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Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems
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Applied Mathematics and Computation 196 (2008) 494–500 www.elsevier.com/locate/amc
Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems q Yan-Fang Xue a
a,b
, Chun-Lei Tang
a,*
Department of Mathematics, Southwest University, Chongqing 400715, People’s Republic of China b College of Mathematics and Information Science, Xinyang Normal University, Xinyang City, Henan 464000, People’s Republic of China
Abstract Some multiplicity results are obtained for periodic solutions of the nonautonomous superquadratic second-order discrete Hamiltonian systems D2 uðt 1Þ þ rF ðt; uðtÞÞ ¼ 0 8t 2 Z by using critical point theory, especially, a three critical points theorem proposed by Brezis and Nirenberg. 2007 Elsevier Inc. All rights reserved. Keywords: Discrete Hamiltonian systems; Periodic solution; Critical points; (PS) condition; Local linking
1. Introduction and main results Consider the nonlinear second-order discrete Hamiltonian systems D2 uðt 1Þ þ rF ðt; uðtÞÞ ¼ 0
8t 2 Z;
ð1Þ
where Du(t) = u(t + 1) u(t), D2u(t) = D(Du(t)), F : Z · RN ! R, F(t, x) is continuously differential in x for every t 2 Z and T-periodic in t for all x 2 RN, T is a positive integer, $F(t, x) denotes the gradient of F(t, x) in x, we shall study the existence of T-periodic solution of systems (1). The theory of nonlinear difference equations (including discrete Hamiltonian systems) has been widely used to study discrete models in many fields such as computer science, economics, neural network, ecology, etc. Since the last decade, many scholars studied the qualitative properties of difference equations such as disconjugacy, stability, attractivity, oscillation and boundary value problems (see [1,2,4,6,10,11,17] and references therein). But results on periodic solutions of difference equations are relatively rare and the results usually q
Supported by National Natural Science Foundation of China (No. 10471113) and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, PRC. * Corresponding author. E-mail address: [email protected] (C.-L. Tang). 0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.06.015
Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
495
obtained by analytic techniques or various fixed point theorems (see [5], it is about the first-order linear difference equation). Recently, Guo and Yu developed a new method to study the existence and multiplicity of periodic solutions of difference equations by using critical point theory (see [7–9,18]), which seems to be a very powerful tool to deal with such problems. In all of their articles, they construct a variational framework and get the corresponding function by the matrix theory, here, we also build a suitable variational structure, so that the critical points of the variational functional correspond to the periodic solutions of the difference equation, but the method we used here is the operator theory which is different from theirs. Ref. [9] is about the first-order systems J Duðt 1Þ þ rH ðt; uðtÞÞ ¼ 0
8t 2 Z;
where H(t, x) 2 C1(R · R2N, R) 0N IN J¼ I N 0N is the symplectic matrix. Papers [7,8,18] are relating to the second-order systems (1), among these three articles, [8] is about the subquadratic condition while [7,18] refer to the superquadratic ones. In Ref. [7], they obtain the following theorem for the superquadratic second-order systems (1) with N = 1. Theorem A (see [7]). Suppose that F(t, x) satisfies the following conditions: (i) F(t, x) 2 C1(R · R,R) and there exists a positive integer T, such that for any (t, x) 2 R · R, F(t + T, x) = F(t, x); ðt;xÞ (ii) For any x 2 R, F(t, x) P 0 and Fjxj 2 ! 0 as jxj ! 0; (iii) There exist some constants R > 0, l > 2 such that for any jxj P R ðx; rF ðt; xÞÞ P lF ðt; xÞ > 0: Then (1) with N = 1 possesses at least three periodic solutions with period T. In Ref. [18], they use the linking theorem to prove the existence of three periodic solutions, which is more general than Theorem A, the main result of their paper is as follows. Theorem B (see [18]). Suppose that F(t, x) satisfies: (a) F(t, x) 2 C1(Z · RN, R) and there exists a positive integer T, such that for any (t, x) 2 Z · RN, F(t + T, x) = F(t, x) and For any x 2 RN, F(t, x) P 0; (b) There are constants d0 > 0, a 2 (0, 1 cos(2p/T)) such that F ðt; xÞ 6 ajxj2 for all (t, x) 2 Z · RN and jxj 6 d0; (c) There exist constants R > 0, c > 0, b 2 (2, + 1) when T is even or b 2 (1 + cos(p/T), + 1) when T is odd such that for any jxj P R F ðt; xÞ P bjxj2 c: Then (1) possesses at least three T-periodic solutions. We know that, for the existence of periodic solutions of Hamiltonian systems € uðtÞ þ rF ðt; uðtÞÞ ¼ 0; a:e: t 2 ½0; T ; _ _ Þ ¼ 0; uð0Þ uðT Þ ¼ uð0Þ uðT there have been many studies in the literature using critical point theory which contains the least action principle, the minimax theory, the geometrical index theory and the Morse theory (see [12–16]), some of those
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articles build a variational framework and get the corresponding function by the operator theory. Motivated by this idea, we construct a variation structure for system (1) and obtain the following theorems. Theorem 1. Suppose that F(t, x) satisfies (F1) There exists a positive integer T, such that F(t + T, x) = F(t, x) for all (t, x) 2 Z · RN; (F2) There is a constant d1, such that for any jxj 6 d1 1 2 0 6 F ðt; xÞ 6 k1 jxj ; 2 where k1 = 2(1 cos(2p/T)); (F3) For t 2 Z[1, T], there exists a constant b 2 (2, + 1) when T is even or b 2(1 + cos(p/T), +1) when T is odd such that lim inf jxj!1
F ðt; xÞ jxj2
P b:
Then problem (1) possesses at least three periodic solutions with period T. Remark 1. Obviously, Theorem B is more general than Theorem A. Comparing Theorem B with our Theorem 1, we can see that F(t, x) P 0 for all x 2 RN in Theorem B and this condition is essential for getting the critical points, but we only need F(t, x) P 0 for jxj 6 d1 in (F2). Also, the case where a = k1/2 = 1 cos(2p/T) cannot be hold in Theorem B, but it is allowed in our Theorem 1. There are functions F satisfying Theorem 1 and not fulfilling the results of Theorems A and B. For example 1 F ðt; xÞ ¼ k1 ðjxj2 jxj4 þ jxj6 Þ: 2 By a simple computation, we know that F does not satisfy condition (ii) in Theorem A and condition (b) in Theorem B. We shall prove more general results than Theorem 1. Theorem 2. Suppose that F(t, x) satisfies (F1), (F3) and the following: (F4) There exist some constants d > 0, k 2 Z[0, [T/2] 1], such that 1 1 2 2 kk jxj 6 F ðt; xÞ 6 kkþ1 jxj 2 2 for all jxj 6 d and t 2 Z[1, T], where Z[a, b] :¼ Z \ [a, b] for every a, b 2 Z with a 6 b, kk = 2 2 cos kx, x = 2p/T, T > 2, [ Æ ] denotes the Gauss Function. Then problem (1) possesses at least three periodic solutions with period T. Remark 2. For F(t, x) is continuously differential in x for every t 2 Z, we can find that (F3) is equal to condition (c). Let k = 0 in (F4), we have 1 2 0 6 F ðt; xÞ 6 k1 jxj 2
ð2Þ
for all jxj 6 d and t 2 Z[1, T]. (2) means that (F2) is the special case of (F4). To illustrate our main results, we give a function which satisfies Theorem 2 and does not fulfill the results of Theorem 1. For example 1 F ðt; xÞ ¼ jxjm lnð1 þ jxj2 Þ þ kjxj2 ; 2 where m > 2, 1 cos(2p/T) = k1 < k < 3.
Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
497
2. Proof of theorems In order to apply the critical point theory, we introduce a variational structure which is different from [7– 9,18], from this framework structure, we can reduce the problem of finding T-periodic solutions of (1) to the one of seeking the critical points of a corresponding functional. First, we shall state some basic notations. For any given positive integer T, HT is defined by H T ¼ fu : Z ! RN juðt þ T Þ ¼ uðtÞ; t 2 Zg: HT can be equipped with the inner product T X ðuðtÞ; vðtÞÞ 8u; v 2 H T hu; vi ¼ t¼1
by which norm k kH T can be induced by !12 T X 2 kuk ¼ juðtÞj 8u 2 H T ; t¼1
where (Æ , Æ) and j Æ j denote the usual inner product and the usual norm in RN. It is easy to see that (HT, hÆ , Æi) is a finite dimensional Hilbert space and linear homeomorphic to RNT. Then, for any u, v 2 HT, we have a useful equality T T X X ðD2 uðt 1Þ; vðtÞÞ ¼ ðDuðtÞ; DvðtÞÞ: ð3Þ t¼1
t¼1
In fact, it follows from u(t + T) = u(t), v(t + T) = v(t) that T T X X ðD2 uðt 1Þ; vðtÞÞ ¼ ðuðt þ 1Þ 2uðtÞ þ uðt 1Þ; vðtÞÞ t¼1
t¼1
¼
T X
ðuðt þ 1Þ uðtÞ; vðtÞÞ þ
t¼1
¼
T X
T X
ðuðtÞ uðt 1Þ; vðtÞÞ
t¼1
ðDuðtÞ; vðtÞÞ þ
t¼1
¼
T X
T 1 X
ðuðt þ 1Þ uðtÞ; vðt þ 1ÞÞ
t¼0
ðDuðtÞ; vðtÞÞ þ
t¼1
T X
ðDuðtÞ; vðt þ 1ÞÞ ¼
t¼1
T X ðDuðtÞ; DvðtÞÞ: t¼1
Consider the functional u defined on HT by T T X 1X 2 jDuðtÞj þ F ðt; uðtÞÞ: uðuÞ ¼ 2 t¼1 t¼1 Since for any v 2 HT, one has hu0 ðuÞ; vi ¼
T T X X ðDuðtÞ; DvðtÞÞ þ ðrF ðt; uðtÞÞ; vðtÞÞ: t¼1
t¼1
Then u 2 HT is a critical point of u if and only if T T X X ðDuðtÞ; DvðtÞÞ ¼ ðrF ðt; uðtÞÞ; vðtÞÞ: t¼1
t¼1
It follows from (3) and (4) that
T T X X ðD2 uðt 1Þ; vðtÞÞ ¼ ðrF ðt; uðtÞÞ; vðtÞÞ: t¼1
t¼1
ð4Þ
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Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
By the arbitrary of v, we conclude that D2 uðt 1Þ þ rF ðt; uðtÞÞ ¼ 0
8t 2 Z:
Since u 2 HT is T-periodic, and F(t, x) is T-periodic in t, hence u 2 HT is a critical point of u if and only if for any t 2 Z, D2u(t 1) + $F(t, u(t)) = 0. Thus the problem of finding the T-periodic solution for problem (1) is reducing to the one of seeking the critical point of functional u on HT. Now, we discuss the properties of finite dimensional space HT by the operator theory and give a three critical points theorem which belongs to Brezis and Nirenberg. Lemma 1. As a subspace of HT, Nk is defined by N k :¼ fu 2 H T j D2 uðt 1Þ ¼ kk uðtÞg; where kk = 2 2 cos kx, k 2 Z[0, [T/2]], x = 2p/T. Then we claim that (i) Nk ? Nj, k 5 j, k, j 2 Z[0, [T/2]], ½T =2 (ii) H T ¼ k¼0 N k . Proof. (i) By the definition of Nk and (3), for any u 2 Nk, v 2 Nj, k 5 j, k, j 2 Z[0, [T/2]], we obtain kk hu; vi ¼ kk
T X
ðuðtÞ; vðtÞÞ ¼
T X
t¼1
¼
T X
ðD2 uðt 1Þ; vðtÞÞ ¼
t¼1
T X ðDuðtÞ; DvðtÞÞ t¼1
ðuðtÞ; D2 vðt 1ÞÞ ¼ kj hu; vi:
t¼1
Since kk 5 kj, we have hu, vi = 0, then (i) is verified. (ii) For k 2 Z[0, [T/2]], Wk is defined by W k ¼ fa cos kxt þ b sin kxtja; b 2 RN ; k 2 Z½0; ½T =2g; then Wk Nk. In fact, for any w = a cos kxt + b sin kxt 2 Wk, one has D2 wðt 1Þ ¼ wðt þ 1Þ þ 2wðtÞ wðt 1Þ ¼ aðcos kxðt þ 1Þ þ cos kxðt 1ÞÞ bðsin kxðt þ 1Þ þ sin kxðt 1ÞÞ þ 2ða cos kxt þ b sin kxtÞ ¼ 2a cos kxt cos kx 2b sin kxt cos kx þ 2ða cos kxt þ b sin kxtÞ ¼ ð2 2 cos kxÞða cos kxt þ b sin kxtÞ ¼ kk wðtÞ; ½T =2
½T =2
this implies that w 2 Nk, thus Wk Nk, furthermore, we have k¼0 W k k¼0 N k H T . It is easy to get that dimW0 = N; dim Wk = 2N, when k 2 Z, 0 < k < T/2; dim W[T/2] = N, when T is even, thus we have ½T =2 dimk¼0 W k ¼ NT ¼ dimH T . Then we complete the proof of (ii). h ½T =2
Lemma 2. Define H k :¼ kj¼0 N j , H ? k :¼ j¼kþ1 N j , k 2 Z[0, [T/2] 1], then one has T X
jDuðtÞj2 6 kk kuk2
8u 2 H k
ð5Þ
t¼1 T X
jDuðtÞj2 P kkþ1 kuk2
8u 2 H ? k :
ð6Þ
t¼1
Proof. By (3), for any uk 2 Nk, we have T X t¼1
ðDuk ðtÞ; Duk ðtÞÞ ¼
T T X X ðD2 uk ðt 1Þ; uk ðtÞÞ ¼ kk ðuk ðtÞ; uk ðtÞÞ: t¼1
t¼1
ð7Þ
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499
Since kk = 2 2 cos kx, x = 2p/T, k 2 Z[0, [T/2]], we have 0 = k0 < k1 < < k[T/2] 6 4 and k[T/2] = 4 when T is even; kP [T/2] = 2(1 + cos(p/T)) when T is odd. For any u 2 Hk, there exist some constants aj, j 2 Z[0, k], such k that u ¼ j¼0 aj uj , where uj 2 Nj. It follows from Lemma 1 and the above equality (7) that ! T T k k T X k T X k X X X X X X 2 jDuðtÞj ¼ aj Duj ðtÞ; aj Duj ðtÞ ¼ a2j ðDuj ðtÞ; Duj ðtÞÞ ¼ a2j kj ðuj ðtÞ; uj ðtÞÞ t¼1
t¼1
6 kk
j¼0
T X
k X
t¼1
j¼0
j¼0
ðaj uj ðtÞ; aj uj ðtÞÞ ¼ kk
t¼1
j¼0
T X
k X
t¼1
j¼0
aj uj ðtÞ;
k X
! aj uj ðtÞ
t¼1
j¼0 2
¼ kk kuk ;
j¼0
then (5) is verified. By using the same method, we can get (6). h Lemma 3 (see [3]). Let u be a C1 function on X = X1 X2 with u(0) = 0, satisfying (PS) condition and assume that, for some d > 0 uðuÞ P 0 for u 2 X 1 ; kuk 6 d; uðuÞ 6 0 for u 2 X 2 ; kuk 6 d:
ð8Þ ð9Þ
Assume also that u is bounded below and infXu < 0, then u has at least two nonzero critical points. Proof of Theorem 2. We shall apply Lemma 3 to the functional u. By (F4), we can get F(t, 0) = 0, so one can say that u(0) = 0, now, we will specify that u satisfies the rest conditions of Lemma 3 or u has infinite periodic solutions. First, we prove that u satisfies (PS) condition and is bounded from below. Suppose that {uk} is a (PS) sequence for u, that is, u 0 (uk) ! 0 as k ! 1 and {u(uk)} is bounded. In order to get a convergent subsequence of {uk}, we only need to prove that {uk} is bounded by the properties of the finite dimensional space. In fact, by (F3), there exist some constants e > 0, G > 0 such that for any jxj P G 2
F ðt; xÞ P ðb þ eÞjxj : Set C1 ¼ ftjt 2 Z½1; T ; juk ðtÞj P Gg;
C2 ¼ ftjt 2 Z½1; T ; juk ðtÞj < Gg:
Then, we obtain T T X X X 1X 1 jDuk ðtÞj2 þ F ðt; uk ðtÞÞ P k½T =2 kuk k2 þ ðb þ eÞjuk ðtÞj2 þ F ðt; uk ðtÞÞ 2 t¼1 2 t2C1 t2C2 t¼1 X ðF ðt; uk ðtÞÞ ðb þ eÞjuk ðtÞj2 Þ: P ekuk k2 þ
uðuk Þ ¼
t2C2
The continuity of F(t, x) (b + e)jxj2 with respect to (t, x) implies that there exists a positive constant M, such that for any t 2 Z[1, T], jxj < G, F(t, x) (b + e)jxj2 P M. Then we have 2
uðuk Þ P ekuk k TM:
ð10Þ
Since e > 0 and {u(uk)} is bounded, then {uk} is bounded. As a consequence in finite dimensional space HT, {uk} has a convergent subsequence, thus we verified that u satisfies (PS) condition. In addition, from (10), we get that u is bounded from below. Next, we claim that u has a local linking at 0, that is, u satisfies (8) and (9). Also we have inf u2H T uðuÞ 6 0. It follows from (F4) and Lemma 2 that uðuÞ ¼
T T T X X 1X 1 1 kk juðtÞj2 ¼ 0 jDuðtÞj2 þ F ðt; uðtÞÞ P kk kuk2 þ 2 t¼1 2 2 t¼1 t¼1
for all u 2 Hk with kuk < d.
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Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
Similar to above, we conclude that uðuÞ ¼
T T T X X 1X 1 1 jDuðtÞj2 þ F ðt; uðtÞÞ 6 kkþ1 kuk2 þ kkþ1 juðtÞj2 ¼ 0 2 t¼1 2 2 t¼1 t¼1
for all u 2 H ? k with kuk < d. Then u has a local linking at 0, which implies that 0 is a critical point of u. At the same time, we get that inf u2H T uðuÞ 6 0. In the case that inf u2H T uðuÞ < 0, our results follows from Lemma 3 directly. In the case that inf u2H T uðuÞ ¼ 0, from above, we have uðuÞ ¼ inf u2H T uðuÞ ¼ 0 for all u 2 H ? k with kuk < d ? which implies that all u 2 H ? with kuk < d are minimum points of u. Hence all u 2 H with kuk < d are k k solutions of systems (1), and systems (1) has infinite solutions in HT. Therefore, Theorem 2 is verified. Then we complete the proof of our main result. h References [1] C.D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, J. Math. Anal. Appl. 180 (2) (1993) 498– 517. [2] M. Bohner, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199 (3) (1996) 804– 826. [3] H. Brezis, L. Nirenberg, Remarks on finding critical points, Commun. Pure Appl. Math. 44 (8–9) (1991) 939–963. [4] Shao-Zhu Chen, Disconjugacy, disfocality, and oscillation of second order difference equations, J. Differen. Equat. 107 (2) (1994) 383– 394. [5] S. Elaydi, S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac. 37 (3) (1994) 401–413. [6] L.H. Erbe, Peng-Xiang Yan, Disconjugacy for linear Hamiltonian difference systems, J. Math. Anal. Appl. 167 (2) (1992) 355–367. [7] Zhiming Guo, Jianshe Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A 46 (4) (2003) 506–515. [8] Zhiming Guo, Jianshe Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc. 68 (2) (2003) 419–430. [9] Zhiming Guo, Jianshe Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal. 55 (7–8) (2003) 969–983. [10] P. Hartman, Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity, Trans. Am. Math. Soc. 246 (1978) 1–30. [11] J.W. Hooker, W.T. Patula, A second-order nonlinear difference equation: oscillation and asymptotic behavior, J. Math. Anal. Appl. 91 (1) (1983) 9–29. [12] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. [13] R. Michalek, G. Tarantello, Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems, J. Differen. Equat. 72 (1) (1988) 28–55. [14] Chun-Lei Tang, Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Am. Math. Soc. 126 (11) (1998) 3263–3270. [15] Chun-Lei Tang, Xing-Ping Wu, Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2) (2001) 386–397. [16] Zhu-Lian Tao, Chun-Lei Tang, Periodic and subharmonic solutions of second-order Hamiltonian systems, J. Math. Anal. Appl. 293 (2) (2004) 435–445. [17] B.G. Zhang, G.D. Chen, Oscillation of certain second order nonlinear difference equations, J. Math. Anal. Appl. 199 (3) (1996) 827– 841. [18] Zhan Zhou, Jianshe Yu, Zhiming Guo, Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh 134A (2004) 1013–1022.
Applied Mathematics and Computation 196 (2008) 494–500 www.elsevier.com/locate/amc
Multiple periodic solutions for superquadratic second-order discrete Hamiltonian systems q Yan-Fang Xue a
a,b
, Chun-Lei Tang
a,*
Department of Mathematics, Southwest University, Chongqing 400715, People’s Republic of China b College of Mathematics and Information Science, Xinyang Normal University, Xinyang City, Henan 464000, People’s Republic of China
Abstract Some multiplicity results are obtained for periodic solutions of the nonautonomous superquadratic second-order discrete Hamiltonian systems D2 uðt 1Þ þ rF ðt; uðtÞÞ ¼ 0 8t 2 Z by using critical point theory, especially, a three critical points theorem proposed by Brezis and Nirenberg. 2007 Elsevier Inc. All rights reserved. Keywords: Discrete Hamiltonian systems; Periodic solution; Critical points; (PS) condition; Local linking
1. Introduction and main results Consider the nonlinear second-order discrete Hamiltonian systems D2 uðt 1Þ þ rF ðt; uðtÞÞ ¼ 0
8t 2 Z;
ð1Þ
where Du(t) = u(t + 1) u(t), D2u(t) = D(Du(t)), F : Z · RN ! R, F(t, x) is continuously differential in x for every t 2 Z and T-periodic in t for all x 2 RN, T is a positive integer, $F(t, x) denotes the gradient of F(t, x) in x, we shall study the existence of T-periodic solution of systems (1). The theory of nonlinear difference equations (including discrete Hamiltonian systems) has been widely used to study discrete models in many fields such as computer science, economics, neural network, ecology, etc. Since the last decade, many scholars studied the qualitative properties of difference equations such as disconjugacy, stability, attractivity, oscillation and boundary value problems (see [1,2,4,6,10,11,17] and references therein). But results on periodic solutions of difference equations are relatively rare and the results usually q
Supported by National Natural Science Foundation of China (No. 10471113) and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, PRC. * Corresponding author. E-mail address: [email protected] (C.-L. Tang). 0096-3003/$ - see front matter 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.06.015
Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
495
obtained by analytic techniques or various fixed point theorems (see [5], it is about the first-order linear difference equation). Recently, Guo and Yu developed a new method to study the existence and multiplicity of periodic solutions of difference equations by using critical point theory (see [7–9,18]), which seems to be a very powerful tool to deal with such problems. In all of their articles, they construct a variational framework and get the corresponding function by the matrix theory, here, we also build a suitable variational structure, so that the critical points of the variational functional correspond to the periodic solutions of the difference equation, but the method we used here is the operator theory which is different from theirs. Ref. [9] is about the first-order systems J Duðt 1Þ þ rH ðt; uðtÞÞ ¼ 0
8t 2 Z;
where H(t, x) 2 C1(R · R2N, R) 0N IN J¼ I N 0N is the symplectic matrix. Papers [7,8,18] are relating to the second-order systems (1), among these three articles, [8] is about the subquadratic condition while [7,18] refer to the superquadratic ones. In Ref. [7], they obtain the following theorem for the superquadratic second-order systems (1) with N = 1. Theorem A (see [7]). Suppose that F(t, x) satisfies the following conditions: (i) F(t, x) 2 C1(R · R,R) and there exists a positive integer T, such that for any (t, x) 2 R · R, F(t + T, x) = F(t, x); ðt;xÞ (ii) For any x 2 R, F(t, x) P 0 and Fjxj 2 ! 0 as jxj ! 0; (iii) There exist some constants R > 0, l > 2 such that for any jxj P R ðx; rF ðt; xÞÞ P lF ðt; xÞ > 0: Then (1) with N = 1 possesses at least three periodic solutions with period T. In Ref. [18], they use the linking theorem to prove the existence of three periodic solutions, which is more general than Theorem A, the main result of their paper is as follows. Theorem B (see [18]). Suppose that F(t, x) satisfies: (a) F(t, x) 2 C1(Z · RN, R) and there exists a positive integer T, such that for any (t, x) 2 Z · RN, F(t + T, x) = F(t, x) and For any x 2 RN, F(t, x) P 0; (b) There are constants d0 > 0, a 2 (0, 1 cos(2p/T)) such that F ðt; xÞ 6 ajxj2 for all (t, x) 2 Z · RN and jxj 6 d0; (c) There exist constants R > 0, c > 0, b 2 (2, + 1) when T is even or b 2 (1 + cos(p/T), + 1) when T is odd such that for any jxj P R F ðt; xÞ P bjxj2 c: Then (1) possesses at least three T-periodic solutions. We know that, for the existence of periodic solutions of Hamiltonian systems € uðtÞ þ rF ðt; uðtÞÞ ¼ 0; a:e: t 2 ½0; T ; _ _ Þ ¼ 0; uð0Þ uðT Þ ¼ uð0Þ uðT there have been many studies in the literature using critical point theory which contains the least action principle, the minimax theory, the geometrical index theory and the Morse theory (see [12–16]), some of those
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articles build a variational framework and get the corresponding function by the operator theory. Motivated by this idea, we construct a variation structure for system (1) and obtain the following theorems. Theorem 1. Suppose that F(t, x) satisfies (F1) There exists a positive integer T, such that F(t + T, x) = F(t, x) for all (t, x) 2 Z · RN; (F2) There is a constant d1, such that for any jxj 6 d1 1 2 0 6 F ðt; xÞ 6 k1 jxj ; 2 where k1 = 2(1 cos(2p/T)); (F3) For t 2 Z[1, T], there exists a constant b 2 (2, + 1) when T is even or b 2(1 + cos(p/T), +1) when T is odd such that lim inf jxj!1
F ðt; xÞ jxj2
P b:
Then problem (1) possesses at least three periodic solutions with period T. Remark 1. Obviously, Theorem B is more general than Theorem A. Comparing Theorem B with our Theorem 1, we can see that F(t, x) P 0 for all x 2 RN in Theorem B and this condition is essential for getting the critical points, but we only need F(t, x) P 0 for jxj 6 d1 in (F2). Also, the case where a = k1/2 = 1 cos(2p/T) cannot be hold in Theorem B, but it is allowed in our Theorem 1. There are functions F satisfying Theorem 1 and not fulfilling the results of Theorems A and B. For example 1 F ðt; xÞ ¼ k1 ðjxj2 jxj4 þ jxj6 Þ: 2 By a simple computation, we know that F does not satisfy condition (ii) in Theorem A and condition (b) in Theorem B. We shall prove more general results than Theorem 1. Theorem 2. Suppose that F(t, x) satisfies (F1), (F3) and the following: (F4) There exist some constants d > 0, k 2 Z[0, [T/2] 1], such that 1 1 2 2 kk jxj 6 F ðt; xÞ 6 kkþ1 jxj 2 2 for all jxj 6 d and t 2 Z[1, T], where Z[a, b] :¼ Z \ [a, b] for every a, b 2 Z with a 6 b, kk = 2 2 cos kx, x = 2p/T, T > 2, [ Æ ] denotes the Gauss Function. Then problem (1) possesses at least three periodic solutions with period T. Remark 2. For F(t, x) is continuously differential in x for every t 2 Z, we can find that (F3) is equal to condition (c). Let k = 0 in (F4), we have 1 2 0 6 F ðt; xÞ 6 k1 jxj 2
ð2Þ
for all jxj 6 d and t 2 Z[1, T]. (2) means that (F2) is the special case of (F4). To illustrate our main results, we give a function which satisfies Theorem 2 and does not fulfill the results of Theorem 1. For example 1 F ðt; xÞ ¼ jxjm lnð1 þ jxj2 Þ þ kjxj2 ; 2 where m > 2, 1 cos(2p/T) = k1 < k < 3.
Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
497
2. Proof of theorems In order to apply the critical point theory, we introduce a variational structure which is different from [7– 9,18], from this framework structure, we can reduce the problem of finding T-periodic solutions of (1) to the one of seeking the critical points of a corresponding functional. First, we shall state some basic notations. For any given positive integer T, HT is defined by H T ¼ fu : Z ! RN juðt þ T Þ ¼ uðtÞ; t 2 Zg: HT can be equipped with the inner product T X ðuðtÞ; vðtÞÞ 8u; v 2 H T hu; vi ¼ t¼1
by which norm k kH T can be induced by !12 T X 2 kuk ¼ juðtÞj 8u 2 H T ; t¼1
where (Æ , Æ) and j Æ j denote the usual inner product and the usual norm in RN. It is easy to see that (HT, hÆ , Æi) is a finite dimensional Hilbert space and linear homeomorphic to RNT. Then, for any u, v 2 HT, we have a useful equality T T X X ðD2 uðt 1Þ; vðtÞÞ ¼ ðDuðtÞ; DvðtÞÞ: ð3Þ t¼1
t¼1
In fact, it follows from u(t + T) = u(t), v(t + T) = v(t) that T T X X ðD2 uðt 1Þ; vðtÞÞ ¼ ðuðt þ 1Þ 2uðtÞ þ uðt 1Þ; vðtÞÞ t¼1
t¼1
¼
T X
ðuðt þ 1Þ uðtÞ; vðtÞÞ þ
t¼1
¼
T X
T X
ðuðtÞ uðt 1Þ; vðtÞÞ
t¼1
ðDuðtÞ; vðtÞÞ þ
t¼1
¼
T X
T 1 X
ðuðt þ 1Þ uðtÞ; vðt þ 1ÞÞ
t¼0
ðDuðtÞ; vðtÞÞ þ
t¼1
T X
ðDuðtÞ; vðt þ 1ÞÞ ¼
t¼1
T X ðDuðtÞ; DvðtÞÞ: t¼1
Consider the functional u defined on HT by T T X 1X 2 jDuðtÞj þ F ðt; uðtÞÞ: uðuÞ ¼ 2 t¼1 t¼1 Since for any v 2 HT, one has hu0 ðuÞ; vi ¼
T T X X ðDuðtÞ; DvðtÞÞ þ ðrF ðt; uðtÞÞ; vðtÞÞ: t¼1
t¼1
Then u 2 HT is a critical point of u if and only if T T X X ðDuðtÞ; DvðtÞÞ ¼ ðrF ðt; uðtÞÞ; vðtÞÞ: t¼1
t¼1
It follows from (3) and (4) that
T T X X ðD2 uðt 1Þ; vðtÞÞ ¼ ðrF ðt; uðtÞÞ; vðtÞÞ: t¼1
t¼1
ð4Þ
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Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
By the arbitrary of v, we conclude that D2 uðt 1Þ þ rF ðt; uðtÞÞ ¼ 0
8t 2 Z:
Since u 2 HT is T-periodic, and F(t, x) is T-periodic in t, hence u 2 HT is a critical point of u if and only if for any t 2 Z, D2u(t 1) + $F(t, u(t)) = 0. Thus the problem of finding the T-periodic solution for problem (1) is reducing to the one of seeking the critical point of functional u on HT. Now, we discuss the properties of finite dimensional space HT by the operator theory and give a three critical points theorem which belongs to Brezis and Nirenberg. Lemma 1. As a subspace of HT, Nk is defined by N k :¼ fu 2 H T j D2 uðt 1Þ ¼ kk uðtÞg; where kk = 2 2 cos kx, k 2 Z[0, [T/2]], x = 2p/T. Then we claim that (i) Nk ? Nj, k 5 j, k, j 2 Z[0, [T/2]], ½T =2 (ii) H T ¼ k¼0 N k . Proof. (i) By the definition of Nk and (3), for any u 2 Nk, v 2 Nj, k 5 j, k, j 2 Z[0, [T/2]], we obtain kk hu; vi ¼ kk
T X
ðuðtÞ; vðtÞÞ ¼
T X
t¼1
¼
T X
ðD2 uðt 1Þ; vðtÞÞ ¼
t¼1
T X ðDuðtÞ; DvðtÞÞ t¼1
ðuðtÞ; D2 vðt 1ÞÞ ¼ kj hu; vi:
t¼1
Since kk 5 kj, we have hu, vi = 0, then (i) is verified. (ii) For k 2 Z[0, [T/2]], Wk is defined by W k ¼ fa cos kxt þ b sin kxtja; b 2 RN ; k 2 Z½0; ½T =2g; then Wk Nk. In fact, for any w = a cos kxt + b sin kxt 2 Wk, one has D2 wðt 1Þ ¼ wðt þ 1Þ þ 2wðtÞ wðt 1Þ ¼ aðcos kxðt þ 1Þ þ cos kxðt 1ÞÞ bðsin kxðt þ 1Þ þ sin kxðt 1ÞÞ þ 2ða cos kxt þ b sin kxtÞ ¼ 2a cos kxt cos kx 2b sin kxt cos kx þ 2ða cos kxt þ b sin kxtÞ ¼ ð2 2 cos kxÞða cos kxt þ b sin kxtÞ ¼ kk wðtÞ; ½T =2
½T =2
this implies that w 2 Nk, thus Wk Nk, furthermore, we have k¼0 W k k¼0 N k H T . It is easy to get that dimW0 = N; dim Wk = 2N, when k 2 Z, 0 < k < T/2; dim W[T/2] = N, when T is even, thus we have ½T =2 dimk¼0 W k ¼ NT ¼ dimH T . Then we complete the proof of (ii). h ½T =2
Lemma 2. Define H k :¼ kj¼0 N j , H ? k :¼ j¼kþ1 N j , k 2 Z[0, [T/2] 1], then one has T X
jDuðtÞj2 6 kk kuk2
8u 2 H k
ð5Þ
t¼1 T X
jDuðtÞj2 P kkþ1 kuk2
8u 2 H ? k :
ð6Þ
t¼1
Proof. By (3), for any uk 2 Nk, we have T X t¼1
ðDuk ðtÞ; Duk ðtÞÞ ¼
T T X X ðD2 uk ðt 1Þ; uk ðtÞÞ ¼ kk ðuk ðtÞ; uk ðtÞÞ: t¼1
t¼1
ð7Þ
Y.-F. Xue, C.-L. Tang / Applied Mathematics and Computation 196 (2008) 494–500
499
Since kk = 2 2 cos kx, x = 2p/T, k 2 Z[0, [T/2]], we have 0 = k0 < k1 < < k[T/2] 6 4 and k[T/2] = 4 when T is even; kP [T/2] = 2(1 + cos(p/T)) when T is odd. For any u 2 Hk, there exist some constants aj, j 2 Z[0, k], such k that u ¼ j¼0 aj uj , where uj 2 Nj. It follows from Lemma 1 and the above equality (7) that ! T T k k T X k T X k X X X X X X 2 jDuðtÞj ¼ aj Duj ðtÞ; aj Duj ðtÞ ¼ a2j ðDuj ðtÞ; Duj ðtÞÞ ¼ a2j kj ðuj ðtÞ; uj ðtÞÞ t¼1
t¼1
6 kk
j¼0
T X
k X
t¼1
j¼0
j¼0
ðaj uj ðtÞ; aj uj ðtÞÞ ¼ kk
t¼1
j¼0
T X
k X
t¼1
j¼0
aj uj ðtÞ;
k X
! aj uj ðtÞ
t¼1
j¼0 2
¼ kk kuk ;
j¼0
then (5) is verified. By using the same method, we can get (6). h Lemma 3 (see [3]). Let u be a C1 function on X = X1 X2 with u(0) = 0, satisfying (PS) condition and assume that, for some d > 0 uðuÞ P 0 for u 2 X 1 ; kuk 6 d; uðuÞ 6 0 for u 2 X 2 ; kuk 6 d:
ð8Þ ð9Þ
Assume also that u is bounded below and infXu < 0, then u has at least two nonzero critical points. Proof of Theorem 2. We shall apply Lemma 3 to the functional u. By (F4), we can get F(t, 0) = 0, so one can say that u(0) = 0, now, we will specify that u satisfies the rest conditions of Lemma 3 or u has infinite periodic solutions. First, we prove that u satisfies (PS) condition and is bounded from below. Suppose that {uk} is a (PS) sequence for u, that is, u 0 (uk) ! 0 as k ! 1 and {u(uk)} is bounded. In order to get a convergent subsequence of {uk}, we only need to prove that {uk} is bounded by the properties of the finite dimensional space. In fact, by (F3), there exist some constants e > 0, G > 0 such that for any jxj P G 2
F ðt; xÞ P ðb þ eÞjxj : Set C1 ¼ ftjt 2 Z½1; T ; juk ðtÞj P Gg;
C2 ¼ ftjt 2 Z½1; T ; juk ðtÞj < Gg:
Then, we obtain T T X X X 1X 1 jDuk ðtÞj2 þ F ðt; uk ðtÞÞ P k½T =2 kuk k2 þ ðb þ eÞjuk ðtÞj2 þ F ðt; uk ðtÞÞ 2 t¼1 2 t2C1 t2C2 t¼1 X ðF ðt; uk ðtÞÞ ðb þ eÞjuk ðtÞj2 Þ: P ekuk k2 þ
uðuk Þ ¼
t2C2
The continuity of F(t, x) (b + e)jxj2 with respect to (t, x) implies that there exists a positive constant M, such that for any t 2 Z[1, T], jxj < G, F(t, x) (b + e)jxj2 P M. Then we have 2
uðuk Þ P ekuk k TM:
ð10Þ
Since e > 0 and {u(uk)} is bounded, then {uk} is bounded. As a consequence in finite dimensional space HT, {uk} has a convergent subsequence, thus we verified that u satisfies (PS) condition. In addition, from (10), we get that u is bounded from below. Next, we claim that u has a local linking at 0, that is, u satisfies (8) and (9). Also we have inf u2H T uðuÞ 6 0. It follows from (F4) and Lemma 2 that uðuÞ ¼
T T T X X 1X 1 1 kk juðtÞj2 ¼ 0 jDuðtÞj2 þ F ðt; uðtÞÞ P kk kuk2 þ 2 t¼1 2 2 t¼1 t¼1
for all u 2 Hk with kuk < d.
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Similar to above, we conclude that uðuÞ ¼
T T T X X 1X 1 1 jDuðtÞj2 þ F ðt; uðtÞÞ 6 kkþ1 kuk2 þ kkþ1 juðtÞj2 ¼ 0 2 t¼1 2 2 t¼1 t¼1
for all u 2 H ? k with kuk < d. Then u has a local linking at 0, which implies that 0 is a critical point of u. At the same time, we get that inf u2H T uðuÞ 6 0. In the case that inf u2H T uðuÞ < 0, our results follows from Lemma 3 directly. In the case that inf u2H T uðuÞ ¼ 0, from above, we have uðuÞ ¼ inf u2H T uðuÞ ¼ 0 for all u 2 H ? k with kuk < d ? which implies that all u 2 H ? with kuk < d are minimum points of u. Hence all u 2 H with kuk < d are k k solutions of systems (1), and systems (1) has infinite solutions in HT. Therefore, Theorem 2 is verified. Then we complete the proof of our main result. h References [1] C.D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, J. Math. Anal. Appl. 180 (2) (1993) 498– 517. [2] M. Bohner, Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl. 199 (3) (1996) 804– 826. [3] H. Brezis, L. Nirenberg, Remarks on finding critical points, Commun. Pure Appl. Math. 44 (8–9) (1991) 939–963. [4] Shao-Zhu Chen, Disconjugacy, disfocality, and oscillation of second order difference equations, J. Differen. Equat. 107 (2) (1994) 383– 394. [5] S. Elaydi, S. Zhang, Stability and periodicity of difference equations with finite delay, Funkcial. Ekvac. 37 (3) (1994) 401–413. [6] L.H. Erbe, Peng-Xiang Yan, Disconjugacy for linear Hamiltonian difference systems, J. Math. Anal. Appl. 167 (2) (1992) 355–367. [7] Zhiming Guo, Jianshe Yu, Existence of periodic and subharmonic solutions for second-order superlinear difference equations, Sci. China Ser. A 46 (4) (2003) 506–515. [8] Zhiming Guo, Jianshe Yu, The existence of periodic and subharmonic solutions of subquadratic second order difference equations, J. London Math. Soc. 68 (2) (2003) 419–430. [9] Zhiming Guo, Jianshe Yu, Periodic and subharmonic solutions for superquadratic discrete Hamiltonian systems, Nonlinear Anal. 55 (7–8) (2003) 969–983. [10] P. Hartman, Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity, Trans. Am. Math. Soc. 246 (1978) 1–30. [11] J.W. Hooker, W.T. Patula, A second-order nonlinear difference equation: oscillation and asymptotic behavior, J. Math. Anal. Appl. 91 (1) (1983) 9–29. [12] J. Mawhin, M. Willem, Critical point theory and Hamiltonian systems, Applied Mathematical Sciences, vol. 74, Springer-Verlag, New York, 1989. [13] R. Michalek, G. Tarantello, Subharmonic solutions with prescribed minimal period for nonautonomous Hamiltonian systems, J. Differen. Equat. 72 (1) (1988) 28–55. [14] Chun-Lei Tang, Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Am. Math. Soc. 126 (11) (1998) 3263–3270. [15] Chun-Lei Tang, Xing-Ping Wu, Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2) (2001) 386–397. [16] Zhu-Lian Tao, Chun-Lei Tang, Periodic and subharmonic solutions of second-order Hamiltonian systems, J. Math. Anal. Appl. 293 (2) (2004) 435–445. [17] B.G. Zhang, G.D. Chen, Oscillation of certain second order nonlinear difference equations, J. Math. Anal. Appl. 199 (3) (1996) 827– 841. [18] Zhan Zhou, Jianshe Yu, Zhiming Guo, Periodic solutions of higher-dimensional discrete systems, Proc. Roy. Soc. Edinburgh 134A (2004) 1013–1022.