Minimal period solutions in asymptotically linear Hamiltonian system with symmetries

In this paper, applying the Maslov-type index theory for periodic orbits and brake orbits, we study the minimal period problems in asymptotically linear Hamiltonian systems with different symmetries. For the asymptotically linear semipositive even Hamiltonian systems, we prove that for any given \begin{document}$ T>0 $\end{document} , there exists a central symmetric periodic solution with minimal period \begin{document}$ T $\end{document} . Moreover, if the Hamiltonian systems are also reversible, we prove the existence of a central symmetric brake orbit with minimal period being either \begin{document}$ T $\end{document} or \begin{document}$ T/3 $\end{document} . Also we give some other lower bound estimations for brake orbits case.


Introduction.
Let H ∈ C 2 (R 2n , R) and let us consider the following probleṁ where J = 0 −I n I n 0 and I is the n × n identity matrix. Denote by | · | and ·, · the standard norm and inner product in R 2n , respectively. Set Let z be a solution of (1.1). (z, T ) is called a periodic solution of (1.1) if z(0) = z(T ). In 1980, H. Amann and E. Zehnder proved the existence of nontrivial periodic solution of asymptotic linear system with constant B nondegenerate at zero and infinity (cf. [1], [2]). Their results was extended to the situation of nonconstant B (cf. [5], [17] and references therein). In 1990, Y. Long removed the assumption of non-degeneracy at zero (cf. [24], [26]). The nondegeneracy at infinity was removed by Chang-Liu-Liu (cf. [7]) and Fei-Qiu (cf. [14]), independently.
For system (1.1), P. Rabinowitz raised the question that whether there is a nonconstant solution with minimal period T for any given T > 0 in 1978. In [8], F. Clarke and I. Ekeland proved the existence of periodic solutions with minimal period T for any given T > 0 under √ 2-pinching condition for convex system. In [11], I. Ekeland and H. Hofer proved the existence of solutions with minimal period T for any given T > 0 under the strictly convex assumption. We refer to [3], [23] for various results of convex Hamiltonian system. In [25], Y. Long considered the system without convex assumption (cf. [26]). In [13], G. Fei and Q. Qiu showed that the lower bound of the minimal period is T 2n for semi-positive definite system. In [31], the second author of this paper proved that the minimal period is T if H is even for semi-positive and super-quadratic Hamiltonian system.
For the minimal period problem of autonomous Hamiltonian system under the assumption of asymptotically linear, D. Dong and Y. Long proved the existence of solution z with minimal period T under the assumption that the Hamiltonian system is positive definite and nondegenerate with i(z) ≤ n + 1 (cf. [9]). Let (1.2) Given an asymptotically linear Lagrangian system, (z, T ) is called a brake orbit if (z, T ) is the solution of (1.2) and B satisfies (B1). C. Liu gave the multiplicity results of the solutions of certain asymptotically linear Hamiltonian systems with a Lagrangian boundary condition in [18]. The second author of this paper dealt with the case of symmetric periodic brake orbit solutions in [30]. For the minimal period problem of (1.2), C. Liu proved the existence of a nonconstant solution of (1.2) with minimal period no less than T /2 for positive definite Hamiltonian system under super quadratic condition in [20]. In [32], the second author of this paper extended the result to the semi-positive case.
We define the following conditions, where (AH2) is the symmetric requirement, 2) has a non-constant brake solution with minimal period no less than T 2n+2 ; 2. In addition, if also (AH4) holds, equation (1.2) has a non-constant brake orbit with minimal period no less than T 2 ; 3. Moreover, if also (AH2) and (B2) hold, equation (1.2) has a non-constant symmetric brake solution with minimal period no less than T 3 . 1.2. Sketch of the proof. For asymptotically linear Hamiltonian system, in order to apply Saddle-point reduction Theorem (Theorem B.3) to varies boundary problems, we first truncate the Hamiltonian function suitably to satisfy (CH). Combining Theorem B.3, Theorem B.4 and Theorem B.11, we obtain the existence of a non-constant symmetric periodic orbit (z, T ) (resp. brake orbit, symmetric brake orbit) for the truncated Hamiltonian function, which is the solution of (1.1) (resp. (1.2)) if we carefully choose truncated function under the assumption (AH1).
Next, we give an estimation of the minimal period of (z, T ). Suppose that z = z p τ , where z τ is also a solution of (1.1) (resp. (1.2)) and z τ has minimal period τ = T /p for some positive integer p. Combining Bott-type iteration formula (i.e., Theorem A.11, Theorem A.7) with Lemma A.10 (resp. Lemma A.8), we obtain the relation of the Masloy-type indices between z and z τ , which gives an upper bound of p for semi-positive Hamiltonian system. In addition, the upper bound of p is related to the minimal period τ = T /p of z τ under the assumption that z = z p τ . Thus z is a non-constant solution of (1.1) (resp. (1.2)) with minimal period T /p. Furthermore, by applying Lemma A.13 to (3.13) (resp. (3.20), (3.21), (3.24)), we remove some exceptions of the estimation of p and thus we obtain more precise upper bound of p. Finally, we obtain the expected minimal period under varies boundary conditions and we complete the proof.
1.3. Main difficulty. There are two difficulties we need to overcome: the first one is the relation between Maslov-type index and the Morse index of symmetric periodic solution of (1.1) (resp. brake solution, symmetric brake solution of (1.2)) obtained by the variational methods. We overcome it by truncating the Hamiltonian function to apply the saddle point reduction and homological link theorem, and thus we obtain the existence of symmetric T -periodic solution of (1.1) (resp. brake orbit solution, symmetric brake orbit solution of (1.2)) with expected Maslov-type index information.
The second one is the precise estimation of the iteration number p in the last step of the proof of Theorem 1.1 (resp. Theorem 1.2). More precisely, we need to exclude the case k = 1 (resp. r = 1 for Theorem 1.2). Indeed, combining Lemma A.10 (resp. Lemma A.8) and Theorem A.11 (resp. Theorem A.7), we obtain a rough estimation of p under the assumption of (AH2)-(AH3) (resp. (AH2) and (AH3)). Based on the pioneer works of Long on i ω index for symplectic paths (cf. [26, P. 204]) and Lemma A.13, we obtain that the inequality (3.13) (resp. (3.20), (3.21) and (3.24)) is strict. We finally obtain the upper bound of the iteration number and hence we obtain the lower bound of the minimal period.
Organization: In Section 2, we establish the basic setting for the application of Theorem B.3 for Hamiltonian systems under various boundary conditions. In Section 3 we prove the main results of this paper based on previous preparations. In Appendix A, we give a brief introduction of Maslov-type index theory and associated iteration theory for symplectic paths starting at identity matrix under various boundary conditions. Appendix B.1 contains 4 parts: the first part is a simple introduction of Saddle Point Reduction Theorem, the second part is the application of Saddle Point Reduction Theorem to Hamiltonian systems, the third part establishes the relations between Maslov-type indices and the Morse indices of a truncated Morse function under saddle point reduction approach and the last part aims to introduce the minimax theorem in critical point theorem.
Throughout this paper, let N, Z, Q, R, C and U denote the set of natural integers, integers, rational numbers, real numbers, complex numbers and the unit circle in C, respectively.
2. The background setting. In order to apply Theorem B.3 for symmetric orbits, brake orbits and symmetric brake orbits, we give an introductions of following notations and the basic properties. The details can be found in [6,Chapter 4], [26,Chapter 4,Chapter 6] and references therein.
Let S T = R/(T Z), L = L 2 (S T , R 2n ) and W = W 1,2 (S T , R 2n ). Definê whereL and L (resp.Ŵ and W ) are equipped with L 2 -norm (resp. W 1,2 -norm). For Hilbert spaces L (resp.L, L) and W (resp.Ŵ , W ), let and let ·, · L and ·, · W be the inner product in L and W , respectively. In addition, W ,Ŵ and W are dense subspaces of L,L and L, respectively. Furthermore, by the standard bootstrap argument (cf. [26,Chapter 4]), a weak solution of (1.1) is actually a classical solution. Let A = J d dt : L → L with dom(A) = W and letÂ = A|L, A = A| L ,Â = A|L ∩L with domain respectively given by dom(Â) =Ŵ , dom(A) = W , dom(Â) =Ŵ ∩ W . By [26, P.93-P.94], A is self-adjoint on W with closed image and the spectrum of A is point spectrum, which each nonzero element of σ(A 0 ) has the multiplicity 2n. ThusÂ (resp. A andÂ) is self-adjoint onŴ (resp. W andŴ ∩ W ) with each nonzero spectrum of multiplicity 2n (resp. n and n). Let and ). Let P 0 : L → X 0 = ker A = R 2n be the projection map and letP 0 = P 0 |L (resp.
3.1. Proof of Theorem 1.1. The proof is divided into 4 steps.
Step 1: In order to apply the saddle point reduction frame work, we truncate H into H k such that it satisfies (CH) (cf. Section B.2) under the assumption (AH1). By (AH1), there exists a constant Λ 0 > 0 such that 0 ≤ B ≤ Λ 0 I and Choose a constant k 0 ≥ 5 such that For k 0 ≤ k ∈ Z, let χ k ∈ C ∞ (R, R) be a cut-off function such that where b k ≥ 1 is chosen to satisfy 0 ≤ −χ (r) ≤ 2 r for k < r < k + b k . Set Then H k ∈ C 2 (R × R 2n , R) and the followings hold: 3: H k (t, z) satisfies (AH1) and

4:
There is a constant c(H, k) such that We let Step 2: We show thatâ k satisfies (PS)-condition (cf. Definition B.9). We first prove that for any critical point z k off k , there is a constant k 1 ≥ k 0 such that which means the critical points off k is uniformly bounded withâ k =f k •û k . In addition, u k and u −1 k is bounded with u −1 k : Im(u k ) → Z k . Combining (3.5), Theorem B.3 with the fact that dom(u k ) is finite dimensional, we obtain thatâ k satisfies (PS)-condition. Indeed, we first show the C 0 -norm of the critical point z k off k is bounded from above, then we apply (3.3) to obtain that the W 1,2 -norm is bounded from above. Let z k be a critical point off k . If (Â − B)z k L = 0, then by Hölder inequality, we get which yields that z k W 1,2 ≤ M 3 for some constant M 3 . Otherwise, by the properties listed in Section 2 ofÂ, there is a constant α = α(z k ) > 0 such that which means that for any critical point z k off k , z k L ≤ M 4 , for some constant M 4 (independent of k). By (3.6), we obtain Step 3: We aim to prove that there exists a non-constant solution z k of (1.1) such that Then there exists a constant ρ > 0 small enough such thatâ where Z ± are the eigenspaces of A corresponding to the eigenvalues belonging to R ± andẐ ± =Ẑ ∩ Z ± , respectively. In addition, there exists y ∈Ẑ + with y = 1 such that for some constant λ 1 > 0, Such y does exist since we require i −1 B, T 2 ≥ 1 by (B2). More precisely, according to Theorem B.6, the existence of such y is obvious. Let where R 0 > 0 will be determined later. Thusâ k (z) ≤ 0 for z = z − ∈Ẑ − . Furthermore, for z = ry + z − ∈ ∂Q y with sufficiently large r, combining (AH1), (3.9) and Theorem B.4 implieŝ Therefore, combining (3.9), (3.10) with Theorem B.11, there exists a critical point z k ∈Ẑ with its Morse index m − (â k ) at z k and the nullity m 0 (â k ) at z satisfying Then by Theorem B.
) > 0, we have that z k is a non-constant solution. In addition, by choosing k ≥ k 1 sufficiently large in step 1, such z k is T -periodic symmetric solution of (1.1).
Step 4: Suppose (z, T ) is a symmetric solution of (1.1) obtained from step 3 with minimal period τ . Then T /τ ≡ p ∈ Z + and there exists a symmetric solution z τ of (1.1) with minimal period τ such that z(t) = z p τ (t). Next we prove that p = 1, which implies τ = T .
Denote by γ zτ and γ z the associated symplectic paths of z τ and z, respectively. Then γ z = γ p zτ and its Maslov-type indices satisfy In addition, since (AH3) holds, we have where the equality holds by Theorem A.11. Moreover, according to Theorem A.11, where the first inequality holds by Lemma A.13 and the second inequality holds by (3.12). Thus if p = 2k + 1 ≥ 1, combining Lemma A.10 and (3.11), contradicts the fact Thus we complete the proof of Theorem 1.1.

3.2.
Proof of Theorem 1.2. We give the proof of conclusions 1 and 2 of Theorem 1.2 in 3 steps: Step 1: Under the assumption of (AH1), (AH2) and (B1), we truncate H into H k in a similar way as in Section 3.1 such that H k satisfy (3.1)-(3.4). Furthermore, since B satisfies (B1), we obtain that H k satisfies (CH) and (AH2) . Let f k be given in Section 3.1 and let f k = f k | W . Suppose (a k , u k ) is given in Theorem B.3 associated for the case of (A, f, L, Z, W ) = (A, f k , Z, L, W ) for c(H) = c(H, k). Similarly, the following two facts hold: • For any critical point z k of f k , there is a constant k 1 ≥ k 0 such that z k W 1,2 ≤ k 1 . • a k satisfies (PS)-condition on Z. Indeed, the above two facts can be obtained by a similar argument in Section 3.1. We omit the proof for simplicity.
Step 2: We aim to show that there exists a non-constant solution z k of the solution of (1.2) such that its Maslov-type indices satisfy where R 0 > 0 will be determined later. Thus Furthermore, for z = ry + z 0 + z − ∈ ∂Q y , since P − w(z) + z 0 + z − is orthogonal to Ay ∈ Z + , taking sufficiently large r, we obtain Therefore, applying (3.16) and (3.17) to Theorem B.11, there exists a critical point z k ∈ Z of a k with its Morse index m − (a k , z) and the nullity m 0 (a k , z) satisfying Then by Theorem B.7, i L0 (z) ≤ 1 ≤ i L0 (z) + ν L0 (z). Furthermore, since f k (x) = f k (u k (z)) > 0, z k is non-constant solution. By taking sufficiently large k 1 , z k is actually a brake solution of (1.2).
Step 3: Suppose (z, T ) is brake solution of (1.2) obtained from step 2 and z(t) = z p τ (t) (up to a time shift by Remark A.3) for some z τ with minimal period τ = T /p. Next we give an estimation of p. More precisely, we show that p ≤ 2n + 2 under the assumption of 1 of Theorem 1.2. If, in addition, (AH4) holds, we prove that p ≤ 2. Thus we complete the proof of 1 and 2 of Theorem 1.2.
Denote by γ zτ and γ z the symplectic paths associated with z τ and z, respectively.
Thus equation (1.2) has a non-constant brake solution with minimal period no less than T /2, the proof of conclusion 2 of Theorem 1.2 is complete.
In the following, we prove conclusion 3 of Theorem 1.2. The proof is divided into 2 steps.
Step 1: Under the assumption of Theorem 1.2, we truncate H in a similar way as in Section 3.1 to make H k satisfy (3.1)-(3.4). Furthermore, since B ∈ L s (R 2n ) satisfies (B1), we obtain that H k satisfies (CH), (AH2) and (AH2) . Suppose that f k is the same as in Section 3.1. Let • For any critical point z k of (a k ) −1 , there is a constant k 1 ≥ k 0 such that z k W 1,2 ≤ k 1 .
• There exists a non-constant symmetric brake orbit z k of (1.2) such that its Maslov-type indices satisfy Step 2: Suppose (z, T ) is the symmetric brake solution of (1.2) obtained by (3.23) with minimal period τ . Then T /τ ≡ p ∈ Z + and there is a symmetric brake solution of (1.2) with minimal period τ such that z(t) = z p τ (t). Under the assumption of Theorem 1.2, we show that p ≤ 3, and hence the minimal period of (z, T ) ≥ T /3. Indeed, let γ zτ and γ z be the symplectic paths associated with z τ and z, respectively. Then γ z = γ p zτ . Besides, by (3.23), By (AH3), the followings hold Then combining (AH1)-(AH3) with Theorem A.11, we have
A. Appendix: Maslov-type indices. Let where we often omit T if R ≥0 for P T (2n). Let J be given in Introduction, J is an almost complex structure of R 2n , i.e., Let R 2n be equipped with the standard symplectic form ω 0 such that ω 0 (x, y) = Jx, y , thus J is compatible with ω 0 in the sense that for x, y ∈ R 2n , ω 0 (Jx, Jy) = ω 0 (x, y) and ω 0 (x, Jx) > 0 for x = 0.
(A.2) Suppose (z, T ) is the solution of (1.2) with the associated symplectic path γ z ∈ P T /2 (2n). Define Suppose γ B ∈ P T (2n) is the solution of (A.2) with H = B for B ∈ C(S T , L s (R 2n )). Such γ B is called the symplectic path associated with B. For ω ∈ U, define For any x i , y i ∈ R ki with i = 1, 2, define (x 1 , y 1 ) (x 2 , y 2 ) = (x 1 , x 2 , y 1 , y 2 ). For any the symplectic sum is given by 2. Suppose that γ is the fundamental solution ofẋ(t) = JB(t)x(t) with symmetric matrix for every t ∈ R Then we have provided that b 11 (t) is strictly positive definite for all t ∈ R. 3. For γ ∈ P T (2n), we have where M ε (P ) is given by P T sin(2ε)I n − cos(2ε)I n − cos(2ε)I n − sin(2ε)I n P + sin(2ε)I n cos(2ε)I n cos(2ε)I n − sin(2ε)I n .
The statement 1 of Proposition A.2 can be found in [4, Property IV (Symplectic Additivity)]. The statement 2 can be found in [19,Lemma 5.1], which is obtained by applying the method developed in [10], [16] and [28] for positive-definite Hamiltonian system. The statement 3 concerns the difference between i L0 and i L1 (cf. [32,Theorem 2.3]), which mainly relies on the homotopy invariant property of Maslov-type index.
Remark A.3. For a brake solution (z, T ) of (1.2), let γ z be the symplectic path associated with z. Then is a brake solution of (1.2) and the symplectic path associated with y is given by Proof. Since z T 2 + t = N z T 2 − t and JN = −N J, we havė . .
The following lemma shows the relation of Maslov-type indices between z and y under the assumption of Remark A.3.
Moreover, if z t + T 2 = −z(t), then for ω ∈ U, we have i ω Lj (γ y ) = i ω Lj (γ z ). Proof. We have proved that in [12], here we give a proof for completeness. First, we have Further more, by Proposition A.2, the followings hold To prove the first statement, it is sufficient to show Indeed, we have M ε (M ) = M T sin(2ε)I n − cos(2ε)I n − cos(2ε)I n − sin(2ε)I n M + sin(2ε)I n cos(2ε)I n cos(2ε)I n − sin(2ε)I n and N sin(2ε)I n cos(2ε)I n cos(2ε)I n − sin(2ε)I n N = sin(2ε)I n − cos(2ε)I n − cos(2ε)I n − sin(2ε)I n .
We define k-th iteration in brake orbit boundary sense as follows: Definition A.5 (cf. [22,Definition 2.9]). For any γ ∈ P T (2n), the k-th iteration γ k of γ in brake orbit boundary sense is defined byγ | [0,kT ] withγ : R → P(2n): Remark A.6. The k-th iteration γ k of γ with γ ∈ P T (2n) in periodic sense was defined in [26,Chapter 8]. Without loss of generality, if z is a solution of (1.1), the iteration of associated symplectic path γ z ∈ P T (2n) means the iteration in the periodic boundary sense (cf. [26, P. 177]). If z is a solution of (1.2), the iteration of associated symplectic path γ z ∈ P T 2 (2n) means the iteration in the brake orbit boundary sense (cf. Definition A.5).
For a Hilbert space L, let A be a self-adjoint operator with dom(A) ⊂ L and Φ ∈ C 1 (L, R) with dΦ = F, Φ(0) = 0. Assume the followings (C): There exists c − < c + such that c± ∈ σ(A) and σ(A) ∩ [c − , c + ] contains at most finitely many eigenvalues with finite multiplicities.
where · L is the norm in L. By assumption 1, there exists a sufficiently small ε > 0 such that 0 ∈ σ(A ε ) where A ε = A + εId. Next we apply the Lyapunov-Schmidt reduction for A ε . Let {E λ } be the spectral resolution of A ε and let Then the followings hold 1. R, S − , S + are pairwise commuting. 2. R and S ± are injective restricted on Z and L ± , respectively.
The norm · V in V is given by Then V 0 and V ± are isomorphic to Z and L ± , respectively. For x = x − + z + x + ∈ L − ⊕ Z ⊕ L + , we define two functionals f and Φ ε as follows: Then by the assumption of Φ, f is C 1 on L. Let F ε = F + εId. The critical point x of f satisfies which implies x ± = ± S ± P ± F ε (v),

In addition, (B.2) is equivalent
By (C), there is a constant c ε > c + + ε such that
Then by (CH), f ∈ C 1 (W, R) and f is Gateaux differentiable with f and df are given by f (z) = Az − g (z) and df (z)[y] = Ay − dg (z)[y]. Let A 0 and X 0 be given as in Section 2. We apply Theorem B.1 to the case of (A, f, L, Z, dom(A)) = (J d dt , f, L 2 (S T , R 2n ), Z, W 1,2 (S T , R 2n )) with respect to c ± = ±c(H). More precisely, we have Theorem B.3 (cf. [26,Theorem 4.4.1]). Suppose that H satisfies (CH). Then there exists a function a ∈ C 2 (Z, R) and an injection map u ∈ C 1 (Z, L) so that u : Z → W = dom(A) and 1. The map u can be written as u(z) = w(z) + z, where P w(z) = 0.
Remark B.5. We denote the restrictions of g toL, L andL ∩ L as followŝ g = g|L, g = g| L andĝ = g|L ∩L .
Then for the case of symmetric periodic orbits, brake orbits and symmetric brake orbits, similar results can be obtained by applying Theorem B.3 and Theorem B.4 to