj2l n . where M C ðT Þ C ðT Þ Now we define At ðhÞ and Et ðhÞ as following: def

At ðhÞ ¼



Dx ry St ½HðSt ½KðhÞÞ

Dy ry St ½HðSt ½KðhÞÞ

þ jNjC 0 ðTn Þ jKjC 1 ðTn Þ 2jjHjC l ðUÞ jMK tlþ1 6 jMK;H tlþ2 ; where M K;H is a constant depending on jKjC l ðTn Þ ; jNjC 0 ðTn Þ ; jHjC l ðUÞ . Hence if we take sufficiently large t5 > t4 such that



Dx rx St ½HðSt ½KðhÞÞ Dy rx St ½HðSt ½KðhÞÞ

þ jNjC 0 ðTn Þ jKjC 1 ðTn Þ 2jjHjC l ðUÞ tlþ2 jjKjC l ðTn Þ 2jNjC 0 ðTn Þ

;

jMK;H  jhEi1 j  tlþ2 < 1=2; for all t > t 5 , the inequality

def

Et ðhÞ ¼ Nt ðhÞDSt ½KðhÞT ½At ðhÞJ  JAt ðhÞDSt ½KðhÞNt ðhÞ

jhEt  Eij  jhEi1 j < 1=2

and prove that hEt i is invertible for sufficiently large t. Performing some computation, we have

holds. From the Neumann’s series theorem, hEt i is invertible for all t > t5 and it satisfies

jDSt ½KðhÞNt ðhÞ  DKðhÞNðhÞjC 0 ðTn Þ 6 jDSt ½KðhÞNt ðhÞ  DSt ½KðhÞNðhÞjC 0 ðTn Þ þ jDSt ½KðhÞNðhÞ 6 jDSt ½KðhÞjC0 ðTn Þ  jN t ðhÞ  NðhÞjC 0 ðTn Þ þ jDSt ½KðhÞ

jJ rSt ½HðSt ½KðhÞÞ  @ x St ½KðhÞjC 0 ðTn þt1 Þ

 DKðhÞjC 0 ðTn Þ  jNðhÞjC 0 ðTn Þ 6 jjKjC 0 ðTn Þ 2j

ð9Þ

Therefore for all t P t5 ; St ½K is non-degenerate. And from Lemma 3, the following holds:

 DKðhÞNðhÞjC0 ðTn Þ

jKj2C l ðTn Þ jNj2C 0 ðTn Þ t lþ1

jhEt i1 j 6 jhEi1 j  ð1 þ jMK;H jhEi1 jtlþ2 Þ:

þ jjKjC l ðTn Þ t

lþ1

jNjC0 ðTn Þ

6 jð2jjKj3Cl ðTn Þ jNj2C0 ðTn Þ þ jKjCl ðTn Þ jNjC0 ðTn Þ Þt lþ1 6 jMK tlþ1 ;

6 jðJ rSt ½HðSt ½KðhÞÞ  St ½J rHðKÞðhÞÞjC 0 ðTn þt1 Þ þ jðSt ½J rHðKÞðhÞ  St ½@ x KðhÞÞjC 0 ðTn þt1 Þ ^ H;K tlþ1 þ jjðJ rHðKÞðhÞ  @ x KðhÞÞj 0 n : 6 j00 M C ðT Þ

ð10Þ

that is,

jDSt ½KðhÞNt ðhÞ  DKðhÞNðhÞjC 0 ðTn Þ 6 jM K tlþ1 ; n

jKj3C l ðTn Þ jNj2C 0 ðTn Þ

~ K ; 2j where M K ¼ max M

ð7Þ

o þ jKjC l ðTn Þ jNjC 0 ðTn Þ .

From the definition of AðhÞ,

 At ðhÞ  AðhÞ ¼

Second step: For sufficiently large t, a couple of St ½H and St ½K satisfies the assumption of Theorem 1 and, from this couple we obtain the approximate solution series inductively.

Dx ry St ½HðSt ½KðhÞÞ  Dx ry HðKðhÞÞDy ry St ½HðSt ½KðhÞÞ  Dy ry HðKðhÞÞ Dx rx St ½HðSt ½KðhÞÞ þ Dx rx HðKðhÞÞ  Dy rx St ½HðSt ½KðhÞÞ þ Dy rx HðKðhÞÞ

At first we prove the following proposition by induction.

and from Lemma 2, the following holds:

jDx ry St ½HðSt ½KðhÞÞ  Dx ry HðKðhÞÞjC 0 ðTn Þ 6 jDx ry St ½HðSt ½KðhÞÞ  Dx ry HðSt ½KðhÞÞjC 0 ðTn Þ

Proposition 4. For fixed m P 1, there exists a sm P 1 such that Hm ¼ Ssm ½H and K m ¼ Ssm ½K satisfies the following conditions.

þ jDx ry HðSt ½KðhÞÞ  Dx ry HðKðhÞÞjC 0 ðTn Þ 6 jjHjC l ðUÞ t lþ2 þ jHjC2 ðTn Þ jjKjCl ðTn Þ t l 6 jjHjCl ðUÞ t lþ2 : where we retake j as jð1 þ jKjC l ðTn Þ Þ. Similarly, estimating other element of At  A, we obtain

jAt  AjC 0 ðTn Þ 6 jjHjCl ðUÞ t lþ2 :

 ;

A1ðmÞ: K m 2 AðTn þ qm ; C 1 Þ; 6 rm

jDK m jC 0 ðTn þqm Þ ! m 1 X j 1 qm ¼ sm ; rm ¼ r 2 j¼0

ð8Þ

Using (7) and (8), the following holds: T

jEt  EjC l ðTn Þ ¼ jNt ðhÞDSt ½KðhÞ ½At ðhÞJ  JAt ðhÞDSt ½KðhÞNt ðhÞ

n

A2ðmÞ: K m ðT Þ  Usm ; jK m  KjC0 ðTn Þ 6 gm

m X gm ¼ g 2j j¼1

T

 NðhÞDKðhÞ ½AðhÞJ  JAðhÞDKðhÞNðhÞjC 0 ðTn Þ 6 jðN t ðhÞDSt ½KðhÞT  NðhÞDKðhÞT Þ½At ðhÞJ  JAt ðhÞDSt ½KðhÞN t ðhÞjC 0 ðTn Þ þ jNðhÞDKðhÞT ½At ðhÞJ  JAt ðhÞ  AðhÞJ  JAðhÞDSt ½KðhÞNt ðhÞjC 0 ðTn Þ

A3ðmÞ: Hm 2 AðUsm þ 2r m qm ; C 2 Þ; 6 jM H ðM H ¼ jHjC l ðTn Þ Þ

jf m jC 2 ðUs

m þ2rm qm Þ

!

W.H. Jong, J.C. Paek / Chaos, Solitons & Fractals 68 (2014) 114–122

A4ðmÞ: jNm jC 0 ðTn þqm Þ 6 2jNjC 0 ðTn Þ

m Y

r c4 km q4 m jem jC 0 ðTn þqm Þ < 1;

ð1 þ 2j Þ

j¼1 1

1

jhEm i j 6 jhEi j

m Y

119

r c2 km q2 m jem jC 0 ðTn þqm Þ < minfr m qm ; gg:

j

ð1 þ 2 Þ

Therefore from Theorem 1, there exists a non-degenerate function K mþ1 2 AðTn þ qmþ1 ; C 1 Þ such that

j¼1

where N m and Em are defined in Definition 6, by replacing Hm ; K m with H; K. Then there exists two constant ~ k and k depending on r; n; g; jHjCl ðTn Þ ; jNjC 0 ðTn Þ ; jhEi1 j such that, if

c4 ~kqmð4rþ1Þ jem jC0 ðTn þqm Þ < minf1; r; gg

ð11Þ

exists a non-degenerate K mþ1 2 AðTn þ qmþ1 ; C 1 Þ then there qm qmþ1 ¼ 2 such that

J rHm ðK mþ1 ðhÞÞ ¼ @ x K mþ1 ðhÞ;

ð12Þ

r jK mþ1  K m jC 0 ðTn þqmþ1 Þ 6 c2 ~kq2 m jem jC 0 ðTn þqm Þ

ð13Þ

and ð2rþ1Þ jDK mþ1  DK m jC0 ðTn þqmþ1 Þ 6 c2 ~kqm jem jC0 ðTn þqm Þ

ð14Þ

(where em ðhÞ ¼ J rHm ðK m ðhÞÞ  @ x K m ðhÞ). Furthermore, if Hmþ1 ¼ S2sm ½H and 2 ð2rþ1Þ 2mþ1 kðql1 jem jC 0 ðTn þqm Þ Þ < minf1; r; gg; m þ c qm

ð15Þ

then Hmþ1 and K mþ1 satisfy properties A1ðm þ 1Þ A4ðm þ 1Þ and the following inequality holds:

jemþ1 jC0 ðTn þqmþ1 Þ 6 jMH ql1 m :

ð16Þ

J rHm ðK mþ1 ðhÞÞ ¼ @ x K mþ1 ðhÞ; r jK mþ1  K m jC 0 ðTn þqmþ1 Þ 6 c2 km q2 m jem jC 0 ðTn þqm Þ

and ð2rþ1Þ jDK mþ1  DK m jC 0 ðTn þqmþ1 Þ 6 c2 km qm jem jC 0 ðTn þqm Þ :

k implies (13) and And above two inequalities and km < ~ (14). Next we prove that Hmþ1 and K mþ1 satisfy A1ðm þ 1Þ– A4ðm þ 1Þ under the condition (15). From (14) and (15), we obtain ð2rþ1Þ jDK mþ1 jC 0 ðTn þqmþ1 Þ 6 jDK m jC 0 ðTn þqm Þ þ c2 ~kqm jem jC 0 ðTn þqm Þ ð2rþ1Þ 6 r m þ c2 ~kqm jem jC0 ðTn þqm Þ

6 r m þ 2ðmþ1Þ r 6 rm þ 2m r 6 rmþ1 : Hence A1ðm þ 1Þ holds. Using (13), we obtain

jK mþ1  KjC 0 ðTn Þ 6 jK mþ1  K m jC0 ðTn Þ þ jK m  KjC 0 ðTn Þ r 6 c2 km q2 m jem jC 0 ðTn þqm Þ þ gm r 6 gm þ c2 ~kq2 m jem jC 0 ðTn þqm Þ rþ1Þ 6 gm þ c2 ~kqð2 jem jC0 ðTn þqm Þ m

Proof. First we prove the existence of K mþ1 2 AðTn þ qmþ1 ; C 1 Þ satisfying (12)–(14) under the properties A1ðmÞ–A4ðmÞ and (11). Define

 km ¼ a jHm jC 2 ðUs

m

 þ2r m qm Þ ;jDK m jC 0 ðTn þqm Þ ; jN m jC 0 ðTn þqm Þ ;jhEm i j ; 1

6 gm þ 2ðmþ1Þ g 6 gmþ1 ; so the property A2ðm þ 1Þ holds. A1ðm þ 1Þ implies

jImðK mþ1 ÞjC 0 ðTn þqmþ1 Þ 6 qmþ1 jDK mþ1 jC 0 ðTn þqmþ1 Þ 6 r mþ1 qmþ1 < 2r qmþ1 : From the above inequality and A2ðm þ 1Þ, we obtain

K mþ1 ðTn þ qmþ1 Þ  Usm þ 2r qm : 1

~k ¼ aðjM f ; 2r; 2ejNj 0 n ; ejhEi jÞ; C ðT Þ then, from the following inequalities

jHm jC2 ðUs

m þ2r m qm Þ

6 jMH ;

jDK m jC0 ðTn þqm Þ 6 r m ¼ r

m1 X

2

j

jDK mþ1 ðhÞT DK mþ1 ðhÞ  Nm ðhÞ1 jC 0 ðTn þqmþ1 Þ

< 2r;

j¼0

jNm jC 0 ðTn þqm Þ

m Y 6 2jNjC 0 ðTn Þ ð1 þ 2j Þ 6 2ejNjC 0 ðTn Þ ; j¼1

jhEm i1 j 6 2jhEi1 j

m Y

Hence Hmþ1  K mþ1 is well defined on Tn þ qmþ1 . A3ðm þ 1Þ follows from Lemma 2. Now we prove A4ðm þ 1Þ. Using A1ðm þ 1Þ and (14), we obtain

ð1 þ 2j Þ 6 2ejhEi1 jC0 ðTn Þ ;

j¼1

one can obtain

km < ~k (where a is a polynomial with positive coefficient in proposition 2). Hence (11) implies

¼ jDK mþ1 ðhÞT DK mþ1 ðhÞ  DK m ðhÞT DK m ðhÞjC 0 ðTn þqmþ1 Þ 6 jDK mþ1 ðhÞT DK mþ1 ðhÞ  DK mþ1 ðhÞT DK m ðhÞjC 0 ðTn þqmþ1 Þ þ jDK mþ1 ðhÞT DK m ðhÞ  DK m ðhÞT DK m ðhÞjC 0 ðTn þqmþ1 Þ 6 jDK mþ1 ðhÞT jC 0 ðTn þqmþ1 Þ  jDK mþ1 ðhÞ  DK m ðhÞjC 0 ðTn þqmþ1 Þ þ jDK mþ1 ðhÞT  DK m ðhÞT jC 0 ðTn þqmþ1 Þ  jDK m ðhÞjC 0 ðTn þqmþ1 Þ ð2rþ1Þ ð2rþ1Þ 6 2r  c2 ~kqm jem jC 0 ðTn þqm Þ þ c2 ~kqm jem jC 0 ðTn þqm Þ  2r ð2rþ1Þ 6 4r  c2 ~kqm jem jC 0 ðTn þqm Þ :

k¼~ k24 rejNjC 0 ðTn Þ and if Define ^ ð2rþ1Þ 2mþ1 c2 ^kqm jem jC0 ðTn þqm Þ 6 1

ð17Þ

120

W.H. Jong, J.C. Paek / Chaos, Solitons & Fractals 68 (2014) 114–122 2 ~ 2r jAmþ1  Am jC0 ðTn Þ 6 jM H ðql2 m þ jHjC l ðUÞ c kqm jem jC 0 ðTn þqm Þ Þ

then the following inequality holds ð2rþ1Þ 4r  c2 ~kqm jem jC0 ðTn þqm Þ jNm jC0 ðTn þqm Þ

ð20Þ

ð2rþ1Þ 6 4r  c2 ~kqm jem jC0 ðTn þqm Þ 2ejNjC 0 ðTn þqm Þ

Using (19) and (20), we obtain

ð2rþ1Þ 6 ~k24 rejNjC0 ðTn þqm Þ c2 qm jem jC 0 ðTn þqm Þ =2

jEmþ1  Em jC l ðTn Þ ¼ jNmþ1 ðhÞDK mþ1 ðhÞT ½Amþ1 ðhÞJ  JAmþ1 ðhÞDK mþ1 ðhÞNmþ1 ðhÞ

ð2rþ1Þ 6 ^kc2 qm jem jC 0 ðTn þqm Þ =2 6 1=2:

T

Hence from Neumann’s series theorem, N mþ1 exists and satisfies ð2rþ1Þ jNmþ1  Nm jC 0 ðTn þqmþ1 Þ 6 jN m jC 0 ðTn þqmþ1 Þ ^kc2 qm jem jC 0 ðTn þqm Þ :

ð18Þ Using (17) and (18), the first inequality of A4ðm þ 1Þ holds as following:

jNmþ1 jC 0 ðTn þqmþ1 Þ 6 jNmþ1 jC0 ðTn þqmþ1 Þ þ jNmþ1  Nm jC 0 ðTn þqmþ1 Þ 6 jNmþ1 jC0 ðTn þqm Þ ð1

6 jðN mþ1 ðhÞDK mþ1 ðhÞT  N m ðhÞDK m ðhÞT Þ½Amþ1 ðhÞJ  JAmþ1 ðhÞDK mþ1 ðhÞNmþ1 ðhÞjC0 ðTn Þ T

þ jN m ðhÞDK m ðhÞ ½Amþ1 ðhÞJ  JAmþ1 ðhÞ  Am ðhÞJ  JAm ðhÞDK mþ1 ðhÞNmþ1 ðhÞjC0 ðTn Þ þ jN m ðhÞDK m ðhÞT ½Am ðhÞJ  JAm ðhÞðDK mþ1 ðhÞN mþ1 ðhÞ  DK m ðhÞN m ðhÞÞjC 0 ðTn Þ rþ1Þ 6 ^k0 c2 qð2 jem jC0 ðTn þqm Þ 2jjHjCl ðUÞ r2ejNjC 0 ðTn Þ m

þ 2ejNjC 0 ðTn Þ r jM H ðql2 m

ð2rþ1Þ þ ^kc2 qm jem jC 0 ðTn þqm Þ Þ ðmþ1Þ

6 jNmþ1 jC0 ðTn þqm Þ ð1 þ 2 6 2jNjC0 ðTn Þ

 N m ðhÞDK m ðhÞ ½Am ðhÞJ  JAm ðhÞDK m ðhÞN m ðhÞjC 0 ðTn Þ

r þ jHjCl ðUÞ c2 ~kq2 m jem jC 0 ðTn þqm Þ Þr2ejNjC 0 ðTn Þ ð2rþ1Þ þ 2ejNjC 0 ðTn Þ r2jjHjC l ðUÞ ^k0 c2 qm jem jC 0 ðTn þqm Þ

Þ

mþ1 X

2 ð2rþ1Þ 6 kðql2 jem jC 0 ðTn þqm Þ Þ; m þ c qm

j¼1

where  k is a constant depending on r; jNjC 0 ðTn Þ ; jHjC l ðUÞ ; ^0 ; ~ k k. Define k ¼ maxf kjhEi1 je; ~ kg, then from (14) we have

ð1 þ 2j Þ:

From (14) and the second inequality of A4ðm þ 1Þ, we obtain

jDK mþ1 ðhÞNmþ1 ðhÞ  DK m ðhÞNðhÞjC 0 ðTn þqmþ1 Þ

2 ð2rþ1Þ 2mþ1 kðql2 jem jC 0 ðTn þqm Þ Þ < 1: m þ c qm

Hence the following inequality holds:

6 jDK mþ1 ðhÞNmþ1 ðhÞ  DK mþ1 ðhÞNm ðhÞjC 0 ðTn þqmþ1 Þ

jEmþ1  Em jC l ðTn Þ jhEm i1 j

þ jDK mþ1 ðhÞNm ðhÞ  DK m ðhÞNm ðhÞjC 0 ðTn þqmþ1 Þ

2 ð2rþ1Þ jem jC 0 ðTn þqm Þ ÞjhEi1 je 6 kðql2 m þ c qm

6 jDK mþ1 ðhÞjC 0 ðTn þqmþ1 Þ  jNmþ1 ðhÞ  Nm ðhÞjC 0 ðTn þqmþ1 Þ

2 ð2rþ1Þ jem jC 0 ðTn þqm Þ Þ 6 6 kðql2 m þ c qm

þ jDK mþ1 ðhÞ  DK m ðhÞjC0 ðTn þqmþ1 Þ  jNm ðhÞjC 0 ðTn þqmþ1 Þ

The Neumann’s series theorem implies that hEmþ1 i is invertible and satisfies

ð2rþ1Þ 6 rjNmþ1 jC 0 ðTn þqmþ1 Þ ^kc2 qm jem jC 0 ðTn þqm Þ ð2rþ1Þ þ c2 ~kqm jem jC 0 ðTn þqm Þ jN m ðhÞjC0 ðTn þqm Þ

jhEmþ1 i1  hEm i1 j 6 jhEm i1 jkðql2 m

rþ1Þ 6 2ejNjC0 ðTn Þ ð^kr þ ~kÞc2 qð2 jem jC0 ðTn þqm Þ m

rþ1Þ þ c2 qð2 jem jC0 ðTn þqm Þ Þ m

rþ1Þ 6 ^k0 c2 qð2 jem jC0 ðTn þqm Þ : m

6 2ðmþ1Þ jhEm i1 j:

Hence define ^ k0 ¼ 2ejNjC 0 ðTn Þ ð^ kr þ ~ kÞ, then the following inequality holds:

So we obtain

jhEmþ1 i1 j 6 jhEmþ1 i1 j þ jhEmþ1 i1 j

jDK mþ1 ðhÞNmþ1 ðhÞ  DK m ðhÞNðhÞjC 0 ðTn þqmþ1 Þ rþ1Þ 6 ^k0 c2 qð2 jem jC0 ðTn þqm Þ : m

1 : 2

ð19Þ

From Lemma 2 and (13), we obtain

jDx ry Hmþ1 ðK mþ1 ðhÞÞ  Dx ry Hm ðK m ðhÞÞjC 0 ðTn Þ 6 jDx ry Hmþ1 ðK mþ1 ðhÞÞ  Dx ry Hm ðK mþ1 ðhÞÞjC0 ðTn Þ þ jDx ry Hm ðK mþ1 ðhÞÞ  Dx ry Hm ðK m ðhÞÞjC 0 ðTn Þ 2 ~ 2r 6 jjHjC l ðUÞ ql2 m þ jjHjC l ðUÞ c kqm jem jC 0 ðTn þqm Þ 2 ~ 2r 6 jMH ðql2 m þ jHjC l ðUÞ c kqm jem jC 0 ðTn þqm Þ Þ:

Similarly, estimating other element of Amþ1  Am , we have

6 jhEm i1 j  ð1 þ 2ðmþ1Þ Þ 6 jhEi1 j

m Y ð1 þ 2j Þ j¼1

and hence A4ðm þ 1Þ holds. Finally, using Lemma 2 and (12) and (16) is proved as follows:

jemþ1 jC 0 ðTn þqmþ1 Þ ¼ jJrHmþ1 ðK mþ1 ðhÞÞ  @ x ðK mþ1 ðhÞÞjC 0 ðTn þqmþ1 Þ 6 jJrHmþ1 ðK mþ1 ðhÞÞ  JrHm ðK mþ1 ðhÞÞjC 0 ðTn þqmþ1 Þ þ jJrHm ðK mþ1 ðhÞÞ  @ x ðK mþ1 ðhÞÞjC 0 ðTn þqmþ1 Þ 6 jJrHmþ1 ðK mþ1 ðhÞÞ  JrHm ðK mþ1 ðhÞÞjC 0 ðTn þqmþ1 Þ l1 6 jjHjC l ðTn Þ ql1 m ¼ jM H qm :

The proposition has been proved.

h

121

W.H. Jong, J.C. Paek / Chaos, Solitons & Fractals 68 (2014) 114–122

Using this proposition inductively, we obtain the approximate solution series. Define

^ K;H ; jM H g; c ¼ 2l maxfj; j0 g maxð4k; ~kÞ maxf1; M

 l1

cðq Þ

ð21Þ

< minf1; r; gg

ð22Þ

And define

s1 ¼ q1 H1 ¼ Ss1 ½H; K 1 ¼ Ss1 ½K: 1 ; Then from (2), (3), (5), (8), H1 and K 1 satisfy the properties A1ð1Þ–A4ð1Þ. Using (2), (10) and (21), we obtain

c4 ~kq1ð4rþ1Þ je1 jC0 ðTn þq1 Þ ð4rþ1Þ

6 c4 ~kq1

^ H;K slþ1 þ jje0 j 0 n Þ ðj00 M 1 C ðT Þ

ð4rþ1Þ 00 ^ 4 ~ ð4rþ1Þ 6 c4 ~kq1 j MH;K ql1 jje0 jC0 ðTn Þ 1 þ c kq1 c 4 c ð4rþ1Þ 6 c ðqÞð4rþ1Þ ðqÞl1 þ c4 q1 je0 jC 0 ðTn Þ 2 2 c 4 c 4 ð4rþ1Þ lð4rþ2Þ 6 c ðqÞ þ c q1 je0 jC0 ðTn Þ 6 minf1; r; gg: 2 2

Hence (11) holds for m ¼ 1. And from (2), (10), (21) and (22), we have ð2rþ1Þ

2 22 kðql1 1 þ c q1 2

6 2 kðq

l1 1

2

je1 jC 0 ðTn þq1 Þ Þ

ð2rþ1Þ ð 1

^ H;K slþ1 þ jje0 j 0 n ÞÞ j00 M 1 C ðT Þ

þc q

ð2rþ1Þ

2 2 6 22 kql1 1 þ 2 kc q1 ð2rþ1Þ

þ 22 kc2 q1

jje0 jC0 ðTn Þ

c 2 lð4rþ2Þ c 4 ð4rþ1Þ 6 2 kql1 þ c q1 je0 jC0 ðTn Þ 1 þ c q1 4 4 6 minf1; r; gg; so (15) also holds for m ¼ 1. Therefore from Proposition 4, there exists a non-degenerate function K 2 2 AðTn þ q2 ; C 1 Þ such that

J rH1 ðK 2 ðhÞÞ ¼ @ x K 2 ðhÞ:

r jK 2  K 1 jC 0 ðTn þq2 Þ 6 ~kc2 q2 je1 jC 0 ðTn þq1 Þ 1 r l1 6 cc2 q2 ðq1 þ je0 jC 0 ðTn Þ Þ: 1

ð23Þ

Now we will prove that for m P 2, if Hm ; K m satisfy A1ðmÞ–A4ðmÞ and (16) for m  1, then (11) and (15) holds for m. Using (21)–(23), we obtain

c kq

4

lð4rþ2Þ m

6 cc q

4 ~

ð4rþ1Þ m

< c kq

4

6 cc ðqÞ

lð4rþ2Þ

6 minf1; r; gg

2 ð2rþ1Þ 2mþ1 kðql1 jem jC 0 ðTn þqm Þ Þ m þ c qm 2 ð2rþ1Þ 6 2mþ1 kðql1 jMH ql1 m þ c qm m Þ

ðq Þl1 m1 l1

ð2

Þ

þ 2mþ1 kc2

l1 m

jMH q

and

6 2mþ1 k

r l1 2 lð2rþ1Þ 6 c2 ~kq2 : m jM H qm 6 c c qm

From Lemma 4, fK m gmP1 converges to K 1 2 C lð2rþ1Þ ðTn ; UÞ and K 1 is a solution of (1). And (10), (13), (14) implies, ð2rþmÞ

jK  K 1 jCm ðTn Þ 6 ~cc2 q1

ðql1 1 þ je0 jC 0 ðTn Þ Þ:



Remark 2. In Theorem 2, we assumed standard symplectic forms like Rüssmann, Pöschel, Llave; the reason is that once it is done, the general case will be done through some standard but complicated calculation. [8] is close to our result. The authors of [8] considered finitely differentiable symplectic maps without action-angle variables. In this article, we considered finitely differentiable Hamiltonian vector fields without action-angle variables. The result of [8] can be applied to periodic vector fields but cannot be applied to nonperiodic vector fields. As de la Llave mentioned in his letter to one of authors, smoothing periodic vector fields is easier.

Theorem 3. Let x 2 Dðc; rÞðr > n  1Þ; 4r þ 3 < l < m0 and U be either a compensated bounded open domain in R2n with C m -boundary, or Tn  U, with U  Rn , a compensated bounded open domain with C m -boundary. Let K 2 C l ðTn ; UÞ satisfies the following properties:

(1) There exists a matrix NðhÞ such that

If H2 ¼ S2s1 ½H; K 2 and H2 satisfy A1ð2Þ–A4ð2Þ and (16) holds for m ¼ 1. And the following inequality holds:

ð4rþ1Þ jem jC 0 ðTn þqm Þ m

r jK mþ1  K m jC 0 ðTn þqmþ1 Þ 6 c2 ~kq2 m jem jC 0 ðTn þqm Þ

In the case of parameterized Hamiltonian, similar result holds. We can prove it in the similar way.

^ H;K slþ1 j00 M 1

2

4 ~

J rHm ðK mþ1 ðhÞÞ ¼ @ x K mþ1 ðhÞ and

take q ð0 < q < 1Þ such that q 6 t 1 5 and such that

cc4 ðq Þlð4rþ2Þ < minf1; r; gg

Hence (11) and (15) hold for m. Therefore there exists a function K mþ1 2 AðTn þ qmþ1 ; C 1 Þ such that

ðq Þlð2rþ2Þ lð2rþ2Þ

ð2m1 Þ

c c l1 lð2rþ2Þ 6 ðq Þ þ c2 ðq Þ 6 minf1;r; gg: 2 2

NðhÞðDKðhÞT DKðhÞÞ ¼ I: 0 B (2) E0 ðhÞ ¼ @

 1

@ N0 ðhÞDKðhÞT J @k rHk ðKðhÞÞ k¼k0 C   A

T @

DKðhÞ J @k rHk ðKðhÞÞ k¼k 0

has a maximum range. Assume that g ¼ distðKðTn Þ; @UÞ=2 > 0 and H 2 U  B ! R, with B  R2n satisfies as following: for each x 2 U, the map Hðx; Þ is C 2 and for each k 2 B, the map Hk ¼ Hð; kÞ 2 C l ðUÞ. Define e0 ðhÞ ¼ J rHk0 ðKðhÞÞ  @ x KðhÞ (h 2 Tn ; k0 2 BÞ. Then there exist two constants c > 0 and q < 1 depending on r; n; l; jHjC l ðTn Þ ; jKjC l ðTn Þ ; jNjC l ðTn Þ ; j < E0 >1 j such that given 0 < q1 < q if ð4rþ1Þ

cc4 q1

je0 jC0 ðTn Þ < minf1; r; gg

then there exists a solution of (1), K 1 2 C lð2rþ1Þ ðTn ; UÞ.

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