 Applied Mathematics, 2011, 2, 846-853 doi:10.4236/am.2011.27114 Published Online July 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Multiple Periodic Solutions for Some Classes of First-Order Hamiltonian Systems Mohsen Timoumi Department of Mat hematics, Faculty of Sciences, Monastir, Tunisia E-mail: Mohsen.Timoumi@fsm.rnu.tn Received April 15, 2011; revised May 19, 2011; accepted May 22, 2011 Abstract Considering a decomposition 2NAB of 2N, we prove in this work, the existence of at least geometrically distinct periodic solutions for the first-order Hamiltonian system 1dimA,t xte t0Jx tH when the Hamiltonian ,Htuv is periodic in  and its growth at infinity in v is at most like or faster than ,tuav, 0a1, and is a forcing term. For the proof, we use the Least Action Principle and a Generalized Saddle Point Theorem. e Keywords: Hamiltonian Systems, Partial Nonlinearity, Multiple Periodic Solutions, Critical Point Theory1. Introduction Consider the nonautonomous first-order Hamiltonian system ,0JxtHt x te t where , is a continuous function, periodic  in the first variable and differentiable with respect to the second 2:NH T ,, ,tx Htx0Tvariable with continuous derivative  ,,HHtx txx, 2:eN is a continuous Tperiodic function with mean value zero and J is the standard symplectic matrix: 00NNIJI NI being the identity matrix of order . NUsing variational methods, there have been many pa-pers devoted to the existence of periodic solutions for (࣢), we refer the readers to [1-5] and the references therein. However, there are few papers discussing the multiplicity of periodic solutions for (࣢) (see [6-9]). Under the assumptions that H is periodic in 1,,pxx, where , 12pN 112,,Nxxx and there exists such that 0, ;T2fL 2,,,..0NHtxft xaet,1. (1.1)  20,d, 0, TNpHtxtasxx  (1.2) the author has shown in  that system (࣢) possesses at least 1p geometrically distinct periodic solutions. The first goal of this note is to generalize the existence result of multiple periodic solutions obtained above to the sublinear case. Precisely, consider a decomposition 2NAB of 2N with 121space, ,,space, ,ppiiAeeBeeNi1 where 02pN2N and 12iiN is the standard basis of and let us denote AP esp. e(rBP) the pro-jection of 2Nto inA (resp. B). We obtain the fol-lowing result Theorem 1.1 Asse that H satisfies um0HH is periodic the variables. in1,,piixx; 1H There exist α א [0, 1] and two T–periodic functions 1(0, ;)1aL T and such that 20, ;bLT   2,,,NBHtx atPxbt xaet ..0,1, (H2) Either 1) 201,d ,THtx tasxx Bx  or 2) 201,d ,THtx tasxx Bx Then the Hamiltonian system (࣢) possesses at least 1pT periodic solutions geometrically distinct. M. TIMOUMI 847 Example 1.1 Let be a periodic and continuously differentiable function. Consider the Ham-iltonian: 2:Na 3212π,, sin2Htrptp arT  (1.3) Then H satisfies conditions 02HH with and 0NA0NB. It is easy to see that conditions 12,HH don’t cover some sublinear cases like  22212π,, cos,2ln 2,, NparHtrp tTpartrp (1.4) The second goal of this paper is to study the existence of multiple periodic solutions for (࣢) when the Hamilto-nian H satisfies a nonlinearity condition which covers the cases like (1.4). Precisely, we will require the nonlinearity to have a partial growth at infinity faster than ,0 1x Our second main result is: Theorem 1.2 Consider a nonincreasing positive func-tion 0, ,C  with the properties: lim inf0sss , 0, ,sssass and assume that H atisfies 0H and the following as-sumptions 3H There exist a positive constant a d a function such that for all and ;T20,gL2Nx.. 0aet,1   ,BB ,HtxaPxP xgt 4H Either 1) 20,d as 1,THtx txx Bxx , 2) 20,d as 1,THtx txx Bxx . Then the system (࣢) possesses at least 1p geo-metrically distinct Tperiodic solutions. Remark 1.1 The Hamiltonian H defined in (1.4) satisfies the conditions 3H,4H introduced above with 21,0ln 2sss, 2. Preliminaries Firstly, let us recall a critical point theorem due to G. Fournier, D. Lupo, M. Ramos and M. Willem . Given a Banach space E and a complete connected Finsler manifold V of class , we consider the space 2CXExV. Let EW Z (topological direct sum) and (nEZn) be a sequence of closed subspaces with nZZ, nWimW, 1dnW . Define nnXExV. For ,X, w/nnX1Cfe denote by ff. Then we have 1,CX for nnDefinition 2.1 Let fall 1. n1,CX. The function f satis-fies the Palais-Smale condition with respect to nX at a level c if every sequence nX satisfying ,,nn nnnxXfx cfx0 has a subsequence which converges in X to a critical point of f. The above property will be referred as the condition with respect to *PS cnX. Theorem 2.1 (Generalized Saddle Point Theorem). Assume that there exist constants and 0r such that 1) f satisfies the c condition with respect to *PSnX for every ,c, 2) ,fwv for every such that ,wvW Vwr, 3) ,fzv for every , ,zvZ V4) ,fwv for every such that ,wv WVwr. Then 1,f contains at least cuplength 1V critical points of f. Consider the Hilbert space 1122,NEHS where T and the continuous quadratic form Q efined in E y 01(). d2TQxJx txtt where ,xy2 inside the sign integral is the inner product of ,Nxy. Let us denote by , , 0E EE respec-tively the subspaces of on which Q is null, negative definite and positive definite. It is well known that these sub-spaces are mutually orthogonal in E221,NLS and in with respect to the bilinear form: E01(, )().()d2 ,,TBxyJxt yttxyE associated to Q. If xE and yE then 0(, )Bxy and ())(Qx yQy(Qx). For 0xxxxE, the expression  1220xQxQx x is an equivalent norm in E. Moreover, the space is compactly embedded in E212,NLS for all 1,s. In particular for all 1,s, there exists a constant 0s such that for all xE, Copyright © 2011 SciRes. AM M. TIMOUMI 848 .ssLxx (2.1) 3. Proof of the Theorems Firstly, let us remark that if xtt is a periodic solution of (࣢), then by replacing by in (࣢) we obtain t ,0JxtHt xtet. So it is clear that the function  ytxt is a pe-riodc solution of the system  ,0JytHt ytet. Moreover, ,Htx satisfies 2Hi (resp. 4Hi) whenever ,Htx satisfies 2Hii (resp. 4Hii ). Hence, in the following, we will assume that H satisfies 2Hi in Theem 1.1 and or4Hi in Theorem 1.2. Associate to the system (࣢) the functional  de-fined on the space , by: E  001d,2TTuJut uttHtuet utt d. It is well known that the functional  is continu-ously differentiable in and critical points of E on corres-pond to the T periodic solutions of the system (࣢), moreover one has E  0,+TuvJu tHtutetvtt d for all . Consider the subspaces ,uv EWE, ZEB of and the quotient space E1,,,ipVAxxeii i which is nothing but the torus pT. We regard the func-tional  as defined on the space XZWV as follows    0001d,2dTTTuvJututtHtutvt tet utt d To find critical points of  we will apply Theorem 2.1 to this functional with respect to the sequence of sub-spaces nnXEV, where for 0n2πˆ:expnmmnExExt mtJuaeT  ... Proof of the Theorem 1.1. Assume 0H, 1H and 2Hi hold. Firstly, let us check the Palais- Smale condition. Lemma 2.1. For all level , the functional c satisfies the  condition with respect to the se-quence . *cPSnNnXProof. Let c and let be a sequence of X such that for all ,nnnuv n, and ,nnuv Xnnnuvc and 0nn nuv as , (3.1) nwhere n is the functional  restricted to nX. Set n0nnnuuuu with nuE, , nuE0nuB. We have the relation 20,dnn nnTnnnuvuuHtuvetu nt (3.2) Since 0nn nuv10c as , there exists a constant such that n1,nnnnnnNu vucu . (3.3) By assumption 1H and Hölder’s inequality, with 1p, 11q, we have  22221'00,d() dTnn nTBn nnBnLLLLHtuvutat PutbtutuaPu b (3.4) Then by (3.2), (3.4) and (2.1), there exist two positive constants , such that 2c3c2nBnucPuc3. (3.5) Observing that a similar result holds for nu: 2nBnucPu c3. (3.6) We conclude from (3.5) and (3.6) that the sequence nu is bounded if and only if the sequence BnPu is bounded. Assume that BnPu is not bounded, we can assume, by going to a subsequence if necessary, that BnPu  as . Since n 01, we con-clude by (3.5) and (3.6) that  0,0 as nnBn BnuunPu Pu. (3.7) Therefore, we have ,,1 as nnBnuyyByByPu .n  (3.8) It follows that 01 as .nBnunPu  (3.9) Consequently, by (3.5), (3.6) and (3.9), we can find a positive constant such that 4cCopyright © 2011 SciRes. AM M. TIMOUMI 849 04,,innucui. (3.10) Now, we apply the fact that nnuv is bounded to get 2222000522000,d.dTnn nnnnTnnnnuu Htuvtuuet uuctuu (3.11) where 3 is a positive constant. Using (3.10) and (3.11), we can find a constant satisfying c6c0220000020006200,,dd,,d,,dTTnnnnnTnnnnTnnnnHtu Htu vttuuHtuHtu vtuHtuHtu vctu(3.12). On the other hand, by the Mean Value Theorem and assumption1H, we have 212220000000,d,.d.TnnnTnnnnnnTBnnnnnnBnnnLLnnnLLHtuHuvtnHtuu u vu u vat P uuubtuuvtaPuuubuuv   (3.13) By considering (3.13) and Sobolev’s embedding E220, ;NLT we can find a constant such that 70c0007,d||Tnnnnn nnnHtuHuvtcuuuu u 1 (3.14) After combining (3.10), (3.12) and (3.14), we get 08200,dTnnHtu tcu (3.15) for some positive constant . However, the condition (3.15) contradicts 8c2Hi because 0nu as . Consequently, nnuv is bounded in X. Going if necessary to a subsequence, we can assume that and . Notice that ,nuu0nu0unvdnnnuu vuuHtv etnnnnu vt0TnQu,,nu vHtuuu  uu (3.16) which implies that n in E. Similarly, nuu in E. It follows that ,,nnuv uv in X and 0uv. So  satisfies the  condition for all *cPSc. The Lemma 3.1 is proved. Now, let us prove that the functional  satisfies the conditions a), b) and c) of Theorem 2.1. a) Let W,uv V. By using the Mean Value Theo-rem, assumptions 0H, 1H and (2.1), we have      2000000,ddddTTTTTTLHtutettuttuttu2002020202012220|| ||dd|| ||,d,.d d,d,d,ddTTTBTBTTTuvuvt etuuHtvHtvu ututuHtvatPubtutetuHtvatPubtutetuHtvatu tt     (3.17) 2910Lc0211dTbetutuucuc  where , 10 , are three positive constants. Since 9c c111c0, then  as ,uv uWu  uniformly in vV. (3.18) b) Let ,uvZV, with . By using the Mean Value Theorem, we get 0uu u2000200000,dd|| ||,d,dTTTTTuvuHtuu vtet utuHtutHtuuvuvet u t(3.19). Copyright © 2011 SciRes. AM M. TIMOUMI 850 By assumption 1H and (2.1), we can find a con-stant such that 12 0c22220000122200012,()d,d11TTTBLLLTLL0dHtuuvuvt etutat Ptuuvbtuvteuuvatu ubecuu u  (3.20) Therefore, by using (3.19) and (3.20) we obtain 200012,d11TuvuHtu tcuu u. (3.21) Now let 2122cd. By assumption 2Hi, there ex- ists a constant such that 13 0c200130,ddTHtu tuc (3.22). So by (3.21) and (3.22), we have 22013012211222200121201212 131112122.uv uduccuu uucu uucucuducuc c    (3.23) Since 2122cd and 01, then  as ,,uv uZu  uniformly in vV. (3.24) Hence by Lemma 3.1 and properties (3.18), (3.24), we deduce that the functional  satisfies all the assump- tions of Theorem 2.1. Therefore the Hamiltonian system (࣢) possesses at least 1pT periodic solutions geometrically distinct. The proof of Theorem 1.1 is com-plete. Proof of Theorem 1.2. Assume 0H,1H and 4Hi hold. The following lemma will be needed for the study of the geometry of the functional . Lemma 3.2. There exist a non-increasing positive function (]0, ,C and a positive constant satisfying the following conditions: 0ci)()0, as ,sss s ii) 20(,)1 ,,BBLtucP uP uuEH iii) 00asu20001,d .THtu tuu  Proof : For uE, let 120, :.BBAtTPut Pu By 3H, we have    2222122012202212220,12221220,dddd||||dsup( )||||.TBBLTBB LBBABB LTABBAsB LHtuaP utPutgttaPutPut tgaPutPuttPut PuttgaPuPuttTsPug So, by (2.1) there exists a positive constant such th 0cat 21122220(,)1.LBBBHtucPuPuPu Take 112221,0ss ss ,  satisfies (ii) and it is clear to see that then satis-).et us define fies (i Next, l2122lim inf.sss 4Hi, for any 0, there exists a positive By Copyright © 2011 SciRes. AM M. TIMOUMI 851 constant s14c uch that 2140,dT.Htx txxc (3.25) which imat for plies th0uB, 00u, 20T001402100 220 002,d .uu cHtu tuu uuu  (3.26) By the definition of , there exists such that fo0Rr all sR 221222() .2ssss s (3.27) Therefore 00142100 220 002,d2THtutcTuu uuu   (3.28) as 0uR and then 000200,dlim .2TuHtu tTuu (3.29) Since  is arbitrary choosen, condition (iii) holds. etonal Now, l us prove the Palais-Smale condition. Lemma 3.3. For all level c, the functi e-that sabe ance in X such fo. (3.30) Set n and n By HöinLemma 2tisfies the *PS condition respect to the squencenXProouc. ,vwith sequenf. Letnnn ,nnuvr all ,nn and u vX  and 0 as nn nnnv cun 0nnnuuuulity, (2.1) and nnuuu..2(ii), we lder’s equaget a positive constant 15c such that 220122015,d,1.Tu vnn n nTnnnnLLnBnBnHtetuutuu Htuvec uPuPu  (3.31) Thus, for n large enough 21521nnnnBn BnuucuPuPu  (3.32) So there exists a positive constant such that .nnnuvuu 16c16 1.nBnBu cPuPun (3.33) By (3.33) and the properties (i) of , we deduce that nu is bounded if and only BnPu is bounded. since Now, is nonincreasing and BnuPu 0max, BuP , we get um, BuuPu (3.34) Combining (3.32) and (3.34), yields fen0inor n large ough 2001521nunnBnBn nnuc uPuPuuu which implies 15 1521.n Bnuc Puc (3.35) Assume that 0015 nncuuBnPu is unbounded, then btoy going a subsequence, if necessary, we can assume that BnPu  as n. Since 0s as s, wcm (3at there exositivstant 17c such that e dedue fro.35) thists a pe con-0017nunncuu (3.36) for large enough. Since the mapn sss is con-tinuos in u0, and goes to  as s, then 0nu as. by the M Theorem nNow, ean Valueity, Hölder’s inequal- and Lemma 3.2(ii), we get 220,,dTnn nHtu vHtut01000121200010000,.dd,dd1d .Tnnn nTnnn nnLnnnBnn BnLHtusuvuvstuvHtusuvt scuvusP uusP us  (3.37) Since 00nBnusPu un for all 0,1sthere ex, we deom (2.1), (3.36) and (3.37) that du-ce frists a positive constant 18c such that   200,,dTnn nHtu vHtut00 0022000 001800||[ 1]1,nn nnnBnLnnn nnnncuvuuu Pucuuuuuuu (3.38) which with (2.1) and (3.36) imply that there exists a positive constant such that 19cCopyright © 2011 SciRes. AM M. TIMOUMI 852 22 00,dTnn nnnTuv uuHtut 00020001902000 00200 019 000022000019,,d.d,d111,d1Tnn n nTnnnnnn nnnn nnnTnnn nnHtuvHtutetutHtut cuuuuu uucuuu uuHtu tuuc uu     (3.39) which, with Lemma 3.2 (iii), imply that as . This contradicts the boundednnnuv ess of nnnuv. So BnPu is boundedAss that . ume0u, then up to a n is unboundedsub-sequence, if sary, we canecesn assume that 0nu as .38), n. As in (3and using (2.1), (3.34) and the fact that ()0s as s , we can find atant such that  cons21 0c20000 000021,,dTnnu vt11.nnnnnnBnLnn nnHt Htucuvu uuPucuvu u   (3.40) Now, since ()0 as ss, then combining (3.33) and (3.34) yields 0000022 1n nnnuuuu Pcuu (3.41) for a positive constant . Therefore there existstive constant such1nn nBnnvu v 22c that a posi-23c00,,.dTnnnn20023 1.nnHtuvH tuetut cuu (3.42) We deduce from (3.41) and (3.42) that therconstant such that e exists a 24 0c200 01,dTnn nnuv cuuHtut24 00200 024 200,d11nTnnnnnHtutuuc uu which implies by Lemma 3.2 (iii) that as . This contradicts the boundedness of nnuv nnnuv. Then 0nu is also bounded and therefore nu is bounded. By a standard argument, we con-clude that nuf of possesses a convergent subse The proo Lemma 3.3 is complete. let -quence.Now,0,,uvuvZ Vu, then as in (3.38) that there exists a positive constant 25c such 000 025, .)d11.Tnnn nBvHtuetu tcuuu uPu0(,Htu (3.43) So, we have for a positive constant c 260026 01,d.Tnuuuc Htut(3.44) Let 20026uvucuu 01, we have  220000 22626 2nnccuuu uu2.u (3.45) By combining (3.44) and (3.45), we get 22026 2602200 26022001,d1Tnnnnnuvcuuc uHtutcuu uu  which implies that uv as uZ, u, uniformly in vV. (3.46) On the other hand, let bB, 0b. By the Mean Value Theorem, we have for uW E 212'2001(,| ddTLTuvbHtbvt s222100100122100122d,.dd)ddddTTBLBTBLBLtb tHtbsu uvbssu buvbabsPu bbsPu bgttsuvbabsPu bbsPu bts g    0,,Htu vHvbt   (3.47) Copyright © 2011 SciRes. AM M. TIMOUMI Copyright © 2011 SciRes. AM 853Take for 0,1s, 0,1 :.BAst bsPubb Mathematical Analysis Application, Vol. 323, No. 15, 2006, pp. 854-863. doi:10.1016/j.jmaa.2005.11.004  P. L. Felmer, “Periodic Solutions of Superquadratic Ha- miltonian Systems,” Journal of Diffuation, Vol. 102, No. 1, 1993, pp. 188-207. By a similar calculation as in the proof of Lemma 3.2, we get for some positive constants and 27ccb 0227,,1Terential Eqdoi:10.1006/jdeq.1993.1027  Z. Q. Ou and C. L. Tang, “Periodic and Subharmonic Class of Superquadratic Hamiltonian Sys-ar Analysis, Vol. 58, No. 3-4, 2004, pp. dHtuvH tbetutcbucbu Solutions for atems,” Nonline which implies that (3.48) 245-258. doi:10.1016/j.na.2004.03.029  C. L. Tang and X. P. Wu, “Periodic Solutions for Second Order Systems with Not Uniformly Coercive Potential,” Journal of Mathematical Analysis Application, Vol. 259, No. 2, 2001, pp. 386-397.doi:10.1006/jmaa.20222701,Tuvu cbucbuHtb t d. (3.49)Since 00.7401  M. Timoumi, “Periodic Solutions for Noncoercive Ham-iltonian Systems,” Demonstratio Mathematica, Vol. 35, No. 4, 2002, pp. 899-913. 0asss, there exists 0b such that 271|| 2cb, which implies that  K. C. Chang, “On the Periodic Nonlinearity and the Mul-tiplicity of Solutions,” Nonlinear Analysis, Vol. 13, No. 5 1989, pp. 527-537. doi:10.1016/0362-546X(89)90062-X  S. X. Chen, X. Wu and F. Zhao, “New Exis2uv011,d.2Tucb uHtbt  So we have tence and Multiplicity Theorems of Periodic Solutions for Non- Autonomous Second Order Hamiltonian Systems,” Mathematical and Computer Modeling, Vol. 46, No. 3-4, 2007, pp. 550-556. doi:10.1016/j.mcm.2006.11.019 uv as uW, u, uniformly in vV (.3.50) Thus, Lemma 3.3 and properties (3.46), (that the functional  I. Ekeland and J. M. Lasry, “On the Number of Periodic Trajectories for a Hamiltonian Flow on a Convex Energy Surface,” Annals of Mathematics, Vol. 112, No. 2, 1980, pp. 283-319.doi:10.2307/13.50) imply  ) satisfies all the assumGeneralized Saddle Point Theorem. Therefore the Hampossesses at least peri-etrically distinct. Theheo-rem 1.2 is complete. 4. References  tions Autonomous Hamiltoned Period,” Journal of ptions of the iltonian system (࣢odic solutions geom 971148  M. 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