 Advances in Pure Mathematics, 2011, 1, 90-94 doi:10.4236/apm.2011.13020 Published Online May 2011 (http://www.scirp.org/journal/apm) Copyright © 2011 SciRes. APM Periodic Solutions to Non-Autonomous Second-Order Dynamical Systems An-Min Mao, Miao-Miao Yang School of Mathemat i cal Sci ences, Qufu Normal University, Qufu, China E-mail: maoam@163. com Received January 17, 2011; revised March 7, 2011; accepted March 12, 2011 Abstract We study the multiple existence of periodic solutions for a second-order non-autonomous dynamical systems ,=0uuVtu (1). Using the method of invariant sets of descending flow and chain of rings theorem, we obtain the existence of seven 2π-periodic solutions. Keywords: Critical Point, Invariant Sets of Descending Flow, (PS) Condition 1. Introduction In this paper, we consider the existence of seven non-trivial solutions for the second order non-autonomous systems ,=0. 0,2π,.NuuVtu tuR  (1) For each NuR the function ,Vtu is periodic in t with period 2π. Problem (1) has been studied by many authors and there is a large literature, see, for example, [1-11] (cf. the references quoted in them). Using the method of invariant sets of descending flow, Z. Liu and J. Sun  got at least four periodic solutions of (1). Via the variational method, which has been most-ly used to prove the existence of solutions of (1), M. Willem, J. Mawhin, S. Li, M. Schechter, C. Tang and others proved existence under various conditions (cf. the reference given in these publications). Also, the fixed point theorems in cones can be chosen to establish the existence of solutions for (1), see . The goal of this paper is to find more periodic solu-tions for problem (1). We get at least seven periodic so-lutions of (1) by using the method of invariant sets of descending flow and Chain of rings Theorem, which is obtained in . Let us give some notations. For two functions u and v defined on 0, 2π and taking their values in NR, we define a partial order by uv if and only if  iiut vt for all 0, 2πt and =1,2, ,iN. If  ,,iii iitVttu for any =1,2, ,iN, there is 0, 2πit such that  <,,iii iitVttu for any =1,2, ,iN, there is 0, 2πit such that  >,.iii iitVttu (H2) There exists a constant >0K such that every entry of the matrix 2,uu tu KVI is nonnegative if u is a function satisfying u or u or u or u, where ,uu tuV is the Hessian matrix of A. M. MAO ET AL. Copyright © 2011 SciRes. APM 91V and I is the NN unit matrix. (H3) There exists >2 and >0R such that, for uR,  0<,, ,uVtuu Vtu where u means the Euclidean norm of u in NR. Theorem 1. If (H1), (H2) and (H3) are satisfied, then problem (1) has at least seven periodic solutions. Remark 1.1. Conditions in Theorem 1 can be satisfied by generic functions. As examples, it can be proved that 42=12=21NiijiijNVu uu u satisfies all the conditions in Theorem 1. One should take 0u and 0,v in which12===,Nuu u 12===Nvv v, u and v are sufficiently small.  and  can be chosen in the same way. Remark 1.2. Our work is based on the results in [1,7]. 2. Preliminary and Lemmas Let H be the Hilbert space of vector functions ut having period 2π and belonging to 1H on 0, 2π, with the following inner product 220,= d,uvuv Kuvt where K is a fixed number satisfying (H2). The corre-sponding norm in H is denoted by H and 222220=, =d.Huuu uKut Let X be the Banach space of N-vector functions ut having period 2π and belonging to 1C on 0, 2π and X is continuously imbedded in H. Define a functional :JHR as  2π201=,d.2JuuVtut Then the critical points of J correspond to the solu-tions of problem (1). Here :== 0KuHJu X. We have  12222d=,duJuuK VtuKut  (2) here 1222ddKt is the inverse operator of 222ddKt with the periodic condition of period 2π. Denote 12222d=,.duAuKVtuKut  Now we will explain that (2) holds: Noting that 2π0,=, d,uJuvuvVtuvt and 2π20,= d,AuvAuvKAuvt then  2π220222220,,= d=d.u vAu vuvKuvAuvKAuvtduvKAuK uv tdt   For 12222d=,duAuKVtuKut  , we have  ,=, ,.Juvuv Auv For 0uX, consider the initial value problem  0d=,0=duuAuuu (3) both in H and in X. Let 0,uu and 0,uu be the unique solution of (3) in H and in X respectively, with maximal right existence interval 00, u and 00, u. Lemma 1. (Lemma 5.1 in ) 00=uu and 00,=,uu uu for all 00or >,inf infnnXXuA uAjjjjJu Ju  A. M. MAO ET AL. Copyright © 2011 SciRes. APM 92 then J has at least 312n critical points; 2) when n is odd, if  1122212 1=1 =1>or >,inf infnnXXXuA uAAjjjjJu Ju  then J has at least 3132n critical points. 3. Proof of Theorem 1 We now give the proof of Theorem 1. Proof. Step 1. First we will prove that J satisfies (PS) condition. (H3) implies the existence of constants 1>0C and 2>0C such that 12,,.NVtuCuCu R (4) Actually let =,gxVtxu,    d1=,= ,d,= ,uugx Vtxuu VtxuxuxxVtxu gxxx i.e. ddgxxgxx. For 1x, we get 11dd,xxgssgs s  lnln 1ln0,gx gx  1,gxg x  ,,.VtxuVtu x thus, 1[0,2π],=,|| ,,1 :=.mintuuVtu VtuVtuuuVtu Cu For <1x, one has 11dd.xxgssgss Similarly,  1,gxg x  ,,.VtxuVtu x then 20, 2π,=|,||,|,1:= .maxtuuVtu VtuVtuuuVtu C If 0kts , for any 0,2πts, we have  122222=,,0.uudAAu KdtVtKVtu Ku And by (H1) 12222222=,0.udAKdtdKVtKdt The maximum principle shows that A. There-fore Au for all 1XuD, that is 1AuD for all 1XuD. Hence 11XADD. In a similar way, Xi iADD, for =2,3,4i. Since 112=,XXXADDuX u 334=,XXXADDuX u  and J is bounded on a bounded set, we get 12 34>,and >.inf infXX XXDD DDJu Ju From Lemma 5 and Remark 1, we know that J has at least 341=72 critical points. Remark 3.1. If (H1) (H2) and the following condition A. M. MAO ET AL. Copyright © 2011 SciRes. APM 94 are satisfied. (H4) There exists 1>0R and positively definite con-stant matrixes A and B with =ABBA such that 1,,, .NuuVtuu RuR AB Since =ABBA, there is an orthogonal matrix T such that =TATTBT are simultaneously diagonal ma-trixes. Let 1=,,NdiagTAT and =TBT 1,,Ndiag and assume also that >0i for =1,2, ,iN and that 2=1 ,=0,1, 2,=.Niii nn  Then (1) has at least seven periodic solutions.  shows Ju satisfies the (PS) condition under (H4). From the proof of Theorem 1, we can get this con-clusion. 4. Acknowledgements We should like to express our appreciation to the referees for suggesting how to improve our paper. 5. References  Z. Liu and J. Sun, “Invariant Sets of Descending Flow in Critical Point Theory with Applications to Nonlinear Differential Equations,” Journal of Differential Equations, Vol. 172, No. 2, 2001, pp. 257-299. doi:10.1006/jdeq.2000.3867  M. Schechter, “Periodic Non-Autonomous Second-Order Dynamical Systems,” Journal of Differential Equations, Vol. 223, No. 2, 2006, pp. 290-302. doi:10.1016/j.jde.2005.02.022  Z. Y. Wang, J. H. Zhang and Z. T Zhang, “Periodic Solu-tions of Second Order Non-Autonomous Hamiltonian Systems with Local Superquadratic Potential,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 70, No. 10, 2009, pp. 3672-3681. doi:10.1006/jmaa.1995.1002  S. J. Li and M. Willem, “Applications of Local Linking to Critical Point Theory,” Journal of Mathematical Anal-ysis and Applications, Vol. 189, No. 1, 1995, pp. 6-32.  A. Ambrosetti and P. H. Rabinowitz, “Dual Variational Methods in Critical Point Theory and Applications,” Journal of Functional Analysis, Vol. 14, No. 4, 1973, pp. 349-381. doi:10.1016/0022-1236(73)90051-7  J. X. Sun and J. L. Sun, “Existence Theorems of Multiple Critical Points and Applications,” Nonlinear Analysis, Vol. 61, No. 8, 2005, pp. 1303-1317. doi:10.1016/j.na.2005.01.082  J. R. Graef, L. J. Kong and H. Y. Wang, “Existence, Mul-tiplicity and Dependence on a Parameter for a Periodic Boundary Value Problem,” Journal of Differential Equa-tions, Vol. 245, No. 5, 2008, pp. 1185-1197. doi:10.1016/j.jde.2008.06.012  F. Zhao and X. Wu, “Existence and Multiplicity of Peri-odic Solutions for Non-Autonomous Second-Order Sys-tems with Linear Nonlinearity,” Nonlinear Analysis, Vol. 60, No. 2, 2005, pp. 325-335.  J. Mawhin and M. Willem, “Critical Point Theory and Hamiltonian Systems,” Springer, Berlin, New York, 1989.  P. H. Rabinowitz, “Minimax Methods in Critical Point Theory with Applications to Differential Equations,” Expository Lectures from the CBMS Regional Confer-ence, Series in Mathematics, American Mathematical So-ciety, Vol. 65, 1986.  C. L. Tang, “Periodic Solutions of Nonautonomous Sec-ond-Order Systems with Sublinear Nonlinearity,” Pro-ceedings of the American Mathematical Society, Vol. 126, No. 11, 1998, pp. 3263-3270. doi:10.1090/S0002-9939-98-04706-6  H. Y. Wang, “Periodic Solutions to Non-Autonomous Second-Order Systems,” Nonlinear Analysis: Theory, Methods & Applications, Vol. 71, No. 3-4, 2009, pp. 1271-1275.