A note on existence of a unique periodic solution of non-autonomous second-order Hamiltonian systems

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Journal of the Franklin Institute 347 (2010) 781–794 www.elsevier.com/locate/jfranklin

A note on existence of a unique periodic solution of non-autonomous second-order Hamiltonian systems$ Lie-Hui Zhanga, Yong Wanga,b, a

State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China b School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, PR China Received 10 June 2009; received in revised form 8 February 2010; accepted 1 March 2010

Abstract In this study, we shall be concerned with a non-autonomous second-order Hamiltonian system. Some criteria for guaranteeing the existence and uniqueness of T-periodic classical solution of this system are presented by employing the least action principle, the Saddle Point Theorem and some analysis techniques. & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. MSC: primary 34C25; 47J30 Keywords: Hamiltonian system; Periodic solution; Existence; Uniqueness

1. Introduction and main results In this note, we shall be concerned with the existence and uniqueness of T-periodic classical solution of the following non-autonomous second-order Hamiltonian system: € ¼ rF ðt; uðtÞÞ; uðtÞ $

ð1:1Þ

This work was supported by the Open Fund (PLN1003) of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation (Southwest Petroleum University), the National Program on Key Basic Research Project (973 Program) (Grant No.: 2006CB705808), the State major science and technology special projects during the 11th five-year plan (Grant No.: 2008ZX05054) and the SWPU Science and Technology Fund of China. Corresponding author. E-mail addresses: [email protected] (L.-H. Zhang), [email protected] (Y. Wang). 0016-0032/$32.00 & 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2010.03.002

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where T40, F : R  RN -R is T-periodic in t for each x 2 RN , rF ðt; xÞ9rx F ðt; xÞ ¼ ð@F =@x1 ; . . . ; @F =@xN Þ and F additionally satisfies the following hypothesis: (H0) F(t,x) is continuous from RNþ1 to R, rF ðt; xÞ 2 CðRNþ1 ; RN Þ, and there exist a 2 CðRþ ; Rþ Þ and b 2 L1 ð0; T; Rþ Þ such that jF ðt;xÞjraðjxjÞbðtÞ and

jrF ðt; xÞjraðjxjÞbðtÞ

for all t 2 ½0; T and x 2 RN . As is known, a Hamiltonian system is a system of differential equations which can model the motion of a mechanical system. An important and interesting question is under what conditions the Hamiltonian system can support periodic solutions. During the past few years, many authors have contributed to the theory of such system as regards the existence of periodic solutions, (cf. [2,3,11]) and the references therein. In his pioneering work [12] of 1978, Rabinowitz established the existence of periodic solutions of Eq. (1.1) with rF ðt; xÞ ¼ rF ðxÞ under the Ambrosetti–Rabinowitz condition (cf. [1]), that is, there exist m42 and L40 such that ðrF ðt;xÞ;xÞrmF ðt;xÞo0 for all t 2 R and x 2 RN with jxjZL. Subsequently, the existence of periodic solutions of Eq. (1.1) was studied by Mawhin and Willem [9,10], Long [8], Tang [15], Ye and Tang [16], Jiang [5,6], Schechter [14] and others (cf. the references given in these publications) under various conditions. However, all of these authors studied only the existence of T-periodic solutions, little was done concerning the existence and uniqueness of T-periodic solution of Eq. (1.1). To our knowledge, only Li and Liu [7] considered a similar case, where the authors obtained a sufficient condition for guaranteeing the existence of a unique hyperbolic 2pperiodic solution of the following Hamiltonian system: € uðtÞAuðtÞ þ rGðuðtÞÞ ¼ pðtÞ; where G 2 C 2 ðRN ; RN Þ, p 2 CðR=2pZ; RÞ and A is a symmetric and positive definite matrix; but for Eq. (1.1), the uniqueness is not yet clear. Thus it is essential to continue to study the periodic solutions of Eq. (1.1) in this case. The main purpose of this paper is to establish some criteria for guaranteeing the existence and uniqueness of T-periodic classical solution of Eq. (1.1), by employing the least action principle, the Saddle Point Theorem and some analysis techniques. We remark that our results are different from the ones in [7]. The following notation will be used throughout the rest of this paper. Z 1 T uðtÞ dt; JuJ1 ¼ max juðtÞj; u ¼ t2½0;T T 0 0

Fi ðt;xÞ9@F ðt;xÞ=@xi

where x ¼ ðx1 ; . . . ;xN Þ and

i ¼ 1; . . . ;N:

Set HT1 9fu : ½0; T-RN ju is absolutely continuous; 2

N

u_ 2 L ð0; T; R Þg and ~ 1 9fu 2 H 1 j u ¼ 0g; H T T

uð0Þ ¼ uðTÞ and

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which are two real Hilbert spaces with the norm Z T 1=2 Z T 2 2 _ juðtÞj dt þ juðtÞj dt : JuJ ¼ 0

0

Here and subsequently, (; ): RN  RN -R denotes the standard inner product in RN and j  j is the induced norm. Our main theorems are as follows. Theorem 1.1. Assume (H0) holds. Also assume that there exist f ; g 2 L1 ð0; T; Rþ Þ, 0oar1 and b40 such that (H1) (H2) (H3) (H4)

ðrF ðt; xÞrF ðt; yÞ; xyÞ40, for all t 2 R and x and y 2 RN with xay; jrF ðt; xÞjrf ðtÞjxja þ gðtÞ;  a1  1 b =12C0 40; 2 2 RT limjxj-1 inf½jxj2a 0 F ðt; xÞ dtC1 40, for all x 2 RN ; 8 < 0; where C0 ¼ T R T f ðtÞ dt; : 6 0

0oao1; a ¼ 1;

and

2a1 T C1 ¼ b

Z

2

T

f ðtÞ dt

:

0

Then (1.1) has a unique T-periodic classical solution in H1T. Theorem 1.2. Assume that there exist f ; g 2 L1 ð0; T; Rþ Þ and 0oar1 such that (H0) and (H2) hold. Also assume that there exist K0 40, K4 40 and g40 such that (H5) ½Fi ðt; xÞFi ðt; yÞðxi yi Þa0 for all t 2 R and x and y 2 RN with xi ayi i ¼ 1,y,N; 0 0 (H6) jFi ðt; xÞFi ðt; yÞjrK0 jxi yi j for all t 2 R and x and y 2 RN i ¼ 1,y,N; (H7) K0 ðT 2 =4pÞo1; (H8) 12 K1 40; pffiffiffiffiffiffiffiffiffiffiffiffiffiffi (H9) K2 40, K4 4 K3 =K2 ; RT (H10) limjxj-1 sup½jxj2a 0 F ðt; xÞ dt þ K5 o0 for all x 2 RN ; 0

0

8 0oao1; < 0; where K1 ¼ T R T f ðtÞ dt; a ¼ 1; : 12 0 K3 ¼

2a1 T g

Z

2

T

f ðtÞ dt 0

and K5 ¼ K42

K2 ¼ 1

2a1 g 2K1 ; 12

  1 2a1 g þ þ 2K1 þ K3 : 2 12

Then (1.1) has a unique T-periodic classical solution in H1T. Remark 1.3. It appears that the conditions of Theorems 1.1 and 1.2 above are a bit long and cumbersome, but one can see that once the parameters T, f(t), a and K0 were obtained, it might be not difficult to choose some appropriate parameters b and g to calculate the other parameters (C0, C1, K1, K2 etc) so as to check the linking conditions of Theorems 1.1 and 1.2.

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As the applications, two examples will be provided in Section 4. On the other hand, our results are different from the ones in [7] and may be the infrequent ones in the literature. Moreover, on the side of the existence of T-periodic solutions of Eq. (1.1), the relevant results contained in the Theorems 1.1 and 1.2 improve and generalize the Theorems 1–4 of Tang [15], that is: if 0oao1, our conditions are weaker than the ones of the Theorems 1 and 2 of Tang [15]; if a ¼ 1, our results have new meaning (for more comments, see Remarks 2.5 and 2.7). In Section 2 we introduce some lemmas which will help us to obtain the main theorems, the proofs of which are given in Section 3. The paper concludes with two illustrative examples which are provided to demonstrate the applications of our results. 2. Lemmas In this section, we shall introduce some lemmas which will help us to obtain our main theorems. First, let us recall some known results. For u 2 HT1 , let u~ ¼ uu, then the following is easily proved (cf., e.g. [10, Proposition 1.3, p. 9]). Lemma 2.1 (Mawhin and Willem [10, Proposition 1.3]). If u 2 HT1 , then Z T Z T2 T ~ 2 dtr 2 _ 2 dt ðWiringer’s inequalityÞ juðtÞj juðtÞj 4p 0 0 and ~ 21 r JuJ

T 12

Z

T

_ 2 dt ðSobolev inequalityÞ: juðtÞj

0

Define the functional j on H1T by Z T Z 1 T 2 _ jðuÞ9 juðtÞj dt þ ½F ðt; uðtÞÞF ðt; 0Þ dt: 2 0 0 Then it follows from (H0) that j is continuously differential and weakly lower semicontinuous on H1T (cf. [10, pp. 12–13]). On the other hand, one has Z T Z T _ ðuðtÞ; v_ ðtÞÞ dt þ ðrF ðt; uðtÞÞ; vðtÞÞ dt /j0 ðuÞ; vS ¼ 0

0 0

for all u; v 2 HT1 , where j denotes the Fre´chet derivative of j. Moreover, it is clear that u0 2 HT1 is a critical point of j if and only if u0 is a T-periodic classical solution of Eq. (1.1) when rF ðt; xÞ 2 CðRNþ1 ; RN Þ (cf., the proof process of [13, Theorem 6.10, pp. 40–41]). Next two lemmas will be introduced to help us in obtaining the existence of T-periodic classical solutions of Eq. (1.1). The first one is the so-called Saddle Point Theorem (cf., e.g. [13, Theorem 4.6, p. 24]). Lemma 2.2 (Rabinowitz [13, Theorem 4.6]). Let E ¼ V  X , where E is a real Banach space and V af0g and is finite dimensional. Suppose j 2 C 1 ðE; RÞ, satisfies (PS) condition, and (A1) there is a constant e and a bounded neighborhood D of 0 in V such that jj@D re, and (A2) there is a constant r4e such that jjX Zr.

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Then j possesses a critical value cZr. Moreover, c can be characterized as c ¼ inf maxjðhðuÞÞ; h2G u2D

where G ¼ fh 2 CðD; EÞjh ¼ id on @Dg. Lemma 2.3 (Mawhin and Willem [10, Theorem 1.1, p. 3]). If j is weakly lower semicontinuous on a reflexive Banach space X and has a bounded minimizing sequence, then j has a minimum on X. Now we present two lemmas on the existence of T-periodic classical solutions of Eq. (1.1). Lemma 2.4. Assume (H0) holds. Also assume that there exist f ; g 2 L1 ð0; T; Rþ Þ, 0oar1 and b40 such that (H2)–(H4) hold. Then Eq. (1.1) has at least one T-periodic classical solution in H1T. Proof. It follows from (H2) and Lemma 2.1 that, for all u 2 HT1 , Z T  Z T         ~ ~ ½F ðt; uðtÞÞF ðt; uÞ dt ¼  ðrF ðt; u þ suðtÞÞ; uðtÞÞ ds dt  0 0 Z Z TZ 1 a ~ ~ f ðtÞju þ suðtÞj juðtÞj ds dt þ r 0

0

r

a

Z 0

0

Z

T

~ 1 f ðtÞ dt þ JuJ Z

1

~ gðtÞjuðtÞj ds dt

0

~ a1 ÞJuJ ~ 1 r2 ðjuj þ JuJ a

Z

T

2

T

gðtÞ dt 0

T 2a1 b 2a1 T 2a ~ 21 þ f ðtÞ dt JuJ juj T b 0 Z T Z T ~ aþ1 ~ 1 þ2a JuJ f ðtÞ dt þ JuJ gðtÞ dt 1 0

0

Z T ðaþ1Þ=2 Z 2a1 b T _ 2 dt þ C1 juj2a þ C2 _ 2 dt juðtÞj juðtÞj r 12 0 0 Z T 1=2 _ 2 dt þC3 juðtÞj ; ð2:1Þ 0

 T ðaþ1Þ=2 R T RT sð 0 f ðtÞ dtÞ2 , C2 ¼ 2a 12 where C1 ¼ 0 f ðtÞ dt and C3 is some positive constant. Now we shall consider two cases as follows: Case (i): If 0oao1, we have from Eq. (2.1) that, for all u 2 HT1 , Z Z T Z T Z T 1 T 2 _ jðuÞ ¼ juðtÞj dt þ ½F ðt; uðtÞÞF ðt; uÞ dt þ F ðt; uÞ dt F ðt; 0Þ dt 2 0 0 0 0  Z T Z T ðaþ1Þ=2 Z T 1=2 1 2a1 b _ 2 dtC2 _ 2 dt _ 2 dt juðtÞj juðtÞj C3 juðtÞj Z  2 12 0  0 Z T Z0 T þjuj2a juj2a F ðt; uÞ dtC1  F ðt; 0Þ dt: ð2:2Þ 2a1 T b

0 2

Since JuJ-1 if and only if ðjuj þ Eq. (2.2) that jðuÞ- þ 1

RT 0

0

_ juðtÞj dtÞ1=2 -1, we get from (H3), (H4) and

as u 2 HT1 and JuJ-1:

2

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This implies that there exists a constant r40 such that jðuÞ4jð0Þ when JuJ4r. Set V ð0; rÞ ¼ fu 2 HT1 jJuJrrg and we know that V(0,r) is weakly compact and weakly closed since H1T is reflexive. Let m ¼ inf u2V ð0;rÞ jðuÞ then we have mrjð0Þ, which leads to m ¼ inf u2V ð0;rÞ jðuÞ ¼ inf u2HT1 jðuÞ. Noticing that V(0,r) is weakly compact and weakly closed, there exists a sequence ðun Þ  V ð0; rÞ such that jðun Þ converges to m ¼ inf u2HT1 jðuÞ (i.e., jðun Þ!inf u2HT1 jðuÞ), which implies j has a bounded minimizing sequence. Also noticing that j is continuously differential and weakly lower semi-continuous on H1T, we can immediately get from Lemma 2.3 that j has a minimum on H1T and consequently has a critical point in H1T. Case (ii): If a ¼ 1, we have from Eq. (2.1) that, for all u 2 HT1 , Z T      ½F ðt; uðtÞÞF ðt; uÞ dt   0

r



b T þ 12 6

Z

Z

T

f ðtÞ dt 0

T

_ 2 dt þ C1 juj2 þ C3 juðtÞj

0

Z

T

1=2 _ 2 dt juðtÞj

0

and Z T Z T Z T Z 1 T _ 2 dt þ juðtÞj ½F ðt; uðtÞÞF ðt; uÞ dt þ F ðt; uÞ dt F ðt; 0Þ dt 2 0 0 0 0  Z T Z T 1=2 Z 1 b T T _ 2 dtC3 _ 2 dt Z   f ðtÞ dt juðtÞj juðtÞj 2 12 6 0 0 0   Z T Z T 2 2 þjuj juj F ðt; uÞ dtC1  F ðt; 0Þ dt: ð2:3Þ

jðuÞ ¼

0

Since JuJ-1 if and only if ðjuj2 þ Eq. (2.3) that

RT 0

0

_ 2 dtÞ1=2 -1, we get from (H3), (H4) and juðtÞj

jðuÞ- þ 1 as u 2 HT1 and JuJ-1: By the same discussion as the case (i), we conclude that j has a minimum on H1T and consequently has a critical point in H1T. This completes the proof. & Remark 2.5. According to the proof of Lemma 2.4, if the condition (H0) is replaced by the weaker condition (A) in [15], Eq. (1.1) has at least one T-periodic weak solutions in H1T. If 0oao1, let b ¼ 3 such that (H3) holds and simultaneously we can see that the condition (H4) is weaker than the condition (3) of Theorem 1 in [15]. This implies that Lemma 2.4 improves Theorem 1. On the other hand, if a ¼ 1, Lemma 2.4 is different from the Theorems 3 and 4 in [15] and is also different from the relevant results in [5,6,8–10, 12–14,16]. Lemma 2.6. Assume (H0) holds. Also assume that there exist f ; g 2 L1 ð0; T; Rþ Þ, 0oar1 and g40 such that (H2) and (H8)–(H10) hold. Then Eq. (1.1) has at least one T-periodic classical solution in H1T. ~ 1 , Lemma 2.2 will be applied to proving this lemma. First Proof. Note that HT1 ¼ RN  H T we prove that j satisfies the (PS) condition. Assume that (un) is a (PS) sequence of j, i.e., 0 j ðun Þ-0 as n-1 and fjðun Þg is bounded. Similar to the proof process of Eq. (2.1), we

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have that, for all n, Z   

T

0

r

  ðrF ðt; un ðtÞÞ; u~ n ðtÞÞ dt Z

2a1 g 12

T

ju_ n ðtÞj2 dt þ K3 ju n j2a þ K6

Z

0

T

ju_ n ðtÞj2 dt

ðaþ1Þ=2

Z

T

ju_ n ðtÞj2 dt

þ K7

0

1=2 ;

ð2:4Þ

0

T ðaþ1Þ=2 where K3 is the same as the one in Theorem 1.2 of this paper, K6 ¼ 2a ð12 Þ and K7 is some positive constant. Now we shall consider two cases as follows: Case (i): If 0oao1, we have from Eq. (2.4) that, for large enough n,

RT 0

f ðtÞ dt

0

Ju~ n JZ/j ðun Þ; u~ n S Z

T

ju_ n ðtÞj2 dt þ

¼ 0

Z

T

ðrF ðt; un ðtÞÞ; u~ n ðtÞÞ dt 0

Z T ðaþ1Þ=2 Z T 1=2  Z T 2a1 g Z 1 ju_ n ðtÞj2 dtK3 ju n j2a K6 ju_ n ðtÞj2 dt K7 ju_ n ðtÞj2 dt : 12 0 0 0

Z ¼ K2

T

ju_ n ðtÞj2 dtK3 ju n j2a K6

Z

0

T

ju_ n ðtÞj2 dt

ðaþ1Þ=2

Z

T

ju_ n ðtÞj2 dt

K7

0

1=2 :

ð2:5Þ

0

where K2 is the same as the one in the Theorem 1.2 of this paper. On the other hand, we have from Lemma 2.1 that, for all n,  2 1=2 Z T 1=2 T 2 þ1 ju_ n ðtÞj dt : Ju~ n Jr 4p2 0

ð2:6Þ

Together with Eqs. (2.5) and (2.6) and (H9) we obtain that, for large enough n, Z T 1=2 K4 ju n ja Z ju_ n ðtÞj2 dt ;

ð2:7Þ

0

Moreover, similar to the proof process of Eqs. (2.1) and (2.4), we have that, for all n, Z T      ½F ðt; u ðtÞÞF ðt; u Þ dt n n   0

r

2a1 g 12

Z 0

T

ju_ n ðtÞj2 dt þ K3 ju n j2a þ K6

Z

T

ju_ n ðtÞj2 dt 0

ðaþ1Þ=2

Z þ K7

T

1=2 ju_ n ðtÞj2 dt :

0

ð2:8Þ Then we can get from Eqs. (2.7) and (2.8) that, for large enough n, Z Z T Z T Z T 1 T jðun Þ ¼ ju_ n ðtÞj2 dt þ ½F ðt; un ðtÞÞF ðt; u n Þ dt þ F ðt; u n Þ dt F ðt; 0Þ dt 2 0 0 0 0 Z T ðaþ1Þ=2  Z T 1 2a1 g ju_ n ðtÞj2 dt þ K3 ju n j2a þ K6 ju_ n ðtÞj2 dt r þ 2 12 0 0

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Z

T 2

1=2

Z

Z

T

T

þK7 ju_ n ðtÞj dt þ F ðt; u n Þ dt F ðt; 0Þ dt 0 0  0 a1  1 2 g rK42 þ ju n j2a þ K3 ju n j2a þ K4aþ1 K6 ju n jaðaþ1Þ þ K4 K7 ju n ja 2 12 Z T Z T þ F ðt; u n Þ dt F ðt; 0Þ dt 0   Z T 0 F ðt; u n Þ dt þ K5 þ K4aþ1 K6 ju n jaðaþ1Þ ¼ ju n j2a ju n j2a Z0 T Z T a þK4 K7 ju n j þ F ðt; u n Þ dt F ðt; 0Þ dt; 0

ð2:9Þ

0

where K5 is the same as the one in Theorem 1.2 of this paper. Note that fjðun Þg is bounded and here 0oao1, (H10) and Eq. (2.9) imply that (u n ) is bounded. Thus (un) is bounded in H1T by Eqs. (2.6) and (2.7). Going if necessary to a subsequence, we can assume that u, n u in H1T. Employing Proposition 1.2 in Mawhin and Willem [10], one can see that un -u uniformly on [0,T]. Therefore 0

0

/j ðun Þj ðuÞ; un uS-0 and Z

T

ðrF ðt; un ÞrF ðt; uÞ; un uÞ dt-0 0

as n-1. Moreover, a direct computation shows that Z T 0 0 _ 2L2  /j ðun Þj ðuÞ; un uS ¼ Ju_ n uJ ðrF ðt; un ÞrF ðt; uÞ; un uÞ dt: 0

_ L2 -0 as n-1. Thus Jun uJ-0 as n-1 and j satisfies the (PS) It follows that Ju_ n uJ condition. By Eq. (2.9) and (H10), we have jðuÞ-1

as juj-1 in RN ;

which implies that the condition (A1) of Lemma 2.2 is satisfied. ~ 1, On the other hand, similar to the proof process of Eq. (2.1), we have that, for all u 2 H T Z   

0

T

 Z   ½F ðt; uðtÞÞF ðt; 0Þ dt ¼ 

T

Z

0

1 0

Z TZ 1 f ðtÞjsuðtÞja juðtÞj ds dt þ gðtÞjuðtÞj ds dt 0 0 0 0 Z T Z T rJuJaþ1 f ðtÞ dt þ JuJ gðtÞ dt 1 1 r

Z

T

Z

  ðrF ðt; suðtÞÞ; uðtÞÞds dt

1

0

r2a K6

Z 0

0

T

ðaþ1Þ=2 Z _ 2 dt juðtÞj þ K8

T

1=2 _ 2 dt juðtÞj ;

0

ð2:10Þ where K6 is the same as the one in Eq. (2.4) and K8 is some positive constant.

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1

~ , It follows that, for all u 2 H T jðuÞ ¼

1 2

Z

Z

1 2

T

_ 2 dt þ juðtÞj

Z

T

½F ðt; uðtÞÞF ðt; 0Þ dt

0

0

Z

T

_ 2 dt2a K6 juðtÞj

Z

0

T

_ 2 dt juðtÞj

ðaþ1Þ=2

Z

0

Note that JuJ-1 if and only if ð implies

RT 0

T

K8

_ 2 dt juðtÞj

1=2 :

ð2:11Þ

0 1

~ and here 0oao1, Eq. (2.11) _ 2 dtÞ1=2 -1 in H juðtÞj T

~ 1: as JuJ-1 in H T

jðuÞ- þ 1;

This leads that the condition (A2) of Lemma 2.2 is satisfied. Consequently j has a critical point in H1T. Case (ii): If a ¼ 1, we get from Eq. (2.4) that, for all n, Z T   Z T 1=2 Z T   2 2 2  r g þ 2K1 ~ _ _ ðrF ðt; u ðtÞÞ; u ðtÞÞ dt j u ðtÞj dt þ K ju j þ K j u ðtÞj dt ; n n n 3 n 7 n   12 0 0 0 ð2:12Þ where K1 is the same as the one in the Theorem 1.2 of this paper. It follows that, for large enough n, 0

Ju~ n JZ/j ðun Þ; u~ n S Z

T

ju_ n ðtÞj2 dt þ

¼

Z

0

T

ðrF ðt; un ðtÞÞ; u~ n ðtÞÞ dt 0

Z T 1=2  Z T g ju_ n ðtÞj2 dtK3 ju n j2 K7 ju_ n ðtÞj2 dt Z 1 2K1 12 0 0 Z

T

ju_ n ðtÞj2 dtK3 ju n j2 K7

¼ K2 0

Z

T

ju_ n ðtÞj2 dt

1=2 ð2:13Þ

0

This together with Eq. (2.6) and (H9) implies that, for large enough n, Z

T

ju_ n ðtÞj2 dt

K4 ju n jZ

1=2 :

ð2:14Þ

0

On the other hand, similar to the proof process of Eqs. (2.1), (2.4) and (2.12), we have that, for all n, Z   

0

T

  Z T 1=2 Z T  g þ 2K1 ½F ðt; un ðtÞÞF ðt; u n Þ dtr ju_ n ðtÞj2 dt þ K3 ju n j2 þ K7 ju_ n ðtÞj2 dt : 12 0 0

ð2:15Þ

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Now we can get from Eqs. (2.14) and (2.15) that, for large enough n, Z Z T Z T Z T 1 T jðun Þ ¼ ju_ n ðtÞj2 dt þ ½F ðt; un ðtÞÞF ðt; u n Þ dt þ F ðt; u n Þ dt F ðt; 0Þ dt 2 0 0 0 0  Z T Z T 1=2 1 g 2 2 2 ju_ n ðtÞj dt þ K3 ju n j þ K7 ju_ n ðtÞj dt r þ þ 2K1 2 12 0 0 Z T Z T þ F ðt; u n Þ dt F ðt; 0Þ dt 0 0 Z T Z T 1 g þ þ 2K1 ju n j2 þ K3 ju n j2 þ K4 K7 ju n j þ rK42 F ðt; u n Þ dt F ðt; 0Þ dt 2 12 0 0   Z T Z T Z T ¼ ju n j2 ju n j2 F ðt; u n Þ dt þ K5 þ K4 K7 ju n j þ F ðt; u n Þ dt F ðt; 0Þ dt; 0

0

0

ð2:16Þ where K5 is the same as the one in Theorem 1.2 of this paper. Note that fjðun Þg is bounded and here a ¼ 1, (H10) and Eq. (2.16) imply that (u n ) is bounded. Thus (un) is bounded in H1T by Eqs. (2.6) and (2.14). Similar to the Case (i), we conclude that j satisfies the (PS) condition. By Eq. (2.16) and (H10), we have jðuÞ-1

as juj-1 in RN ;

which implies that the condition (A1) of Lemma 2.2 is satisfied. ~1, On the other hand, by Eq. (2.10) we have that, for all u 2 H T Z T  Z T Z 1         ½F ðt; uðtÞÞF ðt; 0Þ dt ¼  ðrF ðt; suðtÞÞ; uðtÞÞ ds dt  0

0

rK1

0

Z

T

_ 2 dt þ K8 juðtÞj 0

Z

T

_ 2 dt juðtÞj

1=2 :

ð2:17Þ

0

where K1 is the same as the one in the Theorem 1.2 of this paper. ~ 1, It follows that, for all u 2 H T Z T Z T 1 _ 2 dt þ juðtÞj ½F ðt; uðtÞÞF ðt; 0Þ dt jðuÞ ¼ 2 0 0  Z T Z T 1=2 1 2 2 _ _ Z K1 juðtÞj dtK8 juðtÞj dt : ð2:18Þ 2 0 0 RT ~ 1 , Eq. (2.18) and (H8) imply _ 2 dtÞ1=2 -1 in H Note that JuJ-1 if and only if ð 0 juðtÞj T ~ 1: jðuÞ- þ 1 as JuJ-1 in H T This leads that the condition (A2) of Lemma 2.2 is satisfied. Consequently j has a critical point in H1T. This completes the proof. & Remark 2.7. Like the Remark 2.5, if the condition (H0) is replaced by the weaker condition (A) in [15], Eq. (1.1) has at least one T-periodic weak solutions in H1T. If 0oao1, let g ¼ 3 such that (H9) holds and simultaneously we can see that the condition (H10) is weaker than the condition (4) of Theorem 2 in [15]. This implies that Lemma 2.6 improves

ARTICLE IN PRESS L.-H. Zhang, Y. Wang / Journal of the Franklin Institute 347 (2010) 781–794

791

Theorem 2. On the other hand, if a ¼ 1, Lemma 2.6 is different from Theorems 3 and 4 in [15] and is also different from the relevant results in [5,6,8–10,12–14,16]. Finally one more lemma will be introduced to help us in obtaining the uniqueness. Lemma 2.8 (Hardy et al. [4]). If x 2 C 2 ðR; RÞ with x(tþT) ¼ x(t), then Z T Z T2 T 2 _ € 2 dt: jxðtÞj dtr 2 jxðtÞj 4p 0 0 3. Proofs of Theorems 1.1. and 1.2 We are now in the position to present the proofs of Theorems 1.1 and 1.2. Proof of Theorem 1.1. By Lemma 2.4, it suffices to prove that Eq. (1.1) has at most one T-periodic classical solution in H1T. Suppose that x(t) and y(t) are two T-periodic classical solutions of Eq. (1.1). Then we have € € ðxðtÞ yðtÞÞðrF ðt; xðtÞÞrF ðt; yðtÞÞÞ ¼ 0

ð3:1Þ

Multiplying x(t)y(t) and (3.1) and then integrating it from 0 to T, we get from (H1) that Z T _ _ _ _ ðxðtÞ yðtÞ; xðtÞ yðtÞÞ dt 0Z 0 Z T € € ðxðtÞ yðtÞ; xðtÞyðtÞÞ dt ¼ 0 Z T ðrF ðt; xðtÞÞrF ðt; yðtÞÞ; xðtÞyðtÞÞ dt ¼ 0

Z0 It follows that x(t) ¼ y(t) for all t 2 ½0; T. This completes the proof.

&

Proof of Theorem 1.2. By Lemma 2.6, it suffices to prove that Eq. (1.1) has at most one T-periodic classical solution in H1T. In order to do this, let us consider a system equivalent to Eq. (1.1): 0

u€ i ðtÞ ¼ Fi ðt; uðtÞÞ;

i ¼ 1; . . . ; N;

ð3:2Þ

where u(t)=(u1(t),y,uN(t)). Now suppose that x(t) and y(t) are two T-periodic classical solutions of Eq. (3.2), where x(t)=(x1(t),y,xN(t)) and y(t)=(y1(t),y,yN(t)). Then for each i 2 f1; . . . ; Ng, we have 0

0

½x€i ðtÞy€i ðtÞ½Fi ðt; xðtÞÞFi ðt; yðtÞÞ ¼ 0:

ð3:3Þ

Setting zi(t) ¼ xi(t)yi(t), we obtain from Eq. (3.3) that 0

0

z€ i ðtÞ½Fi ðt; xðtÞÞFi ðt; yðtÞÞ ¼ 0:

ð3:4Þ

Since zi(t) is a continuous T-periodic function in R, there exist two constants timin 2 ½0; T such that zi ðtimax Þ ¼ max zi ðtÞ ¼ maxzi ðtÞ and zi ðtimin Þ ¼ min zi ðtÞ ¼ minzi ðtÞ: t2½0;T

t2R

t2½0;T

t2R

timax

and ð3:5Þ

ARTICLE IN PRESS L.-H. Zhang, Y. Wang / Journal of the Franklin Institute 347 (2010) 781–794

792

It follows that z_ i ðtimax Þ ¼ x_ i ðtimax Þy_ i ðtimax Þ ¼ 0;

z€ i ðtimax Þr0

ð3:6Þ

z€ i ðtimin ÞZ0:

ð3:7Þ

and z_ i ðtimin Þ ¼ x_ i ðtimin Þy_ i ðtimin Þ ¼ 0; Then, by Eqs. (3.4)–(3.7), 0

0

Fi ðtimax ; xðtimax ÞÞFi ðtimax ; yðtimax ÞÞ ¼ z_ i ðtimax Þr0

ð3:8Þ

and 0

0

Fi ðtimin ; xðtimin ÞÞFi ðtimin ; yðtimin ÞÞ ¼ z_ i ðtimin ÞZ0:

ð3:9Þ

From Eqs. (3.8) and (3.9), we know that there exists ti0 2 ½0; T such that 0

0

Fi ðti0 ; xðti0 ÞÞFi ðti0 ; yðti0 ÞÞ ¼ 0:

ð3:10Þ

Then we get by (H5) and Eq. (3.10) that zi ðti0 Þ ¼ xi ðti0 Þyi ðti0 Þ ¼ 0:

ð3:11Þ

Hence, for any t 2 ½ti0 ; ti0 þ T, we obtain  Z  Z t   t   i j_z i ðsÞj ds z_ i ðsÞdsr jzi ðtÞj ¼ zi ðt0 Þ þ   ti ti 0

and

0

  Z i Z t   t0 þT  i  j_z i ðsÞj ds: z_ i ðsÞ dsr jzi ðtÞj ¼ zi ðt0 þ TÞ þ   ti þT t 0

Now combining the above two inequalities, we get Z 1 T j_z i ðsÞj ds: jzi ðtÞjr 2 0 This, together with the Schwarz inequality, gives Z T 1=2 Z 1 T 1 pffiffiffiffi 2 Jzi J1 ¼ max jzi ðtÞjr j_z i ðtÞj dtr j_z i ðtÞj dt T 2 0 2 t2½ti0 ;ti0 þT 0

ð3:12Þ

Next, multiplying z€ i ðtÞ and Eq. (3.4) and integrating it from 0 to T, we have, by Lemma 2.8, (H6), Eq. (3.12) and the Schwarz inequality, that Z T Z T 0 0 j€z i ðtÞj2 dt ¼ ½Fi ðt; xðtÞÞFi ðt; yðtÞÞ€z i ðtÞ dt 0 0 Z T 0 0 r jFi ðt; xðtÞÞFi ðt; yðtÞÞjj€z i ðtÞj dt 0 Z T jzi ðtÞjj€z i ðtÞj dt rK0 0

Z pffiffiffiffi rK0 T Jzi J1

0

T

j€z i ðtÞj2 dt

1=2

ARTICLE IN PRESS L.-H. Zhang, Y. Wang / Journal of the Franklin Institute 347 (2010) 781–794

rK0

T2 4p

Z

793

T

j€z i ðtÞj2 dt:

ð3:13Þ

0

Since zi(t), z_ i ðtÞ and z€ i ðtÞ are continuous T-periodic functions, we get, by Lemma 2.8, (H7), Eqs. (3.12) and (3.13), that zi ðtÞ ¼ z_ i ðtÞ ¼ z€ i ðtÞ ¼ 0

for all t 2 R;

which implies that xi(t) ¼ yi(t) for all t 2 R and each i 2 f1; . . . ; Ng, i.e., x(t) ¼ y(t) for all t 2 R. This completes the proof. & 4. Two illustrative examples Example 4.1. Consider the existence and uniqueness of 2p-periodic classical solution of the non-autonomous second-order Hamiltonian system: € ¼ rF ðt; xðtÞÞ; xðtÞ

ð4:1Þ

where F ðt; xÞ ¼ 1=p4 jxj2 þ ðsin2 t=2Þx1 and T ¼ 2p. Proof. It is not difficult to verify that conditions (H0)–(H2) of Theorem 1.1 are satisfied if we take a ¼ 1, f ðtÞ ¼ 2=p4 and gðtÞ ¼ sin2 t=2. We now turn to others. For convenience, let p ¼ 3:1416. Setting b ¼ 3 we calculate that C0 ¼ 0.1351 and C1 ¼ 0.0352, which implies that (H3) and (H4) are satisfied. Applying the Theorem 1.1 we conclude that Eq. (4.1) has a unique 2p-periodic classical solution in H1T. Moreover, since rF ðt; 0Þa0, this unique solution is non-trivial. & 1 Example 4.2. Consider the existence and uniqueness of 10 -periodic classical solution of the non-autonomous second-order Hamiltonian system:

€ ¼ rF ðt; xðtÞÞ; xðtÞ 2

ð4:2Þ 2

where F ðt; xÞ ¼ pjxj ðcos 10ptÞx1 and T ¼

1 10.

Proof. It is not difficult to verify that conditions (H0), (H2) and (H5)–(H7) of Theorem 1.2 are satisfied if we take f ðtÞ ¼ 2p, gðtÞ ¼ cos2 10pt, a ¼ 1 and K0 ¼ 2p. We now turn to others. For convenience, let p ¼ 3:1416. Setting g ¼ 3 and K4 ¼ 0.1338 we calculate that K1 ¼ 0.0052, K2 ¼ 0.7396, K3 ¼ 0.0132 and K5 ¼ 0.0268, which implies that (H8)–(H10) 1 -periodic are satisfied. Applying the Theorem 1.2 we conclude that Eq. (4.2) has a unique 10 1 classical solution in HT. Moreover, since rF ðt; 0Þa0, this unique solution is nontrivial. & Acknowledgment The authors are grateful to the referee for their careful reading of the manuscript and helpful suggestions on this work. References [1] A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381.

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[2] A. Can˜ada, P. Dra´bek, A. Fonda, Handbook of Differential Equations: Ordinary Differential Equations, vol. 2, Elsevier, Amsterdam, 2006, pp. 77–146. [3] B. Hasselblatt, A. Katok, Handbook of Dynamical Systems, vol. 1A, Elsevier, Amsterdam, 2002, pp. 1091–1129. [4] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, Reprint of the 1952 edition, Cambridge University Press, London, 1988. [5] M.Y. Jiang, Periodic solutions of second order superquadratic Hamiltonian systems with potential changing sign (I), J. Differential Equations 219 (2005) 323–341. [6] M.Y. Jiang, Periodic solutions of second order superquadratic Hamiltonian systems with potential changing sign II, J. Differential Equations 219 (2005) 342–362. [7] G. Li, J. Liu, Existence and uniqueness of hyperbolic periodic solution for a class Hamiltonian systems, Chin. Sci. Bull. 44 (1999) 1184–1187. [8] Y.M. Long, Nonlinear oscillations for classical Hamiltonian systems with bi-even subquadratic potentials, Nonlinear Anal. 24 (1995) 1665–1671. [9] J. Mawhin, M. Willem, Critical points of convex perturbations of some indefinite quadratic forms and semilinear boundary value problems at resonance, Ann. Inst. Henri Poincare´ Anal. Non Lineare´ 3 (1986) 431–453. [10] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. [11] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems: a survey, SIAM J. Math. Anal. 13 (1982) 343–352. [12] P.H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978) 157–184. [13] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics No. 65, Am. Math. Soc., Providence, RI, 1986. [14] M. Schechter, Periodic non-autonomous second-order dynamical systems, J. Differential Equations 223 (2006) 290–302. [15] C.L. Tang, Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Am. Math. Soc. 126 (1998) 3263–3270. [16] Y.W. Ye, C.L. Tang, Periodic solutions for some nonautonomous second order Hamiltonian systems, J. Math. Anal. Appl. 344 (2008) 462–471.