Periodic Solutions to Non-Autonomous Second-Order Dynamical Systems

We study the multiple existence of periodic solutions for a second-order non-autonomous dynamical systems   , = 0 u u V t u   (1). Using the method of invariant sets of descending flow and chain of rings theorem, we obtain the existence of seven 2π -periodic solutions.


Introduction
In this paper, we consider the existence of seven nontrivial solutions for the second order non-autonomous systems     , = 0. 0,2π , .
For each N u R  the function   , V t u is periodic in t with period 2π .
Using the method of invariant sets of descending flow, Z. Liu and J. Sun [1] got at least four periodic solutions of (1).Via the variational method, which has been mostly 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 [12].
The goal of this paper is to find more periodic solutions for problem (1).We get at least seven periodic solutions of (1) by using the method of invariant sets of descending flow and Chain of rings Theorem, which is obtained in [7].
Let us give some notations.For two functions u and v defined on   0, 2π and taking their values in N R , we define a partial order by u v  if and only if and = 1, 2, , i N  , the relationship between u and v will be denoted by u v  .(H1) There are two couples of functions , , , satisfies all the conditions in Theorem 1.One should take 0 u , u and v are sufficiently small.
 and  can be chosen in the same way.

Preliminary and Lemmas
Let H be the Hilbert space of vector functions   u t having period 2π and belonging to 1 H on   0, 2π , with the following inner product where K is a fixed number satisfying (H2).The corresponding norm in H is denoted by H  and Let X be the Banach space of N -vector functions   u t having period 2π and belonging to Then the critical points of J correspond to the solutions of problem (1).Here with the periodic condition of period 2π .

Denote
Now we will explain that (2) holds: Noting that both in H and in be the unique solution of (3) in H and in X respectively, with maximal right existence interval in the H topology for some u K    , the critical set of J , then the limit is also valid in the X topology.
Definition 1. (Chain of rings) (Definition 5 in [7]) Assume that   .i D intersects only with , and form a chain of rings.For the case of = 3 n , 1 A A  =  , we say that 1 D and 2 D form a chain of rings.Lemma 2. (Theorem 4 and Remark 5 in [7]) Assume that H is a Hilbert space, are open convex subsets of X , and form a chain of rings.  then J has at least 3 1 2 n  critical points; 2) when n is odd, if then J has at least  

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 > 0 C and , by the continuousness of V , we can take proper 2 C such that (4) holds.For u R  , by (H3) and ( 4), it follows that, for , we can take proper 3 C such that From these inequalities, we see that as n   , then by (5), one has 2π By ( 6) and ( 7), we obtain and It follows from ( 8) and ( 9) that and here 4 we get 0 as .
are all open subsets of X , and And by (H1)  .In a similar way,   and J is bounded on a bounded set, we get     , = 0,1, 2, = .
Then (1) has at least seven periodic solutions.[1] shows   J u satisfies the (PS) condition under (H4).From the proof of Theorem 1, we can get this conclusion.

Remark 3 . 1 .
If (H1) (H2) and the following condition are satisfied.(H4)There exists 1 > 0 R and positively definite constant matrixes A and B with = AB BA such that