SUB-HARMONICS OF FIRST ORDER HAMILTONIAN SYSTEMS AND THEIR ASYMPTOTIC BEHAVIORS

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1 DISCRETE AND CONTINUOUS Website: DYNAMICAL SYSTEMS SERIES B Volume 3, Number4, November3 pp SUB-HARMONICS OF FIRST ORDER HAMILTONIAN SYSTEMS AND THEIR ASYMPTOTIC BEHAVIORS XIANGJIN XU DEPARTMENT OF MATHEMATICS JOHNS HOPKINS UNIVERSITY, BALTIMORE, MD, 118, USA Abstract. In this paper some new existence results for sub-harmonics are proved for first order Hamiltonian systems with super-quadratic potentials by using two new estimates on C bound for the periodic solutions. Applying the uniform estimates on the sub-harmonics, the asymptotic behaviors of subharmonics is studied when the systems have globally super-quadratic potentials. 1. Introduction and Main Results. In this paper we study the existence of sub-harmonic solutions (i.e. kt-periodic solutions) and the asymptotic behavior of the sub-harmonics for the following first order Hamiltonian system (HS) J u B(t)u = H(t, u), u R N,t R, (1) where B(t) is a given continuous T -periodic and symmetric N N -matrix function, H C 1 (R R N, R) ( ist -periodic) in t, H := u H(t, u) andj is the IN standard symplectic matrix. The first result on subharmonics of I N the system (HS) was obtained by Rabinowitz in his pioneering work [13] for constant matrix B and certain conditions on H. Since then there are many papers on the existence of sub-harmonics of the system (HS), such as [4], [5], [6], [11], [15], [16], [18], [19]. Especially, in [5] and [15], the authors studied the asymptotic behaviors of subharmonics of the system (HS) under certain conditions on B(t) and H(t, u). In this paper, we firstly obtain two a priori estimates on the C bound for the periodic solutions of the modified systems of the system (HS) following the ideas in [17], [18] and [19]. Applying these estimates to the system (HS) and the ideas from [13], we have the following existence results on subharmonics: Theorem 1.1. For T>, supposeh(t, u) C 1 (S T R N, R) satisfies: (H1) there are constants µ> and r > such that <µh(t, u) ( H(t, u),u), u r, (H) H(t, u), for all (t, u) S T R N, H t (t, u) H t (t, u) (H3 ) lim sup u u µ =, or lim inf H (t, u) u u µ =, uniformly in t. H (t, u) (H4) H(t, u) =o( u ), uniformly in t as u, then there is a sequence {k i } N such that k i as i, and corresponding distinct k i T -periodic solutions {u ki } of the system (HS) Mathematics Subject Classification. 37J45,(34C5,47J3). Key words and phrases. Sub-harmonics, Hamiltonian system, asymptotic behaviors. 643

2 644 XIANGJIN XU Remark 1. (H1), (H) and (H4) are usual conditions when one studies the existence of periodic solutions for non-autonomous Hamiltonian systems possessing super-quadratic potentials. There are many papers (cf. [4], [5], [6], [11], [15]) studying the existence of subharmonics of the system (HS) with some growth conditions on H(t, u) at infinity of u variable. It is well known that for autonomous Hamiltonian systems the existence of infinite many periodic solutions was proved in Rabinowitz s pioneering work [1] under condition (H1) only. Here the condition (H3) is on H t (t, u)/( u µ H(t, u)), which measures how far the system (HS) is away from the autonomous system, or one may consider the system (HS) with condition (H3) as a perturbed system with a large perturbation. Especially from condition (H3), one can add any super-quadratic autonomous potential to the system (HS) and the results will still hold. Such a condition on H t (t, u)/h(t, u) as (H3) was first introduced in [1] to study the existence of periodic solutions of the system (HS). Using the same conditions on H(t, u) as in [13], we have a slightly more general existence result on subharmonics for the system (HS) as following: Theorem 1.. Suppose H(t, u) C 1 (S T R N, R) satisfies (H1), (H), (H4) and: (H5) There is a constant c>, such that H(t, u) c( H(t, u),u), u r. then there is a sequence {k i } N such that k i as i, and corresponding distinct k i T -periodic solutions {u ki } of the system (HS). Remark. In [13], B(t) was further assumed to be a constant matrix and the same result as Theorem 1. was proved there. In the second part of this paper, we study the asymptotic behaviors of subharmonics. Firstly we show there is a uniform C 1 bound for the subharmonics of the system (HS) when the system has the globally super-quadratic potential (H1) there is constant µ> such that <µh(t, u) z H(t, u), u >, Theorem 1.3. Suppose H(t, u) satisfies the globally super-quadratic condition (H1), (H)- (H4), then there exists infinitely many distinct sub-harmonics {u k } of the system (HS) and there is a uniform bound M for { u k C 1} k N. Using the uniform bound on the subharmonics from Theorem 1.3 and the ideas in [5] and [13], when B(t) satisfies condition: (B) c σ(a) R, wherea = (J(d/dt)+B(t)) and σ(a) is the spectrum of the self-adjoint operator A on W 1/ (S T, R N ). we have the following result: Theorem 1.4. Assume B(t) satisfies (B) c and H(t, u) satisfies the conditions of Theorem 1.3, then there is a sequence of pairwise geometrically distinct subharmonics {u k (t)} k N C 1 (R, R N ) of the system (HS) such that u k C 1 as k. Using the uniform bound on the subharmonics from Theorem 1.3 and the ideas in [15], when B(t) satisfies condition: (B) σ(a) R =, wherea = (J(d/dt) +B(t)) and σ(a) is the spectrum of the self-adjoint operator A on W 1/ (S T, R N ). we have the following result:

3 SUB-HARMONICS OF HAMILTONIAN SYSTEMS 645 Theorem 1.5. Assume B(t) satisfies condition (B) and H(t, u) satisfies the conditions of Theorem 1.3, then there is a sequence of pairwise geometrically distinct subharmonics {u k (t)} k N C 1 (R, R N ) of the system (HS) such that (i) there are constants m, M > independent of k N such that m kt [ 1 ( J u k,u k ) H(t, u k )] dt M; (ii) moreover {u k (t)} k N is compact in the following sense: for any sequence of integers k n, there exists a subsequence {k ni } i N and ontrivial homoclinic orbit u (t) emanating from such that u kni (t) u (t) in Cloc(R, 1 R N ),asi. Remark 3. in [5], [13] and [15], the authors dealt with the system (HS) with a constant matrix B and conditions on H(t, u) as these in Theorem 1.. Here we use the functional setting for the linearized system as used in [3], by which one can let B(t) depend on t variable. And applying the uniform C 1 bound on the subharmonics of the system (HS), one can directly apply those results in [5], [13] and [15] to our situation. We organize this paper as following: in Section, two a priopi estimates on C bound of the periodic solutions of the modified systems of the system (HS) are proved. In Section 3, Theorem 1.1 and Theorem 1. are proved by applying these estimates. In Section 4, the asymptotic behaviors of sub-harmonic solutions of the system (HS) are studied and Theorem are proved. Acknowledgements: The author of this paper would like to express his thanks to his advisor of Master degree, Professor Yiming Long, for many years instruction and valuable suggestions on his thesis for Master degree when the author studied in Nankai Institute of Mathematics. This paper grows from partial results of the author s thesis for Master degree. The author also want to express his thanks to the referees for their helpful suggestions on this paper.. Two Estimates. In this Section we study the C bound of the periodic solutions of the following modified systems (HS) n J u + H n (t, u) =, (t, u) S T R N. Here {H n } satisfy Proposition.1. For H satisfying (H1), σ (, 1] such that µσ >, andtwo sequences {K n } and {K n} in R, {H n } satisfies (i) K n is monotonous increasing to infinity, as n,andk n <K n, n N. (ii) for any given t S T, H n (t, ) C (R N, R), for every n N. (iii) H n (t, u) =H(t, u), u K n, for every n N; andforsomeλ [σ, 1], such that H n (t, u) =c n u µλ, u K n, for every n N. (iv) <σµh n (t, u) ( H n (t, u),u), u r, for every n N. Remark 4. The truncated results on H(t, u) as Proposition.1 was first proved by Long in [7] and [8], where Long got a better monotone truncated sequence {H n } on H for H(t, u) C in u variable. In next two sections, we don t need the monotone property and we can choose λ = σ =1when we define the truncated sequence {H n } as in [13] by H n (t, u) =χ n ( u )H(t, u)+(1 χ n ( u ))c n u µ

4 646 XIANGJIN XU where χ n (s) =1for s K n, χ n (s) =for s K n, and χ n(s) < for s (K n,k n), andc n (depend on n) be sufficiently large constant. In this truncation, H need only be C 1. Let X := W 1/, (S T, R N ), and define the functional I n : X R by I n (u) = 1 ( Ju B(t)u) udt H n (t, u) dt. It is well known that I n C 1 (X, R), and the periodic solutions of (HS) n are obtained as critical points of the functional I n. Using the ideas in [17], [18], [19], we prove the following two Lemmas on the C bound of the periodic solutions of the following modified systems (HS) n. Lemma.1. Suppose that H satisfies (H1) and (H5), {H n } satisfies Proposition.1, and u(t) is a T-periodic solution of (HS) n such that ( H n (t, u),u) dt C, H n (t, u) dt C, then there is a constant M independent of u and n and depend on C only such that u C M. Proof. Integrating (iv) of Proposition.1 gives H n (t, u) a u µσ b, u R N, where a and b are independent of n by (iii) of Proposition.1. Hence we have C H n (t, u) dt at u µσ dt bt at (min u(t) ) µσ bt. t S T Therefore we have min t ST u(t) C where C is independent of u and n. Without loss generality, we may assume u(t) obtains its minimum at t =,and u() r, t t u(t) u() = d u(s) ds u(s) ds ( B(s)u(s) + H n (s, u(s)) ) ds ds From (H5) and (iii) of Proposition.1 we have H n (t, u) c( H n (t, u),u), for r u K n We first show for large enough n, u C K n If not, by passing a subsequence, for each n N, there exists u n (t) andt n S T, such that u n (t n ) = K n and u n (t) K n for t [,t n ). Let we then have u n (t n ) tn c c d = tn t max { ( H n (t, u),u) }, {(t,u) S T R N H(t,u)<} tn H n (s, u n (s)) ds + B u(s) ds + u n () ( H n (s, u n (s)),u n (s)) ds + c tn u(s) σµ ds + u n () tn ( H n (s, u n (s)),u n (s)) ds + c H(s, u(s)) ds + cdt+ C cc + cdt + C,

5 SUB-HARMONICS OF HAMILTONIAN SYSTEMS 647 where c(may different in each step), d, C and C are independent of u and n. But then K n, as n, which leads to a contradiction. Hence there exists m N, depending only on H and C such that for any n m, u C K n holds. Repeating this argument, we find that for any n m, u(t) cc + cdt + C, t S T. For k<m,from(iii) of Proposition.1, we have H k (t, u) c k ( H k (t, u),u) u r, for some suitable constant c k. By the same argument, u(t) c k C + d k T + C, t S T where c k and d k are determined by H k for k =1,,,m 1. Therefore u C max{cc + cdt + C,c k C + d k T + C,k=1,,,m 1} = M, which completes the proof. Q.E.D. Lemma.. Suppose that H(t, u) satisfies (H1) and (H3), {H n } satisfies Proposition.1, and u(t) is a T-periodic solution of system (HS) n such that H n (t, u)udt C, H n (t, u) dt C. then there is a constant M independent of u and n and dependent on C only such that u C M. Proof. Since H n (t, u) > for u r,wehave C H n (t, u) dt H n (t, u) dt T sup H(t, u). {H n(t,u) } H(t,u)<,t S T Hence we have H n (t, u) dt C + T sup H(t, u) = C, {H n(t,u) } H(t,u)<,t S T where C is a constant independent of n and u. Define H t (t, u) H t (t, u) σ(r) = sup u r,t S T u µ, and δ(r) = inf H(t, u) u r,t S T u µ H(t, u). Then (H3) means lim σ(r) =or lim r δ(r) =. r Case I: Suppose we have lim r σ(r) =. By the definition of σ(r), we have that σ(r) is decreasing to. Fix a large R>r, such that a σ(r) C >. First, we show u C K n for large n. If not, by passing to a subsequence we may assume for each n, there exists u n (t), and b n such that (,b n ) {t S T R< u n (t) <K n } and u n ( ) = R, u n (b n ) = K n. Denote Ĥ n (t, u) = 1 (B(t)u, u)+h n(t, u); Ĥ(t, u) = 1 (B(t)u, u)+h(t, u)

6 648 XIANGJIN XU Therefore we have = = = Ĥ(b n,u n (b n )) Ĥ(,u n ( )) On the other hand, d dtĥn(t, u n (t)) dt [( Ĥn(t, u n (t)), u(t)) + Ĥt(t, u n (t))] dt H t (t, u n (t)) + 1 (B t(t)u n (t),u n (t))] dt σ( u n (t) ) u n (t) µ H(t, u n (t)) dt + B t σ(r)k n µ H(t, u n (t)) dt + c H(t, u n (t)) dt σ(r) CK n µ + c C. u n (t) dt Ĥ(b n,u n (b n )) Ĥ(,u n ( )) a u n (b n ) µ b B u n (b n ) max Ĥ(t, u) u R,t S T = ak n µ B Kn (b + max H(t, u) ) u R,t S T Combine these two formulas, we get that (a σ(r) C)K n µ B Kn b + c C + max H(t, u) u R,t S T Since a σ(r) C >, µ>, and K n as n, the left side tends to infinity, but the right side is a constant independent of u and n. This leads to a contradiction. Hence there exists m N, which is determined by H(t, u) andc only, such that for any n m, u C K n. For n m, if u C doesn t have an n-independent upper bound M, then following the above proof with K n replaced by M n where M n as n,we also get a contradiction. For n<m, as the proof in last part of Lemma.1, we have u(t) c k C + d k T + C, t S T, where c k and d k are determined by H k, k =1,,,m 1. Hence we have u C max{m,c k C + d k T + C,k=1,,,m 1} = M Case II: Suppose we have lim r δ(r) =. We need only to modify the proof of Case I a little. We have (a + δ(r) C)K n µ B Kn b + c C + max H(t, u) u R,t S T where a + δ(r) C >, µ>, and K n as n. Using the same argument as in Case I, we have u C M By combining these two cases, we prove this Lemma. Q.E.D.

7 SUB-HARMONICS OF HAMILTONIAN SYSTEMS 649 Remark 5. Here we proved the Lemmas for general potentials H, which doesn t assume any condition nearby u = on the potential H(t, u). Such kind estimates was first proved in [18] when the author studied the existence of periodic solutions of the first order Hamiltonian systems possessing super-quadratic potentials. In this paper H(t, u) is satisfied, which implies d = d k =and C = C, then the bound M is independent of the period T and depends only on C and H(t, u) from the proofs of the Lemmas. This is one key observation to apply these estimates to get the existence of subharmonics and to get the uniform estimates for subharmonics {u k } in the next sections. 3. Existence of subharmonics. We first show that there exists at least one nonzero T -periodic solution of (HS) under the conditions of Theorem 1.1 and Theorem 1.. as did in [13]. We truncate the potential H(t, u) by{h n (t, u)} as done in [13] satisfying Proposition.1 with λ = σ = 1 to get a sequence of modified systems, and we define I n (u) for these new systems. We use Theorem 1.4 in [1] to obtain the existence of the nonzero critical point of I n. One may check the details of the proof in [13], here we only give a sketched proof. Let X := H 1 (S T, R N ), By [9] and standard spectral theory, there exists a decomposition X = X + X X according to the self-adjoint operator B by extending the bilinear form (Bu,v) = 1 J u vdt (B(t)u, v) dt with dim X =kerb<, dimx + =dimx =. We verify the conditions of Theorem 1.4 in [1] for I n,setx 1 = X +, X = X X and I n (u) = 1 J u udt 1 (B(t)u, u) dt H n (t, u) dt = 1 ( u+ u ) H n (t, u) dt. As the proof of Theorem 6.1 in [14], we have I n C 1 (X, R) andi n satisfies (I1)- (I3) of Theorem 1.4 in [1]. To verify (I4), we construct S = B ρ X 1 which is the same as that in [1]. To obtain Q with r 1 and r independent of n, following the proof of Theorem 1.4 in [13], we let e B 1 X 1 and u = u + u X, then I n (u + se) = s u H n (t, u) dt s u a 3 ( u µ + s µ )+a 4 with n-independent constants a 3 and a 4 which are determined by (iv) of Proposition.1. Choose r 1 so that φ(s) =s a 3 s µ + a 4 () for all s r 1. Choose r large enough as [13], we have I n on Q with Q = {se s r 1 } (B r X ). So from Theorem 1.4 in [1], I n possesses a nonzero critical point u n with I n (u n ) α n >. Now we need to find an n-independent upper bound for { u n C }. In Theorem 1.4 in [1], the critical value c can be characterized as the minimax of I n over an

8 65 XIANGJIN XU appropriate class of sets (cf.[1]). Observe that Q is one of such sets and H n (t, u) satisfies (H), therefor we have I n (u n )=c n sup I n (u) sup (s u H n (t, u) dt) r1. u Q u +u r,s [,r 1] Since u n is a critical point of I n and each H n (t, u) satisfies Proposition.1, we have I n (u n ) = 1 = 1 ( Ju n,u n ) dt H n (t, u n )u n dt H n (t, u n ) dt ( 1 1 µ ) H n (t, u n )u n dt C 1 ( µ 1) H n (t, u n ) dt C Where C 1,C are independent of n. From above we have H n (t, u n )u n dt C, H n (t, u n ) dt H n (t, u n ) dt C for some constant C independent of n. Hence from Lemma.1 and Lemma., we have an n-independent constant M such that On the other hand, we have u n C M, for all n N. H n (t, u) =H(t, u), for u <K n. Hence for large n N such that K n >M, u n is onzero T -periodic solution of the system (HS). Next we show that the system (HS) has infinitely many distinct sub-harmonics. We will follow the ideas in Proof of Theorem 1.36 in [13]. For a given k N, we make the change of variables s = k 1 t.thusifu(t) isakt-periodic solution of the system (HS), η(s) = u(ks) satisfies J dη + k(b(ks)η + H(ks, η)) = (3) ds Since k H(ks, u) satisfies the conditions of our Theorem too, there is a solution η k (s) of (3), which is a critical point of I k (η) = 1 J η ηds k (B(ks)η, η) ds k H(ks, η) ds. Note that η 1 (ks) also satisfies (3), then if η 1 (ks) =η k (s), we have c k = I k (η k )= ki 1 (η 1 )=kc 1. Next we show that c k = I k (η k ) is bounded from above and the upper bound is independent of k. In the proof for the existence of one solution, we have c k r1(k) and the parameter r 1 (k) is determined by condition (). The corresponding condition satisfied by r 1 (k) is φ k (s) =s ka 3 s µ + ka 4,

9 SUB-HARMONICS OF HAMILTONIAN SYSTEMS 651 for all s r 1 (k). It follows that we can let ( ( ) 1 ( ) 1 ) ( ) 1 ( ) 1 µ a4 µ µ a4 µ r 1 (k) max, +. (4) ka 3 a 3 a 3 a 3 Now, for any given m N, if for some k>m, η k (s) =η m (s) holds for all s R, we have that η k (s) askt-periodic function is k l folds of η l (s) aslt-period function and η m (s) asmt -periodic function is m l folds of η l (s) aslt-period function, for some l N such that l k and l m and some corresponding η l (s). Hence we have c k = I k (η k )= k l Il (η l ), c m = I m (η m )= m l Il (η l ), that means c k = k m c m. On the other hand, we have c m > and{c k } is bounded by a k-independent constant from (4). This implies that there are at most finitely many k>msuch that η k (s) =η m (s) for any given m N. Hence Theorem 1.1 and Theorem 1. are proved. In [6], by using the iterated Maslov-type index theory and estimating the Maslovtype indices of the critical points of the direct variational calculus, Liu studied the existence of subharmonic solutions for the system (HS) under some conditions on H. Replacing the condition (H4) in [6] by our condition (H3), we have the following result: Theorem 3.1. Suppose H C (S T R N, R) and satisfies (H1)-(H4), and (H6) B(t) is a symmetric continuous matrix with period T, B C ω for some ω>, andb(t) is a semi-positive definite matrix for all t [,T]. then for each integer 1 k< π ωt there is a kt-periodic nonconstant solution u k of the system (HS). If all {u k } are non-degenerate, u j and u pj are geometrically distinct for p>1. Especially for B(t), for each integer k 1 there is a kt-periodic nonconstant solution u k of the system (HS). If all {u k } are non-degenerate, u j and u pj are geometrically distinct for p>1. If one doesn t assume non-degenerate condition on {u k }, u j and u pj are geometrically distinct for p > n +1. The proof of this Theorem follows the proof of Theorem 1.1 in [6]. The only difference is how to remove the growth restriction on H under our condition (H3), and notice that apply our Lemma., we get a K-independent upper bound for {u k K } for each k N, where uk K is the kt-periodic solution obtained in [6] for the modified system (HS) K. The Theorem is proved. 4. Asymptotic behaviors of subharmonics. In this section the asymptotic behaviors of the sub-harmonics of the system (HS) are studied when the potential H(t, u) of the system (HS) satisfies the globally super-quadratic condition (H1). We firstly show that there is a C 1 uniform bound for sub-harmonics {u k } as stated in Theorem 1.3. Proof of Theorem 1.3. From the proofs of Theorem 1.1, we need only to show there exist a uniform bound for all { u n,k C } n,k N where {u n,k } n,k N are those sub-harmonics that we obtain for (HS) n.

10 65 XIANGJIN XU Let c n,k = In(u k n,k ), where In k is the functional defined by H n for kt-periodic solution. From the Proof in Section 3, we have c n,k r1(k), where ( ) 1 ( ) 1 µ a4 µ r 1 (k) + a 3 which are independent of n and k. Hence from the globally super-quadratic condition (H1),wehave C I n,k (u n,k ) = 1 kt kt ( J u n,k B(t)u n,k ) u n,k dt H(t, u n,k ) dt = 1 kt kt H(t, u n,k ) u n,k dt H(t, u n,k ) dt ( 1 1 kt µ ) H(t, u n,k ) u n,k dt which implies kt ( µ kt 1) H(t, u n,k ) dt H n (t, u n,k )u n,k dt C, kt a 3 H n (t, u n,k ) dt C. Here C is independent of k. From Lemma.1, Lemma. and Remark.1, we know there is a constant M independent of n and k such that u n,k C M. And u n,k satisfies the equation of (HS), then there is a uniform C 1 bound M for all sub-harmonics {u n,k } n,k N. Q.E.D. Now we study the asymptotic behaviors of sub-harmonics {u k } k N as k under some further conditions as in Theorem 1.4 and Theorem 1.5. Here we apply the uniform C 1 estimates on subharmonics as in Theorem 1.3 to make use of those argument in [5], [13] and [15]. Since in those paper, B(t) was further assumed to be constant matrix, we need make ew functional setting as done in [3] as following. Let A = (J d dt + B(t)) is the self-adjoint operator acting on L (R, R N ) with the domain D(A) =H 1 (R, R N )andσ(a) is the spectrum of A. We use the norm E, which is defined as u E =( A 1 u + u ) 1, instead of the norm µ in [3], and the Banach space E, which is the completion of the set D(A) =H 1 (R, R N ) under the norm E, instead of the space E µ in [3]. As Section in [3], we have these facts: E has the direct sum decomposition E = E E +, and E is embedded continuous in L ν for any ν [, ) and compactly in L ν loc for any ν [, ). Notice that using our norm E instead of µ, the reader can check the details of these facts following the Proofs in Section in [3]. Set Φ n (u) = H n (t, u) dt R It is easy to check Φ n C 1 (E,R) since each H n satisfies (H1) and (iii) ofproposition.1, and E is embedded continuous in L ν for any ν [, ). We will use

11 SUB-HARMONICS OF HAMILTONIAN SYSTEMS 653 this functional setting to replace those in [5], [13] and [15], where only deal with the case that B(t) is a constant matrix. In [5] and [13], the authors show that the sub-harmonics converge to when the constant matrix B satisfies: (A) c σ(jb) ir, whereσ(a) is the spectrum of the matrix A. Notice that it is the same condition as our condition (B) c when B(t) is a constant matrix. From Theorem 1.3 we have the uniform C 1 bound M for all sub-harmonics {u n,k } n,k N, we can see all sub-harmonics of the system (HS) are the same as those of the modified system (HS) n for some large n. Now fixed a large n, from Proposition.1, the modified system (HS) n satisfies the conditions on H in [5] and [13], and we use the functional setting on A = (J d dt + B(t)) instead of those used in [5] and [13] for constant matrix B, we prove Theorem 1.4. Remark 6. In [13], the author proved that the subharmonics converge to when σ(jb) ir, andin[5], the author proved the subharmonics converge to under condition (A) c. In [15], the author shows that there exists ontrivial homoclinic orbit for the system (HS) as the limit of the subharmonics u k when the constant matrix A satisfies: (A) A is a N N symmetric matrix such that σ(ja) ir =. Notice that it is the same condition as our condition (B) when B(t) is a constant matrix. Now as above fixed a large n, we see that the modified system (HS) n satisfies the conditions of Theorem.1 in [15], and we use the functional setting on A = (J d dt + B(t)) instead of those used in [15] for constant matrix A, andwe prove Theorem 1.5. In the last part of this section, we study when the sub-harmonics {u k (t) = η k (t/k)} obtained in Theorem 1.1 and Theorem 1. are uniformly bounded in C 1. Let Tl 1 k denote the minimal period of η k (t). Then u k (t) =η(kt) has minimal period Tkl 1 k. Theorem 4.1. Under the conditions (H1),(H), (H3)(or (H5)),(H4), if kl 1 k don t tend to infinity along some subsequence, then the functions u k (t) s are uniformly bounded in C 1. Proof. We have c k = I k (u k ) 1 ((Ik ) (u k ),u k ) kt [ ] 1 = (u k(s), H(s, u k )) H(s, u k ) ds = k 1 l k kl From (H1), we have 1 k T { c k k 1 l k ( 1 1 kl µ ) k 1 l k { [ 1 (u k(s), H(s, u k (s))) H(s, u k (s)) ( µ 1) kl 1 k T 1 k T ] ds (u k (s), H(s, u k (s))) ds a 1 kl 1 k T H(s, u k (s)) ds a kl 1 k T } }

12 654 XIANGJIN XU for some constants a 1 and a independent of k. Since kl 1 k kl 1 k H(t, u k )u k dt C, kl 1 k T H(t, u k ) dt C is bounded, we have for some constant C independent of k. From Lemma.1 and Lemma., we have u k M, for some constant M independent of k. And u k satisfies the equation of (HS), then there is a uniform C 1 bound M for all sub-harmonics {u k } k N. Q.E.D. REFERENCES [1] V. Benci and P. H. Rabinowitz, Critcal point theorems for indefinite functionals, Inv. Math. 5. (1979), [] D.G. defigueiredo and Y.H. Ding, Solutions of onlinear Schrödinger equation. Discrete and Continuous Dynamical Systems - Series A Vol 8, No 3, () [3] Y. H. Ding and M. Willem, Homoclinic orbits of a hamiltonian system. Z. angew. Math. Phys. 5 (1999), [4] Ekeland, I. and Hofer, H. Subharmonics for convex non-autonomous Hamiltonian systems. Comm. Pure Appl. Math. 4 (1987) [5] P. L. Felmer, Subharmonic solutions near an equilibrium point for hamiltonian systems, Manuscripta Math. 66 (199) [6] Liu, Chun-Gen, Subharmonic solutions of Hamiltonian systems, Nolinear Analysis. TMA 4 () [7] Y. Long, Multiple solutions of perturbed superquadratic second order Hamiltonian systems. Trans. A.M.S. 311 (1989), [8] Y. Long, Periodic solutions of perturbed superquadratic Hamiltonian systems. Ann. Scuola Norm. Sup. Pisa (199), [9] Y. Long and E. Zehnder, Morse theory for forced oscillatiotically linear Hamiltonian systems, Stochastic Processes, Physics and Geometry. World Sci. Press, (199), [1] Y. Long and Xiangjin Xu, Periodic Solutions for a Class of Nonautonomous Hamiltonian Systems, Nolinear Analysis T.M.A. 41 () [11] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Springer, New York, [1] P.H.Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), [13] P. H. Rabinowitz, On sub-harmonic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 33 (198), [14] P. H. Rabinowitz, Minimax Methods in Critcal Point Theory with Applications to Differential Equations, CBMS 65, A.M.S., Providence, R.I., [15] Tanaka, K. Homoclinic orbits in a first order superquadratic Hamiltonian system: Convergence of sub-harmonic orbits. J. Diff. Eq. 94, (1991) [16] Xiangjin Xu, Sub-harmonic solutions of a class of non-autonomous Hamiltonian systems. Acta Sci. Nat. Univer. Nankai. Vol. 3, No., (1999), [17] Xiangjin Xu, Homoclinic Orbits for First Order Hamiltonian Systems possessing superquadratic potentials, Nonlinear Analysis, TMA 51 () [18] Xiangjin Xu, Periodic solutions of Non-autonomous Hamiltonian systems possessing superquadratic potentials. Nonlinear Analysis, TMA 51 () [19] Xiangjin Xu, Sub-harmonics for first order convex nonautonomous Hamiltonian systems. (submitted) Received September ; revised February 3. address: xxu@math.jhu.edu