Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems

In this paper, for any positive integer $n$, we study the Maslov-type index theory of $i_{L_0}$, $i_{L_1}$ and $i_{\sqrt{-1}}^{L_0}$ with $L_0=\{0\}\times \R^n\subset \R^{2n}$ and $L_1=\R^n\times \{0\} \subset \R^{2n}$. As applications we study the minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltonian systems. For first order nonlinear autonomous reversible Hamiltonian systems in $\R^{2n}$, which are semipositive, and superquadratic at zero and infinity, we prove that for any $T>0$, the considered Hamiltonian systems possesses a nonconstant $T$ periodic brake orbit $X_T$ with minimal period no less than $\frac{T}{2n+2}$. Furthermore if $\int_0^T H"_{22}(x_T(t))dt$ is positive definite, then the minimal period of $x_T$ belongs to $\{T,\;\frac{T}{2}\}$. Moreover, if the Hamiltonian system is even, we prove that for any $T>0$, the considered even semipositive Hamiltonian systems possesses a nonconstant symmetric brake orbit with minimal period belonging to $\{T,\;\frac{T}{3}\}$


Introduction and main results
In this paper, let J = We consider the following problem: A solution (T, x) of (1.1)-(1.2) is a special periodic solution of the Hamiltonian system (1.1). We call it a brake orbit and T the period of x. Moreover, if x(R) = −x(R), we call it a symmetric brake orbit. It is easy to check that if τ is the minimal period of x, there must holds x(t + τ 2 ) = −x(t) for all t ∈ R.
Since 1948, when H. Seifert in [47] proposed his famous conjecture of the existence of n geometrically different brake orbits in the potential well in R n under certain conditions, many people began to study this conjecture and related problems. Let #Õ (Ω) and #J b (Σ) the number of geometrically distinct brake obits in Ω for the second order case and on Σ for the first order case respectively. S.
Bolotin proved first in [7](also see [8]) of 1978 the existence of brake orbits in general setting. K.
In 1989, A. Szulkin in [49] proved that #J b (H −1 (h)) ≥ n, if H satisfies conditions in [43] of Rabinowitz  In [42] of 2006, Long , Zhu and the author of this paper proved that there exist at least 2 geometrically distinct brake orbits on any central symmetric strictly convex hypersuface Σ in R 2n for n ≥ 2. Recently, in [35], Liu and the author of this paper proved that there exist at least [n/2] + 1 geometrically distinct brake orbits on any central symmetric strictly convex hypersuface Σ in R 2n for n ≥ 2, if all brake orbits on Σ are nondegenerate then there are at least n geometrically distinct brake orbits on Σ. For more details one can refer to [42], [35] and the reference there in.
In his pioneering paper [43] of 1978, P. Rabinowitz proved the following famous result via the variational method. Suppose H satisfies the following conditions: (H1 ′ ) H ∈ C 1 (R 2n , R).
Then for any T > 0, the system (1.1) possesses a non-constant T -periodic solution. Because a T /k periodic function is also a T -periodic function, in [43] Rabinowitz proposed a conjecture that under conditions (H1 ′ ) and (H2)-(H4), there is a non-constant solution possessing any prescribed minimal period. Since 1978, this conjecture has been deeply studied by many mathematicians. A significant progress was made by Ekeland and Hofer in their celebrated paper [16] of 1985, where they proved Rabinowitz's conjecture for the strictly convex Hamiltonian system. For Hamiltonian systems with convex or weak convex assumptions, we refer to [2]- [3], [12]- [13], [15]- [17], [41], [20]- [23], and references therein for more details. For the case without convex condition we refer to [37]- [39] and Chapter 13 of [41] and references therein. A interesting result is for the semipositive first order Hamiltonian system, in [18] G. Fei, S.-T. Kim, and T. Wang proved the existence of a T periodic solution of system (1.1) with minimal period no less than T /2n for any given T > 0.
Note that in the second order Hamiltonian systems there are many results on the minimal problem of brake orbits such us [37]- [39] and [50]. For the even first order Hamiltonian system, in [51], the author of this paper studied the minimal period problem of semipositive even Hamiltonian system and gave a positive answer to Rabinowitz's conjecture in that case. In [19], G. Fei, S.-T.
Kim, and T. Wang proved the same result for second order Hamiltonian systems.
So it is natural to consider the minimal period problem of brake orbits in reversible first order nonlinear Hamiltonian systems. In [32], Liu have considered the strictly convex reversible Hamiltonian systems case and proved the existence of nonconstant brake orbit of (1.1) with minimal period belonging to {T, T /2} for any given T > 0.
Since [51], we also hope to obtain some interesting results in the even Hamiltonian system for the minimal period problem of brake orbits.
It can be found in many papers mentioned above that the Maslov-type index theory and its iteration theory play a important role in the study of minimal period problems in Hamiltonian systems. In this paper we study some monotonicity properties of Maslov-type index and apply it to prove our main results.
In this paper we denote by L(R 2n ) and L s (R 2n ) the set of all real 2n×2n matrices and symmetric matrices respectively. And we denote by y 1 · y 2 the usual inner product for all y 1 , y 2 ∈ R k with k being any positive integer. Also we denote by N and Z the set of positive integers and integers respectively.
Then for any T > 0, the system (1.1)-(1.2) possesses a nonconstant T periodic brake orbit solution x T with minimal period no less that T 2n+2 . Moreover, for x = (x 1 , x 2 ) with x 1 , x 2 ∈ R n , denote by H ′′ 22 (x) the second order differential of H with respect to x 2 , if Then the minimal period of x T belongs to {T, T 2 }. In the case n = 1, the result can be better, i.e., the following Theorem 1.2. For n = 1, suppose that H satisfies conditions (H1)-(H4).
As a direct consequence of Theorem 1.3, we have the following Corollary 1.1.
where we see B 0 as an element in C([0, T /2], L s (R 2n ) satisfies condition (B1), under the same assumptions of Theorem 1.2, the system (1.1) possesses a nonconstant T -periodic brake orbit with minimal period no less that T 2n+2 . We can also prove the following Corollary 1.2 of Theorem 1.3. being n × n matrix, assume for all x ∈ R 2n , andĤ satisfies conditions (H1)-(H6). Then for any T > 0, the system (1.1)-(1.2) possesses a nonconstant symmetric brake orbit x T with minimal period no less than is even, then the minimal period of x T is no less than where we see B 0 as an element in C([0, T /4], L s (R 2n ) satisfies condition (B1). Remark 1.3. In section 3, we will show that i L 0 As a direct consequence of Theorem 1.5, we have the following Corollary 1.3. We can also prove the following Corollary 1.4 of Theorem 1.5. Corollary 1.4. If B 0 = 0, then for 0 < T < π ||B 0 || with ||B 0 || being the operator norm of B 0 , under the same condition of Theorem 1.5, the system (1.1) possesses a nonconstant symmetric brake orbit with minimal period belonging to {T, T /3}. This paper is organized as follows. In section 2, we study the Maslov-type index theory of i L 0 , i L 1 and i L 0 √ −1 . We compute the difference between i L 0 (γ) and i L 1 (γ). In Section 3, we study the relation between the Maslov-type index (i L 0 ) satisfies condition (B1) and the Morse indices of the corresponding Galerkin approximation. As applications we get some monotonicity properties of i L 0 (B), i L 1 (B) and i L 0 √ −1 (B) and we prove Theorem 3.2 which is very important in the proof of Theorems 1.4-1.5. In Section 4, based on the preparations in Sections 2 and 3 we prove Theorems 1.1-1.3 and Corollary 1.2. In Section 5, we prove Theorems 1.4-1.5 and corollary 1.4. In Section 6, we give a briefly review of (i ω , ν ω ) index theory with ω ∈ U for symplectic paths starting with identity as appendix.
2 Maslov-type index theory associated with Lagrangian subspaces possess the standard inner product. We define the symplectic structure of F by We denote by Lag(F ) the set of Lagrangian subspaces of F , and equip it with the topology as a subspace of the Grassmannian of all 2n-dimensional subspaces of F .
It is easy to check that, for any M ∈ Sp(2n) its graph By Proposition 6.1 of [35] and Lemma 2.8 and Definition 2.5 of [42], we give the following definition.
Definition 2.2. We define the index function where the definition of sf of spectral flow for the path of bounded self-adjoint linear operators one can refer to [53] and references their in.
By (3.21) of [35], we have Proof. Without loss of generality we can suppose the C 1 path Gr(γ B ) of Lagrangian subspaces intersects V ω regularly (otherwise we can perturb it slightly with fixed endow points such that they intersects regularly and the index dose not change by the homotopy invariant property µ CLM where the definition of intersection form can be found in [46]. We denote by µ BF the maslov index defined by Booss and Furutani in [9]. By the spectral flow formula of Theorem 5.1 in [9] or Theorem 1.5 of [10] (cf. also proof of (2.9) where in the fourth equality we have used Theorem 2.1 in [9] and the property of index µ RS for symplectic paths defined in [46](cf also (2.6)-(2.8) of [52]), in the sixth equality we have used Lemma 2.6 of [42], in the second and seventh equalities we used the symplectic invariance property of index µ BF and µ CLM F respectively.
, L s (R 2n ) and γ B be the symplectic path associated to B. We define (2.11) By Lemma 2.1, in general we give the following definition.
Theorem 2.1. (Theorem 4.1 of [35]) Let γ ∈ P τ (2n) and ω k = e π √ −1/k . For odd k we have and for even k, we have Obviously we also have In order to study the minimal period problem for Even reversible Hamiltonian systems, we need the iteration formula of the Maslov-type index of (i L 0 √ −1 , ν L 0 √ −1 ) for symplectic paths starting with identity. We use Theorem 2.1 to obtain it.
Precisely we have the following Theorem.

21)
and for even k, we have Proof. For odd k, since γ 2k = (γ k ) 2 , by Theorem 2.1 we have Also by Theorem 2.1 we have Since ω k = ω 2 2k , by (
The precise difference of i L 0 (γ) and i L 1 (γ) for γ ∈ P τ with τ > 0 is very important in the proof of the main results of this paper. In this subsection we use the Hörmander index (cf. [14]) to compute it. Note that in [42], in fact we have already proved that |i L 0 (γ) − i L 1 (γ)| ≤ n.
For any P ∈ Sp(2n) and ε ∈ R, we set Then we have the following theorem.
we also have, where ε < 0 and |ε| is sufficiently small.
In the proof of Theorem 3.3 of [42], we have proved that for ε > 0 small enough, there holds where sgn(W 1 , W 3 ; W 2 ) for 3 Lagrangian spaces with W 3 transverses to W 1 and W 2 is introduced in Definition 3.2.3 on p. 67 of [14]. Note that by Claim 1 below, we can prove (2.39) at once.
Choose ε < 0 such that |ε| is sufficiently small, by the discussion of µ CLM F index we have Then by the same proof as above, we have where ε < 0 is small enough. Hence (2.32) holds. The proof of Theorem 2.3 is complete.
We have the following consequence.
Moreover if γ(1) is a orthogonal matrix then there holds Proof. (2.52) holds directly from Theorem 2.3, so we only need to prove (2.53). Since γ(τ ) is an orthogonal and symplectic matrix, we have It is easy to check that for any ε ∈ R, there holds Hence by (2.55) and (2.56), we have Thus for any x ∈ R 2n and λ ∈ R satisfying Since for ε > 0 small enough M ε (γ(τ )) is an invertible symmetric matrix, by (2.60) we have Proof. Since Sp(2n) is path connected, we can choose a path γ ∈ P τ with γ(τ ) = P (0). By Proposition 2.11 of [42] and the definition of µ j for j = 1, 2 in [42], we have So by the Path Additivity and Reparametrization Invariance properties of µ CLM F in [11], we have where the definition of joint path η * ξ is given by (6.1) in Section 6 below. Then by Theorem 2.3 we have Then (2.63) holds from (2.65)-(2.67). The proof of Lemma 2.2 is complete.
Remark 2.1. It is easy to check that for n j × n j symplectic matrix P j with j = 1, 2 and n j ∈ N, we have By direct computation according to Theorem 2.3 and Corollary 2.1, for γ ∈ P τ (2), b > 0, and Also we give a example as follows to finish this section √ −1 and the corresponding Morse indices, and their monotonicity properties.
In [31], Liu studied the relation between the L-index of solutions of Hamiltonian systems with Lboundary conditions and the Morse index of the corresponding functional defined via the Galerkin approximation method on the finite dimensional truncated space at its corresponding critical points.
In order to prove the main results of this paper, in this section we use the results of [31] to study some monotonicity properties of i L 0 and i L 1 . We also study the index i For any τ > 0 and B ∈ C([0, τ /4], L s (R 2n )) (in order to apply the results in this section conveniently Section 5, we always assume B ∈ C([0, τ /4], L s (R 2n )) satisfying condition (B1). We Then since B( τ 2 ) = B(0), we can extend it τ 2 -periodically to R, so we can see B as an element in C(S τ /2 , L s (R 2n )).
a.e. t ∈ R} with the usual norm and inner product denoted by || · || and · respectively.
By the Sobolev embedding theorem, for any s ∈ [1, +∞), there is a constant C s > 0 such that Note that B can also be seen as an element in C(S τ , L s (R 2n )). We define two selfadjoint operators A τ and B τ on E τ by the following bilinear forms Then A τ is a bounded operator on E τ and dim ker A τ = n, the Fredholm index of A τ is zero, Let Γ τ = {P τ,m : m = 0, 1, 2, ...} be the usual Galerkin approximation scheme w.r.t. A τ , just as in [31], i.e., Γ τ is a sequence of orthogonal projections satisfies: Similarly we define two subspaces of a.e. t ∈ R} be the symmetric ones and τ 2 -periodic ones of E τ respectively.
We define two selfadjoint operatorsÂ andB onÊ by the following bilinear forms ThenÂ is a bounded Fredholm operator onÊ and dim kerÂ = 0, the Fredholm index ofÂ is zero.B is a compact operator onÊ.
For any positive integer m, we definê For m ≥ 1, letP m be the orthogonal projection fromÊ toÊ m . Then {P m } is a Galerkin approximation scheme w.r.t.Â.
Proof. The method of the proof here is similar as that of Theorem 2.1 in [51].
For any positive integer m, we defineẼ For m ≥ 1, letP m be the orthogonal projection fromẼ toẼ m . Then {P m } is a Galerkin approximation scheme w.r.t.Ã.
For any y ∈Ê m and z ∈Ẽ m , it is easy to check that Similarly, we have the following A τ − B τ orthogonal decomposition Hence, under above decomposition we have Note that, the space E τ and the operators A τ , B τ and P τ,m are also defined in the same way.
So by the definition we see thatẼ is the τ -periodic extending of E τ from S τ to S 2τ , andẼ m is the τ -periodic extending of E τ,2m from S τ to S 2τ too.
Thus we have By (3.13) and (3.15) we have In the following, we study some monotonicity of the the Maslov-type i L 0 √ −1 index. In this paper, for any two symmetric matrices B 1 and B 2 , we say and then there holds Proof. Let the spaceÊ and the orthogonal projection operatorP m be the ones defined in Section 2. Correspondingly we define the compact operatorsB 1 andB 2 . By Theorem 3.1, for    proof. By Lemma 3.1, we have Then the conclusion holds from the fact that Where γ 0 is the identity symplectic path.
By Theorem 2.1 of [31] and the Remark below Theorem 2.1 in [31] and the similar proof of Lemma 3.1 we have the following lemma.
Choose a C 1 path γ ∈ P pτ It is clear that So by (3.65), there is a σ > 0 small enough such that By the Homotopy invariance with respect to end points and Path additivity properties of µ CLM F index in [11], we have    For T > 0, we set E = W 1/2,2 (S T , R 2n ) with the usual norm and inner product denoted by || · || and · respectively, and two subspaces of E by As in Section 3, we define two selfadjoint operators A T on E T by the same way as (3.3). We also define two selfadjoint operatorsǍ T onĚ T by the following bilinear form: Then A T is a bounded operator on E T and dim ker A T = n, the Fredholm index of A T is zero, anď A T is a bounded operator onĚ T and dim kerǍ T = n, the Fredholm index ofǍ T is zero.
Let P T,m be the orthogonal projection from E T to E T,m andP T,m be the orthogonal projection For z ∈ E T , we define It is well known that f ∈ C 2 (E T , R) whenever, for some s ∈ (1, +∞) and all x ∈ R 2n .
By similar argument of Lemma 4.1 of [51], looking for T -periodic brake orbit solutions of (1.1) is equivalent to look for critical points of f .
In order to get the information about the Maslov-type indices, we need the following theorem which was proved in [24,28,48]. (i) There exist ρ, δ > 0 such that f (w) ≥ δ for any w ∈ W ; (ii) There exist e ∈ ∂B 1 (0) ∩ Y and r 0 > ρ > 0 such that for any w ∈ ∂Q, f (w) < δ where Then (1) f possesses a critical value c ≥ δ, which is given by where Γ = {h ∈ C(Q, E) : h = id on ∂Q}; (2) There exists w 0 ∈ K c ≡ {w ∈ E : f ′ (w) = 0, f (w) = c} such that the Morse index m − (w 0 ) Proof of Theorem 1.3. For any given T > 0, we prove the existence of T -periodic brake solution of (1.1) whose minimal period satisfies the inequalities in the conclusion of Theorem 1.2.
We divide the proof into five steps.
Step 1. We truncate the functionĤ suitably and evenly such that it satisfies the growth condition (4.4). Hence corresponding new reversible function H satisfies condition (4.4).
We follow the method in Rabinowitz's pioneering work [43] (cf. also [18], [44] and [51]). Let Where K will be determined later. Set and where the constant R K satisfies Then H K ∈ C 2 (R 2n , R). SinceĤ satisfies (H3), ∀ε > 0, there is a δ 1 > 0 such thatĤ K (z) ≤ ε|z| 2 for |z| ≤ δ 1 . It is easy to see that H K (z)|z| 4 is uniformly bounded as |z| → +∞, there is an Then f K ∈ C 2 (E T , R) and where B 0T is the selfadjoint linear compact operator on E T defined by Step 2. For m > 0, let f Km = f |E T,m . We show f Km satisfies the hypotheses of Theorem 4.1.
We set For z ∈ Y m , by (4.8), (3.2), and the fact that P T,j B 0T = P T,j B 0T for j > 0, we have where C 2 and C 4 are constants for s = 2, 4 for the Sobolev embedding of inequality (3.2), and they are independent of m and K.
So if choose ε > 0 small enough such that εC 2 2 < 1 4 ||(A T −B 0T ) # || −1 , then there exists ρ = ρ(K) > 0 small enough and δ = δ(K) > 0, which are independent of m, such that where r 1 will be determined later. Let z = z − + z 0 ∈ B r 1 (0) ∩ X m , we have SinceĤ satisfies (H2) we havê where α = min{µ, 4}, a 1 > 0, a 2 are two constants independent of K and m. Then there holds where a 3 and a 4 are constants independent of K and m. By (4.11) and (4.12) we have Since α > 2 there exists a constant r 1 > ρ > 0, which are independent of K and m, such that Then by Theorem 4.1, f Km has a critical value c Km , which is given by Step 3. We prove that there exists a T -periodic brake orbit solution x T of (1.1) which satisfies Note that id ∈ Γ m , by (4.11) and condition (H4), we have Then {c Km } possesses a convergent subsequence, we still denote it by {c Km } for convenience.
So there is a c K ∈ [δ, ] such that c Km → c K .
By the same arguments as in section 6 of [44] we have f K satisfies (P S) * c condition for c ∈ R, i.e., any sequence z m such that z m ∈ E T,m , f ′ Km (z m ) → 0 and f Km (z m ) → c possesses a convergent subsequence in E T . Hence in the sense of subsequence we have By similar argument in [44], x K is a classical nonconstant symmetric T -periodic solution oḟ , Then B K ∈ C([0, T /2], L s (R 2n )) and satisfies condition (B1). Let B K T be the operator defined by the same way of the definition of B 0T . It is easy to show that Then for z ∈ {z ∈ E T : So we have Then by (4.15), (4.16), and (4.17)-(4.19), we have Since x T obtained in Step 3 is a nonconstant and symmetric T -period solution, its minimal period τ = T k for some k ∈ N. We denote by x τ = x T | [0,τ ] , then it is a brake orbit solution of (1.1) with the minimal τ and X T = x k τ being the k times iteration of x τ . As in Section 1, let γ x T and γ xτ be the symplectic path associated to (τ, x) and (T, x T ) respectively. Then γ xτ ∈ C([0, τ 2 ], Sp(2n)) and γ x T ∈ C([0, T 2 ], Sp(2n)). Also we have γ x T = γ k xτ .
Step 4. We prove that We follow the way of the proof of Theorem 1.2 of [18]. By the same way asĚ T andǍ T we can define the spaceĚ τ and the operatorǍ τ on it. Also we can define the orthogonal projectionP τ , m and the subspacesĚ τ,m for m = 0, 1, 2, .... LetB τ be the selfadjoint linear compact operator oň E T defined by: and Y be the orthogonal complement of X in L 1 , i.e., , by (H4) it is easy to see that there exists λ 0 > 0 such that

Thus for any
By choosing ε suitably one can see that there exists 0 < c 0 < 1 with |1 − c 0 | small enough such When ||z 0 || ≤ c 0 , we have ||z − || 2 ≥ 1 − c 2 0 . By (4.21) and (H4) Hence we always have By (4.24) and Theorem 2.1 of [31] and Remark 3.1 and the definition of i L 1 (γ(x τ )), for m large enough, we have 25) which implies that Since x τ is a nonconstant brake solution of (1.1), by the definition of X we have Hence by (4.26) and (4.27) we have Step 5. Finish the proof of Theorem 1.3.
Hence we have proved that k can not be 2(i L 0 (B 0 ) + ν L 0 (B 0 )) + 2n + 3. By the same argument we can prove that k can not be 2(i L 0 (B 0 ) + ν L 0 (B 0 )) + 2n + 4. Thus Claim 2 is proved and the proof of Theorem 1.3 is complete.
Proof of Theorem 1.1. Note that this is the case B 0 = 0 of Theorem 1.3. Then by Theorem 1.3 and the fact that i L 0 (0) = −n and ν L 0 (0) = n, the minimal period of x T is no less than T 2n+2 . In the following we prove that if (1.9) holds then the minimal period of x T belongs to {T, T 2 }. Let x T is the k-time iteration of x τ with τ being the minimal period of x τ and τ = T k . Then by the proof of Theorem 1.3 with B 0 = 0 we have (4.28), (4.29) and (4.30) hold. Since (1.9) holds, by Lemma 3.3 we have (4.56) So by (4.29) if k is odd, we have Hence k ≤ 3. Now we prove that k can not be 3, other wise we have And by Theorem 2.1 and Theorem 6.2 we have Then all the equalities of (4.61) hold. By Lemma 6.2 and 2 of Theorem 6.2 again, there exist p ≥ 0, q ≥ 0 with p + q ≤ n and 0 < θ 1 ≤ θ 2 ≤ ... ≤ θ n−(p+q) ≤ 2π/3 such that where Ω 0 (M ) for a symplectic matrix M is defined in Section 6. By (4.62) we have − 1 / ∈ σ((γ 2 xτ )(τ )). We first prove D is invertible. Otherwise, there exists a n × n invertible matrix P such that Since by (4.2) o f [35] we have by (4.67) and (4.68) we have which contradicts to (4.63). Thus we have proved that D is invertible. Similarly we can prove A is invertible, and Claim 1 is proved.
In fact Since B and D are both invertible, for any ω ∈ C, we have Continue the proof of Theorem 1.1.
If detB > 0, there is a continuous symplectic matrix path joint Hence By (4.80), there exist invertible matrix R such that ).
Thus we have prove that k can not be 3. So if k is odd, it must be 1. By the same proof we have if k is even, it must be 2. Then τ ∈ {T, T 2 }. The proof of Theorem 1.1 is complete. Proof of Corollary 1.2. Since 0 < T < π ||B 0 || , there is ε > 0 small enough such that It is easy to see that So we have Then by (5.40) and Lemma 3.1 and Corollary 3.1 we have Hence by the same proof of Theorem 1.1, the conclusions of Corollary 1.2 holds.
Moreover, if 0 < T < π ||B 0 || or i L 0 (B 0 ) + ν L 0 (B 0 ) = 0, we have τ ∈ {T, T 2 }. Proof of Theorem 1.2. This is the case n = 1 and B 0 = 0 of Theorem 1.3, by the proof Theorem 1.3, for any T > 0 we obtain an T-periodic brake solution x T satisfies If it's minimal period is τ = T /k for some k ∈ N, we denote x τ = x T | [0,τ ] . Then by the proof of Theorem 1.3 we have In the following we prove Theorem 1.2 in 2 steps.
Firstly by the proof of Theorem 1.3 we have We divide the argument into three cases.
In fact, apply Bott-type iteration formula of Theorem 2.1 to the the case of the iteration time equals to 4 and note that by Corollary 3.1 i √ −1 (γ xτ ) ≥ 0. Then by the same argument of Step 1, we can prove that p = 0.
Thus by Steps 1 and 2, Theorem 1.2 is proved.
A natural question is that can we prove the minimal period is T in this way? We have the following remark. For n = 1 and T = 4π, we can not exclude the following case: It is easy to check that γ x T (t) = R(t) for t ∈ [0, 2π]. Hence by Lemma 5.1 of [30] or the proof of Lemma 3.1 of [42] we have In this case the minimal period of x T is T 2 . Similarly for n > 1 we can construct examples to support this remark. In this section we study the minimal period problem for symmeytric brake orbit solutions of the even reversible Hamiltonian system (1.1) and complete the proof of Theorems 1.4-1.5 and Corollary N x(t) a.e. t ∈ R} with the usual W 1/2,2 norm and inner product. CorrespondinglyÊ andẼ are defined to be the symmetric ones and the For z ∈ E T , we define For z ∈Ê, we definef (z) = 1 2 Â z, z − Theorem 1.5. Since the proof of existence of T -periodic symmetric brake orbit solution x T of (1.1) is similar to that of the proof of Theorem 1.3, we will only give the sketch. We divide the proof into several steps.
Step 1. Similarly as Step 1 in the proof of Theorem 1.3, for any K > 0 we can truncate the func-tionĤ suitably and evenly toĤ K such that it satisfies the growth condition (4.4). Correspondingly we obtain a new even and reversible function H K satisfies condition (4.4). Setf Thenf K ∈ C 2 (Ê, R) and whereB 0 is the selfadjoint linear compact operator onÊ defined by Step 2. For m > 0, letf Km =f |Ê m , whereÊ m =P mÊ . Set By the same argument of Step 2 in the proof of Theorem 1.3, we can show thatf K m satisfies the hypotheses of Theorem 4.1. Moreover, we obtain a critical point x Km off Km with critical value and where δ is a positive number depending on K and r 1 > 0 is independent of K and m.
Step 3. We prove that there exists a symmetric T -periodic brake orbit solution x T of (1.1) which satisfies From the proof of Theorem 1.3 we have f K satisfies (P S) * c condition for c ∈ R, by the same proof of Lemma 5.1, we havef K satisfies (P S) * c condition for c ∈ R, i.e., any sequence z m such that z m ∈Ê m ,f ′ Km (z m ) → 0 andf Km (z m ) → c possesses a convergent subsequence inÊ. Hence in the sense of subsequence we have By similar argument as in [44], x K is a classical nonconstant symmetric T -periodic solution oḟ Set B K (t) = H ′′ K (x K (t)), Then B K ∈ C(S T /2 , L s (R 2n )). LetB K be the operator defined by the same way of the definition ofB 0 . It is easy to show that So we have Step 4. Finish the proof of Theorem 1.5.
Since x T obtained in Step 3 is a nonconstant and symmetric T -period brake orbit solution, its minimal period τ = T 4r+s for some nonnegative integer r and s = 1 or s = 3. We now estimate r. We denote by x τ = x T | [0,τ ] , then it is a symmetric period solution of (1.1) with the minimal τ and X T = x 4r+s τ being the 4r + s times iteration of x τ . As in Section 1, let γ x T and γ xτ the symplectic path associated to (τ, x) and (T, x T ) respectively. Then γ xτ ∈ C([0, τ 4 ], Sp(2n)) and γ x T ∈ C([0, T 4 ], Sp(2n)). Also we have γ x T = γ 4r+s xτ , which is the 4r + s times iteration of γ xτ . By (5.21) we have Since x τ is also a nonconstant symmetric periodic solution of (1.1). It is clear that SinceĤ satisfies condition (H5) and B 0 is semipositive, by Corollary 3.1 of [51] (also by Theorem By Corollary 3.2 of [51] (cf. aslo [29]), we have It is easy to see that So by Theorem 6.1 of Bott-type iteration formula we have If r ≥ 1, then by Theorems 2.2 and 6.2 and (5.27) we have where ω 2r = e π √ −1/(2r) as defined in Theorem 2.2.
By Theorem 3.2, we have is even, the equality in (5.31) can not hold. Otherwise, r ≥ 1 and the equality in (5.28) holds i.e., By the definition of ω 2r , we have ω 2j−1 2r = −1 for j = 1, 2, ..., r. So by (5.33) and 2 of Theorem 6.2, we have I 2p ⋄ N 1 (1, −1) ⋄q ⋄ K ∈ Ω 0 (γ 4 xτ (τ )) for some non-negative integers p and q satisfying 0 ≤ p + q ≤ n and K ∈ Sp(2(n − p − q)) with σ(K) ∈ U \ {1} satisfying the condition that all But by (5.26), Lemma 6.1, and Theorem 6.1, we have Then i −1 (γ 4r xτ ) is an even integer, which yields a contradiction to (5.36). So Claim 3 holds, and we have Also a natural question is that can we prove the minimal period is T in this way? We have the following remark. It is easy to check that γ x T (t) = R(t) for t ∈ [0, 3π]. Hence by Theorem 2.1 and Lemma 5.1 of [30] or the proof of Lemma 3.1 of [42] we have In this case the minimal period of x T is T 3 . Similarly for n > 1 we can construct examples to support this remark.
6 Appendix on Maslov-type indices (i ω , ν ω ) We first recall briefly the Maslov-type index theory of (i ω , ν ω ). All the details can be found in [41].