SOME UNITED EXISTENCE RESULTS OF PERIODIC SOLUTIONS FOR NON-QUADRATIC SECOND ORDER HAMILTONIAN SYSTEMS

In this paper, some existence theorems are obtained for periodic solutions of second order Hamiltonian systems under non-quadratic conditions by using the minimax principle. Our results unite, extend and improve those relative works in some known literature.

1. Introduction and main results. Consider the second order Hamiltonian systemü (t) + A(t)u(t) + ∇F (t, u(t)) = 0, a.e. t ∈ R, (1) where A(t) is an N × N real symmetric matrix, and is continuous and T -periodic in t, and F : R × R N → R is T -periodic in t and satisfies the following assumption: (A) F (t, x) is measurable in t for every x ∈ R N and continuously differentiable in x for a.e. t ∈ [0, T ], and there exist a ∈ C(R + , R + ) and b ∈ L 1 (0, T ; R + ) such that for all x ∈ R N and a.e. t ∈ [0, T ].
Then system (2) has at least one nonconstant T -periodic solution.
In 2006, Schechter [18] also obtained some new existence results of nonconstant periodic solutions for system (2). We will not present those theorems here.
There are also lots of results on the existence of periodic solutions of system (1) under the superquadratic condition (see [12], [13], [24] and references therein). In 1995, Li and Willem [13] established so-called abstract local linking theorem and applied it to study the existence of nontrivial periodic solutions for system (1) under condition (F3). In detail, they obtained the following result: Theorem 1.3 (see [13], Theorem 7). Suppose that F satisfies (F2) and (F3). Then one of the following cases occurs: (1) if 0 is not an eigenvalue of −d 2 /dt 2 −Â (with periodic boundary condition), then system (1.1) has at least one nontrivial T -periodic solution; (2) if 0 is an eigenvalue of −d 2 /dt 2 −Â (with periodic boundary condition) and there exists l > 0 such that then system (1) has at least one nontrivial T -periodic solution, whereÂ : In 2005, Luan and Mao [11] improved the abstract local linking theorem obtained by Li and Willem [13] and in [12], they also applied it to study the existence of nontrivial periodic solutions for system (1). They obtained the following different result: Theorem 1.4 (see [12], Theorem A). Suppose that F satisfies (F2) and the following conditions: (F8) there exist positive constants a 1 , a 2 and R such that where σ > 1. Then the conclusions (1) and (2) in Theorem 1.3 hold.
Based on these works, in this paper, we will present several different results. For the sake of convenience, denoteL by the operator −d 2 /dt 2 −Â : H 1 T → H 1 T (with periodic boundary condition), where Theorem 1.8. Suppose that F satisfies (A), (F7) and the following conditions: (H1) there exist positive constants m, ζ, η and ν ∈ [0, 2) such that (H2) there exist consecutive eigenvalues λ k and λ k+1 ofL with λ k < λ k+1 and l 0 > 0 such that and all |x| ≤ l 0 .
Then system (1) has at least one nontrivial T -periodic solution in H 1 T . Moreover, when the eigenvalues locate near 0, we will present one better result. Denote λ −1 and λ +1 by the largest negative eigenvalue and the smallest positive eigenvalue ofL, respectively. Theorem 1.9. Suppose that F satisfies (A), (F7), (H1) and the following condition: (H3) there exists l 1 > 0 such that Then one of the following cases occurs: (C1) if 0 is not an eigenvalue ofL,L has at least one negative eigenvalue and there exists l 2 > 0 such that then system (1) has at least one nontrivial T -periodic solution; (C2) if 0 is not an eigenvalue ofL,L has no negative eigenvalues and then system (1) has at least one nontrivial T -periodic solution; (C3) if 0 is an eigenvalue ofL and there exists l 2 > 0 such that then system (1) has at least one nontrivial T -periodic solution; (C4) if 0 is an eigenvalue ofL,L has no negative eigenvalues and there exists l 2 > 0 such that and (8) holds, then system (1) has at least one nontrivial T -periodic solution; (C5) if 0 is an eigenvalue ofL,L has at least one negative eigenvalue, there exists l 2 > 0 such that (7) and (10) hold, then system (1) has at least one nontrivial T -periodic solution.
One also can find examples satisfying Theorem 1.5 (an example corresponding to Theorem 1.6 can be seen in [10]) but not satisfying Theorem 1.3. For example, let From Remark 1, Remark 2 and Remark 3, we know that Theorem 1.8 and Theorem 1.9 unite and improve Theorem 1.3, Theorem 1.5 and Theorem 1.6. One can verify that (16), (17) Moreover, it is remarkable that in [3], the following condition which is similar to (H1) has been presented: (Ŝ 2 ) there exist p > 2, c 1 , c 2 , c 3 > 0 and ν ∈ (0, 2) such that, for all |z| ≥ r 1 , which is used to consider the existence of homoclinic solutions for the first order Hamiltonian systemż = J∇H(t, z). Ding [3] claimed that (Ŝ 2 ) implies a condition which is similar to (F8). However, our Theorem 1.8 and Theorem 1.9 present that are not necessary when we consider the existence of periodic solutions for system (1). The fact shows that Theorem 1.8 and Theorem 1.9 are different from Theorem 1.4. One can verify that (18) does not satisfy Theorem 1.4.
By Theorem 1.9, we can obtain two useful corollaries. We denote ξ ii (t) (i = 1, · · · , N ) by the main diagonal entries of A(t).
(C2) if 0 is not an eigenvalue ofL, A(t) is semi-negative definite for all t ∈ [0, T ] and (8) holds, then system (1) has at least one nontrivial T -periodic solution.
Corollary 2. Suppose that A(t) ≡ A which is a real symmetric constant matrix, and F satisfies (A), (F7), (H1) and (H3). Then (C3) or one of the following cases occurs: (C1) if 0 is not an eigenvalue ofL, A is not semi-negative definite and (7) holds, then system (1) has at least one nontrivial T -periodic solution.
(C2) if 0 is not an eigenvalue ofL, A is semi-negative definite and (8) holds, then system (1) has at least one nontrivial T -periodic solution.
(C4) if 0 is an eigenvalue ofL, A is semi-negative definite and (8) and (10) hold, then system (1) has at least one nontrivial T -periodic solution.
(C5) if 0 is an eigenvalue ofL, A is not semi-negative definite and (7) and (10) hold, then system (1) has at least one nontrivial T -periodic solution.
Then system (1) has at least one nontrivial T -periodic solution. Furthermore, if A(t) is semipositive definite matrix for all t ∈ [0, T ], then system (1) has at least one nonconstant T -periodic solution.
Remark 6. Remark 3 shows that Theorem 1.10 is different from Theorem 1.7. Moreover, since (F1) is a global condition and (H4) is weaker than (F7), Theorem 1.10 is also different from Theorem 1.8 and Theorem 1.9. Another aim of Theorem 1.10 is to obtain the nonconstant solution. Remark 2 shows that Theorem 1.10 extends and improves Theorem 1.1.
Theorem 1.11. Suppose that F satisfies assumption (A), (F1), (H1) and the following conditions hold: If A(t) is semi-positive definite matrix for all t ∈ [0, T ] and A < 4π 2 T 2 , then system (1) has at least one nonconstant T -periodic solution, where Remark 7. Remark 2 and Remark 3 show that Theorem 1.11 still unites and maybe improves Theorem 1.1 and Theorem 1.2 even if A(t) ≡ 0. Here, we use the word "maybe" before "improves" because we do not find an example satisfying Theorem 1.11 but not satisfying both Theorem 1.1 and Theorem 1.2 when A(t) ≡ 0. We would like to leave this problem to readers. Moreover, it is remarkable that Theorem 1.10 and Theorem 1.11 are different from those results in [18] even if A(t) ≡ 0 because the example (17) does not satisfy them.

Preliminaries. H 1
T is a Hilbert space with the inner product and the norm defined by Then one has (see Proposition 1.3 in [14]) and for some C > 0 and all u ∈ H 1 T , where u ∞ = max t∈[0,T ] |u(t)| (see Proposition 1.1 in [14]). It follows from assumption (A) that the functional ϕ on H 1 T given by It is well known that the solutions of system (1.1) correspond to the critical points of ϕ (see [14]).
In order to prove Theorem 1.10 and Theorem 1.11, we need the following linking method in [19]. Let (E, · ) be a Banach space and Φ be the set of all continuous maps Γ = Γ(t) from E × [0, 1] to E such that 1). Γ(0) = I, the identity map.

Proofs of Theorems.
Proof of Theorem 1.8. We will verify those conditions in Lemma 2.1.
(2) We claim that ϕ maps bounded sets into bounded sets.
Proofs of Corollary 1 and Corollary 2. Note that H − = {0} if and only ifL has no negative eigenvalues. Then the proofs are easy to be completed by Proposition 1, Proposition 2 and Theorem 1.9.
Proof of Theorem 1.10. Let E = H 1 T . We use the same decomposition as in the proof of Theorem 1.9. We first construct A and B which satisfy assumptions in Lemma 2.3. It follows from (F2) that for any given 0 < ε 0 < δ 1 /2, there exists r > 0 such that If u ∈ H + and u = ρ = r C , then by (21), we have u ∞ ≤ r. Let B ρ = {u ∈ H 1 T : u < ρ}. Hence, for all u ∈ ∂B ρ ∩ H + , it follows from (57) and (65) that By (H4) and (F1), we know that there exists K 1 > 0 such that

XINGYONG ZHANG AND XIANHUA TANG
Let u(t) = u − + u 0 + sw 0 (t), where s ≥ 0 and w 0 ∈ H + is an eigenvector corresponding to λ +1 . Since H 0 = ker(I − K) is a finite dimensional space, there exist positive constants K 2 and K 3 such that Consequently, by (54), (56), (68) and the above inequality, we obtain By (F1) and (53), it is easy to obtain that For > ρ, let By Example 3 of section 3.5 in [19], we know that A links B. Moreover, we can choose sufficiently large such that By definitions of Γ and set A, it is easy to know that (1 + u n ) ϕ (u n ) (H 1 T ) * → 0 as n → ∞. Similar to the proof of Theorem 1.8, we can obtain a renamed subsequence such that u n → u strongly in H 1 T . Since ϕ (u n ) (H 1 T ) * → 0, it follows from the continuity of ϕ (·) in H 1 T that ϕ (u) = 0. So u is a T -periodic solution of system (1). Since ϕ is continuous, by (69) and (70), we know that ϕ(u) = c > 0.
If u ∈ R N and A(t) is semipositive, then by (F1), which contradicts ϕ(u) = c > 0. Thus the proof is complete.
Moreover, Since A(t) is semi-positive definite, by (F1), it is easy to obtain that ϕ(x) ≤ 0, ∀ x ∈ R N .
By Example 3 of section 3.5 in [19], we know that A links B. Moreover, we can choose sufficiently large 1 such that The other arguments are similar to Theorem 1.10. We omit the details.