Minimal period problems for brake orbits of nonlinear autonomous reversible semipositive Hamiltonian systems
aa r X i v : . [ m a t h . D S ] O c t Minimal period problems for brake orbits of nonlinear autonomousreversible semipositive Hamiltonian systems
Duanzhi Zhang ∗ School of Mathematical Sciences and LPMC, Nankai UniversityTianjin 300071, People’s Republic of China
Abstract
In this paper, for any positive integer n , we study the Maslov-type index theory of i L , i L and i L √− with L = { } × R n ⊂ R n and L = R n × { } ⊂ R n . As applications we studythe minimal period problems for brake orbits of nonlinear autonomous reversible Hamiltoniansystems. For first order nonlinear autonomous reversible Hamiltonian systems in R n , which aresemipositive, and superquadratic at zero and infinity we prove that for any T >
0, the consideredHamiltonian systems possesses a nonconstant T periodic brake orbit X T with minimal period noless than T n +2 . Furthermore if R T H ′′ ( x T ( t )) dt is positive definite, then the minimal period of x T belongs to { T, T } . Moreover, if the Hamiltonian system is even, we prove that for any T > { T, T } . MSC(2000):
Key words: symmetric, brake orbit, semipositive and reversible, Maslov-type index, minimalperiod, Hamiltonian systems.
In this paper, let J = − I n I n and N = − I n I n , where I n is the identity in R n and n ∈ N . We suppose the following condition ∗ Partially supported by National Science Foundation of China (10801078, 11171314), LPMC of Nankai University.E-mail: [email protected] H ∈ C ( R n , R ) and satisfies the following reversible condition H ( N x ) = H ( x ) , ∀ x ∈ R n . We consider the following problem:˙ x = J H ′ ( x ) , x ∈ R n , (1.1) x ( − t ) = N x ( t ) , x ( T + t ) = x ( t ) , ∀ t ∈ R . (1.2)A solution ( T, x ) of (1.1)-(1.2) is a special periodic solution of the Hamiltonian system (1.1). Wecall it a brake orbit and T the period of x . Moreover, if x ( R ) = − x ( R ), we call it a symmetric brakeorbit . It is easy to check that if τ is the minimal period of x , there must holds x ( t + τ ) = − x ( t )for all t ∈ R .Since 1948, when H. Seifert in [47] proposed his famous conjecture of the existence of n geometri-cally different brake orbits in the potential well in R n under certain conditions, many people beganto study this conjecture and related problems. Let ˜ O (Ω) and ˜ J b (Σ) the number of geometricallydistinct 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.Hayashi in [27], H. Gluck and W. Ziller in [25], and V. Benci in [5] in 1983-1984 proved ˜ O (Ω) ≥ V is C , ¯Ω = { V ≤ h } is compact, and V ′ ( q ) = 0 for all q ∈ ∂ Ω. In 1987, P. Rabinowitz in[45] proved that if H is C and satisfies the reversible conditon, Σ ≡ H − ( h ) is star-shaped, and x · H ′ ( x ) = 0 for all x ∈ Σ, then ˜ J b (Σ) ≥
1. In 1987, V. Benci and F. Giannoni gave a differentproof of the existence of one brake orbit in [6].In 1989, A. Szulkin in [49] proved that ˜ J b ( H − ( h )) ≥ n , if H satisfies conditions in [43] ofRabinowitz and the energy hypersurface H − ( h ) is √ ˜ O (Ω) ≥ n under different pinchingconditions.In [42] of 2006, Long , Zhu and the author of this paper proved that there exist at least 2geometrically distinct brake orbits on any central symmetric strictly convex hypersuface Σ in R n 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 hypersufaceΣ in R n for n ≥
2, if all brake orbits on Σ are nondegenerate then there are at least n geometricallydistinct 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 thevariational method. Suppose H satisfies the following conditions:2H1 ′ ) H ∈ C ( R n , R ).(H2) There exist constants µ > r > < µH ( x ) ≤ H ′ ( x ) · x, ∀| x | ≤ r . (H3) H ( x ) = o ( | x | ) at x = 0.(H4) H ( x ) ≥ x ∈ R n . Then for any
T > , 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 prescribedminimal period. Since 1978, this conjecture has been deeply studied by many mathematicians. Asignificant progress was made by Ekeland and Hofer in their celebrated paper [16] of 1985, wherethey proved Rabinowitz’s conjecture for the strictly convex Hamiltonian system. For Hamiltoniansystems 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 semipositivefirst order Hamiltonian system, in [18] G. Fei, S.-T. Kim, and T. Wang proved the existence of aT periodic solution of system (1.1) with minimal period no less than
T / n for any given T > { T, T / } for any given T > L ( R n ) and L s ( R n ) the set of all real 2 n × n matrices and symmetricmatrices respectively. And we denote by y · y the usual inner product for all y , y ∈ R k with k being any positive integer. Also we denote by N and Z the set of positive integers and integersrespectively.Let Sp(2 n ) = { M ∈ L ( R n ) | M T J M = J } be the 2 n × n real symplectic group. For any τ > P τ = { γ ∈ C ([0 , τ ] , Sp(2 n )) | γ (0) = I n } and S τ = R / ( τ Z ).For any γ ∈ P τ and ω ∈ U , where U is the unit circle of the complex plane C , the Maslov-typeindex ( i ω ( γ ) , ν ω ( γ )) ∈ Z × { , , ... n } was defined by Long in [40]. We have a brief review inAppendix of Section 6.For convenience to introduce our results, we define the following ( B1 ) condition, since theHamiltonian systems considered here are reversible, this condition is natural. (B1) Condition . For any τ > B ∈ C ([0 , τ ] , L s ( R n ) with the n × n matrix square blockform B ( t ) = B ( t ) B ( t ) B ( t ) B ( t ) satisfying B (0) = B (0) = 0 = B ( τ ) = B ( τ ), We will call B satisfies the condition (B1) .Throughout this paper, we denote by L = { } × R n ⊂ R n , L = R n × { } ⊂ R n . (1.3)The definitions of Maslov-type indices ( i L √− ( γ ) , ν L √− ( γ )) and ( i L j ( γ ) , ν L j ( γ )) ∈ Z × { , , ..., n } for j = 0 , γ ∈ P τ (2 n ) with τ > B ∈ C ([0 , τ ] , L s ( R n ) satisfies condition (B1), the definitions of ( i L √− ( B ) , ν L √− ( B )) and ( i L j ( B ) , ν L j ( B )) ∈ Z × { , , ..., n } for j = 0 , γ ∈ P τ (2 n ) can be found in Section 2 and references therein.For any B ∈ C ([0 , τ ] , L s ( R n )), denote by γ B the fundamental solution of the following problem:˙ γ B ( t ) = J B ( t ) γ B ( t ) , (1.4) γ B (0) = I n . (1.5)Then γ B ∈ P τ . We call γ B the symplectic path associated to B . Definition 1.1. If H ∈ C ( R n , R ) is a reversible function, for any x τ be a τ -periodic brake orbitsolution of (1.1), let B ( t ) = H ′′ ( x ( t )), we define γ x τ = γ B | [0 , τ ] and call it the symplectic pathassociated to x τ . We define i L ( x τ ) = i L ( γ x τ ) , ν L ( x τ ) = i L ( γ x τ ) . (1.6)4oreover, if H is even and x τ is a τ -periodic symmetric brake orbit solution of (1.1), let B ( t ) = H ′′ ( x ( t )), we define γ x τ = γ B | [0 , τ ] and call it the symplectic path associated to x τ . We define i L √− ( x τ ) = i L √− ( γ x τ ) , ν L √− ( x τ ) = i L √− ( γ x τ ) . (1.7) Definition 1.2.
For any τ -period and k ∈ N ≡ { , , ... } , we define the k times iteration x k of x by x k ( t ) = x ( t − jτ ) , jτ ≤ t ≤ ( j + 1) τ, ≤ j ≤ k. (1.8)As in [35], for any γ ∈ P τ and k ∈ N ≡ { , , ... } , in this paper the k -time iteration γ k of γ ∈ P τ (2 n )in brake orbit boundary sense is defined by ˜ γ | [0 ,kτ ] with˜ γ ( t ) = γ ( t − jτ )( N γ ( τ ) − N γ ( τ )) j , t ∈ [2 jτ, (2 j + 1) τ ] , j = 0 , , , ...N γ (2 jτ + 2 τ − t ) N ( N γ ( τ ) − N γ ( τ )) j +1 t ∈ [(2 j + 1) τ, (2 j + 2) τ ] , j = 0 , , , ... The followings are our main results of this paper.
Theorem 1.1.
Suppose that H satisfies conditions (H1)-(H4) and(H5) H ′′ ( x ) is semipositive definite for all x ∈ R n .Then for any T > , the system (1.1)-(1.2) possesses a nonconstant T periodic brake orbitsolution x T with minimal period no less that T n +2 . Moreover, for x = ( x , x ) with x , x ∈ R n ,denote by H ′′ ( x ) the second order differential of H with respect to x , if Z T H ′′ ( x T ( t )) dt > , (1.9) then the minimal period of x T belongs to { T, T } . Remark 1.1. (Theorem 1.1 of [32])
Suppose that H satisfies conditions (H1)-(H4) and if x T satisfies(H5 ′ ) R T H ′′ ( X T ( t )) dt > .Then the minimal period of x T belongs to { T, T } . 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).Then for any T > , the system (1.1)-(1.2) possesses a nonconstant T periodic brake orbitsolution with minimal period belong to { T, T } . Consider the minimal period problem for H ( x ) = B x · x + ˆ H ( x ), where B ∈ L s ( R n ). Thisis motivated by [18], [22], and [43], where in [18] B was considered to be semipositive, in [22] and[43] B was considered to be positive.We have the following general result. 5 heorem 1.3. Let n × n be real semipositive matrix B = diag( B , B ) with B and B being n × n matrix. Assume H ( x ) = B x · x + ˆ H ( x ) for all x ∈ R n , and ˆ H satisfies conditions(H1)-(H5).Then for any T > , (1.1) possesses a nonconstant T -periodic brake orbit x T with minimalperiod no less than T i L ( B )+2 ν L ( B )+2 n +2 , where we see B as an element in C ([0 , T / , L s ( R n )) satisfies condition (B1). Remark 1.2.
In section 3, we will show i L ( B ) + ν L ( B ) ≥ Corollary 1.1.
For
T > such that i L ( B ) + ν L ( B ) = 0 , where we see B as an elementin C ([0 , T / , L s ( R n ) satisfies condition (B1), under the same assumptions of Theorem 1.2, thesystem (1.1) possesses a nonconstant T -periodic brake orbit with minimal period no less that T n +2 . We can also prove the following Corollary 1.2 of Theorem 1.3.
Corollary 1.2. If B = 0 , then for < T < π || B || with || B || being the operator norm of B ,under the same condition of Theorem 1.2, possesses a nonconstant T -periodic brake orbit x T withminimal period no less than T n +2 . Moreover , if Z T H ′′ ( x T ( t )) dt > , then the minimal period of x T belongs to { T, T } . Theorem 1.4.
Suppose that H satisfies conditions (H1)-(H5) and(H6) H ( − x ) = H ( x ) for all x ∈ R n .Then for any T > , the system (1.1)-(1.2) possesses a nonconstant symmetric brake orbit withminimal period belonging to { T, T / } . Theorem 1.5.
Let n × n be real semipositive matrix B = diag( B , B ) with B and B being n × n matrix, assume H ( x ) = B x · x + ˆ H ( x ) for all x ∈ R n , and ˆ H satisfies conditions(H1)-(H6). Then for any T > , the system (1.1)-(1.2) possesses a nonconstant symmetric brakeorbit x T with minimal period no less than T i L √− ( B )+ ν L √− ( B ))+7 . Moreover, if i L √− ( B )+ ν L √− ( B ) is even, then the minimal period of x T is no less than T i L √− ( B )+ ν L √− ( B ))+3 where we see B as anelement in C ([0 , T / , L s ( R n ) satisfies condition (B1). Remark 1.3.
In section 3, we will show that i L √− ( B ) ≥
0, hence i L √− ( B ) + ν L √− ( B ) ≥ Corollary 1.3.
For
T > such that i L √− ( B ) + ν L √− ( B ) = 0 , under the same assumptions ofTheorem 1.4, the system (1.1) possesses a nonconstant symmetric brake orbit with minimal periodbelonging to { T, T / } .
6e can also prove the following Corollary 1.4 of Theorem 1.5.
Corollary 1.4. If B = 0 , then for < T < π || B || with || B || being the operator norm of B , underthe same condition of Theorem 1.5, the system (1.1) possesses a nonconstant symmetric brake orbitwith minimal period belonging to { T, T / } . This paper is organized as follows. In section 2, we study the Maslov-type index theory of i L , i L and i L √− . We compute the difference between i L ( γ ) and i L ( γ ). In Section 3, we studythe relation between the Maslov-type index ( i L √− ( B ) , ν L √− ( B )) for B ∈ C ([0 , τ ] , L s ( R n ) satisfiescondition (B1) and the Morse indices of the corresponding Galerkin approximation. As applicationswe get some monotonicity properties of i L ( B ), i L ( B ) and i L √− ( B ) and we prove Theorem 3.2which is very important in the proof of Theorems 1.4-1.5. In Section 4, based on the preparationsin Sections 2 and 3 we prove Theorems 1.1-1.3 and Corollary 1.2. In Section 5, we prove Theorems1.4-1.5 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. ( i L j , ν L j ) with j = 0 , and ( i L √− , ν L √− ) Let F = R n ⊕ R n (2.1)possess the standard inner product. We define the symplectic structure of F by { v, w } = ( J v, w ) , ∀ v, w ∈ F, where J = ( − J ) ⊕ J = − J J . (2.2)We denote by Lag( F ) the set of Lagrangian subspaces of F , and equip it with the topology as asubspace of the Grassmannian of all 2 n -dimensional subspaces of F .It is easy to check that, for any M ∈ Sp(2 n ) its graphGr( M ) ≡ xM x | x ∈ R n is a Lagrangian subspace of F .Let V = { } × R n × { } × R n ⊂ R n , V = R n × { } × R n × { } ⊂ R n . (2.3)7y Proposition 6.1 of [35] and Lemma 2.8 and Definition 2.5 of [42], we give the followingdefinition. Definition 2.1.
For any continuous path γ ∈ P τ (2 n ), we define the following Maslov-type indices: i L ( γ ) = µ CLMF ( V , Gr( γ ) , [0 , τ ]) − n, (2.4) i L ( γ ) = µ CLMF ( V , Gr( γ ) , [0 , τ ]) − n, (2.5) ν L j ( γ ) = dim( γ ( τ ) L j ∩ L j ) , j = 0 , , (2.6)where we denote by i CLMF ( V, W, [ a, b ]) the Maslov index for Lagrangian subspace path pair ( V, W )in F on [ a, b ] defined by Cappell, Lee, and Miller in [11].For ω = e √− θ with θ ∈ R , we define a Hilbert space E ω = E ωL consisting of those x ( t ) in L ([0 , τ ] , C n ) such that e − θtJ x ( t ) has Fourier expending e − θtτ J x ( t ) = X j ∈ Z e jπtτ J a j , a j ∈ C n with k x k := X j ∈ Z τ (1 + | j | ) | a j | < ∞ . For ω = e √− θ , θ ∈ (0 , π ), we define two self-adjoint operators A ω , B ω ∈ L ( E ω ) by( A ω x, y ) = Z h− J ˙ x ( t ) , y ( t ) i dt, ( B ω x, y ) = Z h B ( t ) x ( t ) , y ( t ) i dt on E ω . Then B ω is also compact. Definition 2.2.
We define the index function i L ω ( B ) = I ( A ω , A ω − B ω ) ≡ − sf { A ω − sB ω , ≤ s ≤ } ,ν L ω ( B ) = m ( A ω − B ω ) , ∀ ω = e √− θ , θ ∈ (0 , π ) , where the definition of sf of spectral flow for the path of bounded self-adjoint linear operators onecan refer to [53] and references their in.By (3.21) of [35], we have i L ( B ) ≤ i L ω ( B ) ≤ i L ( B ) + n. (2.7) Lemma 2.1.
For ω = e √− θ with θ ∈ (0 , π ) , let V ω = L × ( e θJ L ) ⊂ R n ≡ F . There holds i L ω ( B ) = µ CLMF ( V ω , Gr( γ B ) , [0 , τ ]) . (2.8)8 roof. Without loss of generality we can suppose the C path Gr( γ B ) of Lagrangian subspacesintersects V ω regularly (otherwise we can perturb it slightly with fixed endow points such that theyintersects regularly and the index dose not change by the homotopy invariant property µ CLMF ),where the definition of intersection form can be found in [46]. We denote by µ BF the maslov indexdefined 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 ofProposition 2.3 of [52]), we havesf { A ω sB ω , ≤ s ≤ } = µ BF (Gr( γ B ) , V ω , [0 , τ ])= µ BF (( I ⊕ e −√− θJ )Gr( γ B ) , ( I ⊕ e −√− θJ ) V ω , [0 , τ ])= µ BF (( I ⊕ e −√− θJ )Gr( γ B ) , V , [0 , τ ])= − m − ( − Γ(( I ⊕ e −√− θJ )Gr( γ B ) , V , X Definition 2.3. Let B ∈ C ([0 , τ ] , L s ( R n ) and γ B be the symplectic path associated to B . Wedefine i L ω ( γ B ) = i L ω ( B ) , (2.10) ν L ω ( γ B ) = ν L ω ( B ) . (2.11)By Lemma 2.1, in general we give the following definition. Definition 2.4. For any γ ∈ P τ (2 n ) and ω = e √− θ with θ ∈ (0 , π ), we define i L ω ( γ ) = µ CLMF ( V ω , Gr( γ B ) , [0 , τ ]) , L ω ( γ ) = dim (cid:16) γ ( τ ) L ∩ e √− θJ L (cid:17) . (2.12)For any γ ∈ P τ (2 n ), we define a new symplectic path ˜ γ ∈ P τ (2 n ) by˜ γ ( t ) = I n , t ∈ [0 , τ ] ,γ (3 t − τ ) , t ∈ [ τ , τ ] ,γ ( τ ) , t ∈ [ τ , τ ] . (2.13)So we can perturb ˜ γ slightly to a C path ˆ γ such that ˆ γ is homotopic to ˜ γ with fixed end pointsand ˆ γ ( t ) = I n for t ∈ [0 , τ ] and ˆ γ ( t ) = γ ( τ ) for t ∈ [ τ , τ ]. Set ˆ B ( t ) = − J ˙ˆ γ ( t )(ˆ γ ( t )) − . So we haveˆ B (0) = ˆ B ( τ ) = 0 . (2.14)Then this ˆ B ∈ C ([0 , τ ] , L s ( R n ) and satisfies condition (B1). Also we have ˆ γ is is homotopic to γ with fixed end points. So we have i (ˆ γ k ) = i ( γ k ) = i ( γ k ˆ B ) , ∀ k ∈ N , (2.15) ν (ˆ γ k ) = ν ( γ k ) = ν ( γ k ˆ B ) , ∀ k ∈ N (2.16)and i L (ˆ γ k ) = i L ( γ k ) = i L ( γ k ˆ B ) , ∀ k ∈ N , (2.17) ν L (ˆ γ k ) = ν L ( γ k ) = ν L ( γ k ˆ B ) , ∀ k ∈ N . (2.18)Also by the property of index µ CLMF and Definition 2.4 have i L √− ( γ k ) = i L √− (ˆ γ k ) = i L √− ( γ k ˆ B ) , ∀ k ∈ N ,ν L √− ( γ k ) = ν L √− (ˆ γ k ) = ν L √− ( γ k ˆ B ) , ∀ k ∈ N . Hence, in [35] the authors essentially proved the following Bott-type iteration formula. Theorem 2.1. (Theorem 4.1 of [35]) Let γ ∈ P τ (2 n ) and ω k = e π √− /k . For odd k we have i L ( γ k ) = i L ( γ ) + ( k − / X i =1 i ω ik ( γ ) ,ν L ( γ k ) = ν L ( γ ) + ( k − / X i =1 ν ω ik ( γ ) , and for even k , we have i L ( γ k ) = i L ( γ ) + i L √− ( γ ) + k/ − X i =1 i ω ik ( γ ) ,ν L ( γ k ) = ν L ( γ ) + ν L √− ( γ ) + k/ − X i =1 ν ω ik ( γ ) . i L ( γ ) ≤ i L √− ( γ ) ≤ i L ( γ ) + n. (2.19) ( i L √− , ν L √− ) In order to study the minimal period problem for Even reversible Hamiltonian systems, we needthe iteration formula of the Maslov-type index of ( i L √− , ν L √− ) for symplectic paths starting withidentity. We use Theorem 2.1 to obtain it.Precisely we have the following Theorem. Theorem 2.2. Let γ ∈ P τ (2 n ) and ω k = e π √− /k . For odd k we have i L √− ( γ k ) = i L √− ( γ ) + ( k − / X i =1 i ω i − k ( γ ) , (2.20) ν L √− ( γ k ) = ν L √− ( γ ) + ( k − / X i =1 ν ω i − k ( γ ) , (2.21) and for even k , we have i L √− ( γ k ) = k/ X i =1 i ω i − k ( γ ) , (2.22) ν L √− ( γ k ) = k/ X i =1 ν ω i − k ( γ ) . (2.23) Proof. For odd k , since γ k = ( γ k ) , by Theorem 2.1 we have i L ( γ k ) = i L ( γ k ) + i L √− ( γ k ) , (2.24) ν L ( γ k ) = ν L ( γ k ) + ν L √− ( γ k ) . (2.25)Also by Theorem 2.1 we have i L ( γ k ) = i L ( γ ) + ( k − / X i =1 i ω ik ( γ ) , (2.26) ν L ( γ k ) = ν L ( γ ) + ( k − / X i =1 ν ω ik ( γ ) , (2.27) i L ( γ k ) = i L ( γ ) + i L √− ( γ ) + k − X i =1 i ω i k ( γ ) , (2.28) ν L ( γ k ) = ν L ( γ ) + ν L √− ( γ ) + k − X i =1 ν ω i k ( γ ) . (2.29)11ince ω k = ω k , by (2.24), (2.28) minus (2.26) yields (2.20). By (2.25), (2.29) minus (2.27)yields (2.21).For even k, by similar argument we obtain (2.22) and (2.23). The proof of Theorem 2.2 iscomplete. i L ( γ ) and i L ( γ ) . The precise difference of i L ( γ ) and i L ( γ ) for γ ∈ P τ with τ > | i L ( γ ) − i L ( γ ) | ≤ n .For any P ∈ Sp(2 n ) and ε ∈ R , we set M ε ( P ) = P T sin 2 εI n − cos 2 εI n − cos 2 εI n − sin 2 εI n P + sin 2 εI n cos 2 εI n cos 2 εI n − sin 2 εI n . (2.30)Then we have the following theorem. Theorem 2.3. For γ ∈ P τ with τ > , we have i L ( γ ) − i L ( γ ) = 12 sgn M ε ( γ ( τ )) , (2.31) where sgn M ε ( γ ( τ )) is the signature of the symmetric matrix M ε ( γ ( τ )) and ε > is sufficientlysmall.we also have, ( i L ( γ ) + ν L ( γ )) − ( i L ( γ ) + ν L ( γ )) = 12 sign M ε ( γ ( τ )) , (2.32) where ε < and | ε | is sufficiently small. Proof. By the first geometrical definition of the Maslov-type index in Section 4 of [11], thereexists an ε > V ∩ e − ε J Gr( γ (0)) = { } , V ∩ e − ε J Gr( γ ( τ )) = { } . (2.33)By definition 2.1, we have i L ( γ ) = µ CLMF ( V , e − ε J Gr( γ ) , [0 , τ ]) − n, (2.34) i L ( γ ) = µ CLMF ( V , e − ε J Gr( γ ) , [0 , τ ]) − n. (2.35)Define γ ( t ) = e − ε J Gr( γ ( t )) and γ ( t ) = e − ε J Gr( γ ( τ − t )) for t ∈ [0 , τ ]. Then γ and γ are twopaths of Lagrangian subspaces of the symplectic space ( F, J ) defined in (2.1) and (2.2). γ connects12 − ε J Gr( γ (0)) and e − ε J Gr( γ ( τ )) and is transversal to V and V . γ connects e − ε J Gr( γ ( τ )) and e − ε J Gr( γ (0)) and is transversal to V and V . Denote by γ the catenation of the paths γ and γ .By Definition 3.4.2 of the H ¨ ormande index s ( M , M ; L , L ) on p. 66 of [14] and (2.34)-(2.35),we have s ( V , V ; e − ε J Gr( γ (0)) , e − ε J Gr( γ ( τ )))= h γ, α i = µ CLMF ( V , γ ) + µ CLMF ( V , γ ) (2.36)= µ CLMF ( V , e − ε J Gr( γ )) − µ CLMF ( V , e − ε J Gr( γ )) (2.37)= i L ( γ ) − i L ( γ ) , (2.38)where α is the Maslov-Arnold index defined in Theorem 3.4.9 on p. 64 of [14]. Since γ and γ aretransversal to V and V (2.36) holds, (2.37) holds from the definition of γ and γ .In the proof of Theorem 3.3 of [42], we have proved that for ε > V , e − ε J Gr( I n ); V ) = 0 , (2.39)where sgn( W , W ; W ) for 3 Lagrangian spaces with W transverses to W and W is introducedin Definition 3.2.3 on p. 67 of [14]. Note that by Claim 1 below, we can prove (2.39) at once. Claim 1. For ε > 0, small enough, there holdssign( V , e − ε J Gr( γ ( τ )); V ) = sgn( M ε ( γ ( τ ))) . (2.40) Proof of Claim 1. In fact, e −J Gr( γ ( τ )) = e εJ e − εJ γ ( τ ) pqpq = cp − sqsp + cq ( c, s ) γ ( τ )( p, q ) T ( − s, c ) γ ( τ )( p, q ) T ; p, q ∈ R n , (2.41)where we denote by c = cos εI n and s = sin εI n . Hence the transformation A : V e −J Gr( I n )satisfies A (0 , − sp − cq, , − ( − s, c ) γ ( τ )( p, q ) T )= ( cp − sq, sp + cq, ( c, s ) γ ( τ )( p, q ) T , ( − s, c ) γ ( τ )( p, q ) T ) , ∀ p, q ∈ R n , (2.42)13here A is introduced in Definition 3.4.3 of sign( M , M ; L ) on p. 67 of [14]. For the convenienceof our computation, we rewrite (2.42) as follows. A − s c pq , − − s c γ ( τ ) pq = c − ss c pq , c s − s c γ ( τ ) pq . (2.43)Then for p , p , q , q ∈ R n , the symmetric bilinear form Q ( V ) : ( x, y ) 7→ J ( Ax, y ) on V defined in Definition 3.4.3 on p. 67 of [14] satisfies: Q ( V ) − s c pq , − − s c γ ( τ ) pq = * (( − J ) ⊕ J ) c − ss c pq , c s − s c γ ( τ ) pq , − s c pq , − − s c γ ( τ ) pq + . = * s c J c − ss c − γ ( τ ) T − s c J c s − s c γ ( τ ) pq , pq + . = * sc − s c − sc + γ ( τ ) T sc s − c − sc γ ( τ ) pq , pq + . (2.44)Let ˜ M ε ( γ ( τ )) = sc − s c − sc + γ ( τ ) T sc s − c − sc γ ( τ ). Then by definition of the sym-metric bilinear form Q ( V ), ˜ M ε ( γ ( τ ) is an invertible symmetric 2 n × n matrix. We define M ε ( γ ( τ )) = 2 ˜ M ε ( γ ( τ )) = ˜ M ε ( γ ( τ )) + ˜ M Tε ( γ ( τ )) . (2.45)Then we have M ε ( γ ( τ )) = γ ( τ ) T sin 2 εI n − cos 2 εI n − cos 2 εI n − sin 2 εI n γ ( τ ) + sin 2 εI n cos 2 εI n cos 2 εI n − sin 2 εI n . (2.46)It is clear that sgn Q ( V ) = sgn ˜ M ε ( γ ( τ )) = sgn M ε ( γ ( τ )) . (2.47)By the definition of sgn( V , e − ε J Gr( γ ( τ )); V ), we havesgn( V , e − ε J Gr( γ ( τ )); V ) = sgn Q ( V ) . (2.48)14hen (2.40) holds from (2.47) and (2.48), and the proof of Claim 1 is complete.Thus by (2.38), (2.39) and Claim 1, we have i L ( γ ) − i L ( γ )= s ( V , V ; e − ε J Gr( γ (0)) , e − ε J Gr( γ ( τ )))= 12 sgn( V , e − ε J Gr( γ ( τ )); V ) − 12 sgn( V , e − ε J Gr( γ (0)); V )= 12 sgn( V , e − ε J Gr( γ ( τ )); V ) − 12 sgn( V , e − ε J Gr( I n ); V )= 12 sgn( V , e − ε J Gr( γ ( τ )); V )= 12 sgn M ε ( γ ( τ )) . Here in the second equality, we have used Theorem 3.4.12 of on p. 68 of [14]. Thus (2.31) holds.Choose ε < | ε | is sufficiently small, by the discussion of µ CLMF index we have i L ( γ ) = µ CLMF ( V , e − ε J Gr( γ ) , [0 , τ ]) − ν L ( γ ) , (2.49) i L ( γ ) = µ CLMF ( V , e − ε J Gr( γ ) , [0 , τ ]) − ν L ( γ ) . (2.50)Then by the same proof as above, we have i L ( γ ) + ν L ( γ ) − i L ( γ ) − ν L ( γ ) = 12 sgn M ε ( γ ( τ )) , (2.51)where ε < Corollary 2.1. (Theorem 2.3 of [35]) For γ ∈ P τ (2 n ) with τ > , there hold | i L ( γ )) − i L ( γ )) | ≤ n, | i L ( γ ) + ν L ( γ ) − i L ( γ ) − ν L ( γ ) | ≤ n. (2.52) Moreover if γ (1) is a orthogonal matrix then there holds i L ( γ ) = i L ( γ ) . (2.53) Proof. (2.52) holds directly from Theorem 2.3, so we only need to prove (2.53). Since γ ( τ ) isan orthogonal and symplectic matrix, we have γ T ( τ ) J γ ( τ ) = J, γ T ( τ ) γ ( τ ) = I n . (2.54)So we have γ ( τ ) J = J γ ( τ ) , γ ( τ ) T J = J γ ( τ ) T . (2.55)15t is easy to check that for any ε ∈ R , there holds J sin 2 εI n ± cos 2 εI n ± cos 2 εI n − sin 2 εI n J = sin 2 εI n ± cos 2 εI n ± cos 2 εI n − sin 2 εI n . (2.56)Hence by (2.55) and (2.56), we have J M ε ( γ ( τ )) J = J γ ( τ ) T sin 2 εI n − cos 2 εI n − cos 2 εI n − sin 2 εI n γ ( τ ) + sin 2 εI n cos 2 εI n cos 2 εI n − sin 2 εI n J = J γ ( τ ) T sin 2 εI n − cos 2 εI n − cos 2 εI n − sin 2 εI n γ ( τ ) J + J sin 2 εI n cos 2 εI n cos 2 εI n − sin 2 εI n J = γ ( τ ) T J sin 2 εI n − cos 2 εI n − cos 2 εI n − sin 2 εI n J γ ( τ ) + J sin 2 εI n cos 2 εI n cos 2 εI n − sin 2 εI n J = γ ( τ ) T sin 2 εI n − cos 2 εI n − cos 2 εI n − sin 2 εI n γ ( τ ) + sin 2 εI n cos 2 εI n cos 2 εI n − sin 2 εI n = M ε ( γ ( τ )) . (2.57)So we have M ε ( γ ( τ )) J = − J M ε ( γ ( τ )) . (2.58)Thus for any x ∈ R n and λ ∈ R satisfying M ε ( γ ( τ )) x = λx. (2.59)By (2.58) we have M ε ( γ ( τ ))( J x ) = − J M ε ( γ ( τ )) x = − λ ( J x ) . (2.60)Since for ε > M ε ( γ ( τ )) is an invertible symmetric matrix, by (2.60) we have m + ( M ε ( γ ( τ ))) = m − ( M ε ( γ ( τ ))) = n (2.61)which yields sgn M ε ( γ ( τ )) = m + ( M ε ( γ ( τ ))) − m − ( M ε ( γ ( τ ))) = 0 . (2.62)Then (2.53) holds from Theorem 2.3. Lemma 2.2. For a symplectic path P : [0 , τ ] → Sp(2 n ) with τ > , if for j = 0 , there holds ν L j ( P ( t )) = constant for all t ∈ [0 , τ ] , then for ε > small enough we have sgn M ε ( P (0)) = sgn M ε ( P ( τ )) . (2.63)16 roof. Since Sp(2 n ) is path connected, we can choose a path γ ∈ P τ with γ ( τ ) = P (0). ByProposition 2.11 of [42] and the definition of µ j for j = 1 , µ CLMF ( V j , Gr( P ) , [0 , τ ]) = 0 , j = 0 , . (2.64)So by the Path Additivity and Reparametrization Invariance properties of µ CLMF in [11], we have i L j ( P ∗ γ ) = µ CLMF ( V j , Gr( P ∗ γ ) , [0 , τ ]) − n = µ CLMF ( V j , Gr( γ ) , [0 , τ ]) + µ CLMF ( V j , Gr( P ) , [0 , τ ]) − n = µ CLMF ( V j , Gr( γ ) , [0 , τ ]) − n = i L j ( γ ) , (2.65)where the definition of joint path η ∗ ξ is given by (6.1) in Section 6 below. Then by Theorem 2.3we have i L ( γ ) − i L ( γ ) = 12 sgn( M ε ( P (0))) , (2.66) i L ( P ∗ γ ) − i L ( P ∗ γ ) = 12 sgn( M ε ( P ( τ ))) . (2.67)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 , and n j ∈ N ,we have M ε ( P ⋄ P ) = M ε ( P ) ⋄ M ε ( P ) , sgn M ε ( P ⋄ P ) = sgn M ε ( P ) + sgn M ε ( P ) . By direct computation according to Theorem 2.3 and Corollary 2.1, for γ ∈ P τ (2), b > 0, and ε > M ε ( R ( θ )) = 0 , for θ ∈ R , (2.68)sgn M ε ( P ) = 0 , if P = ± b or ± − b , (2.69)sgn M ε ( P ) = 2 , if P = ± − b , (2.70)sgn M ε ( P ) = − , if P = ± b . (2.71)17lso we give a example as follows to finish this sectionsgn M ε ( P ) = 2 , if P = ± − − . (2.72) i L , i L , i L √− and the corresponding Morse in-dices, and their monotonicity properties. In [31], Liu studied the relation between the L -index of solutions of Hamiltonian systems with L -boundary conditions and the Morse index of the corresponding functional defined via the Galerkinapproximation 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 studysome monotonicity properties of i L and i L . We also study the index i L √− ( B ) with B being acontinuous symmetric matrices path satisfying condition (B1) defined in Section 1 and the Morseindex of the corresponding functional defined via the Galerkin approximation method. Then asapplications we study some monotonicity properties of i L √− ( B ) which will be important in theproof of Theorems 1.4-1.5 in Section 5 below.For any τ > B ∈ C ([0 , τ / , L s ( R n )) (in order to apply the results in this sectionconveniently Section 5, we always assume B ∈ C ([0 , τ / , L s ( R n )) satisfying condition (B1). Weextend B to [0 , τ ] by B ( τ t ) = N B ( τ − t ) N, ∀ t ∈ [0 , τ . (3.1)Then since B ( τ ) = B (0), we can extend it τ -periodically to R , so we can see B as an element in C ( S τ/ , L s ( R n )).Let E τ = { x ∈ W / , ( S τ , R n ) | x ( − t ) = N x ( t ) a.e. t ∈ R } with the usual norm and innerproduct denoted by || · || and h·i respectively.By the Sobolev embedding theorem, for any s ∈ [1 , + ∞ ), there is a constant C s > || z || L s ≤ C s || z || , ∀ z ∈ E τ . (3.2)Note that B can also be seen as an element in C ( S τ , L s ( R n )). We define two selfadjointoperators A τ and B τ on E τ by the following bilinear forms h A τ x, y i = Z τ − J ˙ x · y dt, h B τ x, y i = Z τ B ( t ) x · y dt. (3.3)18hen A τ is a bounded operator on E τ and dim ker A τ = n , the Fredholm index of A τ is zero,and B τ is a compact operator on E τ .Set E τ ( j ) = (cid:26) z ∈ E τ (cid:12)(cid:12)(cid:12)(cid:12) z ( t ) = exp( 2 jπtτ J ) a + exp( − jπtτ J ) b, ∀ t ∈ R ; ∀ a, b ∈ L (cid:27) . and E τ,m = E τ (0) + E τ (1) + · · · + E τ ( m ) . Let Γ τ = { P τ,m : m = 0 , , , ... } 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:(1) E τ, = P τ, E τ = ker A τ , E τ,m = P τ,m E τ is finite dimension for m ≥ P τ,m → x as m → ∞ for any x ∈ E τ ;(3) P τ,m A τ = A τ P τ,m , ∀ m ≥ d > 0, we denote by M + d ( · ), M − d ( · ) and M d ( · ) the eigenspace corresponding to the eigenvalue λ belong to [ d, + ∞ ), ( −∞ , − d ] and ( − d, d ) respectively, and M + ( · ), M − ( · ) and M ( · ) the positive,negative and null subspace of of the selfadjoint operator defining it respectively. For any boundedselfadjoint linear operator on E , We denote L = ( L | ImL ) − , and we also denote by P τ,m LP τ,m =( P τ,m LP τ,m ) | E τ,m : E τ,m → E τ,m .Similarly we define two subspaces of E τ by ˆ E = { x ∈ E | x ( t + τ ) = − x ( t ) , a.e. t ∈ R } and˜ E = { x ∈ E | x ( t + τ ) = x ( t ) , a.e. t ∈ R } be the symmetric ones and τ -periodic ones of E τ respectively.We define two selfadjoint operators ˆ A and ˆ B on ˆ E by the following bilinear forms h ˆ Ax, y i = Z τ − J ˙ x · y dt, h ˆ Bx, y i = Z τ B ( t ) x ( t ) · y ( t ) dt. (3.4)Then ˆ A is a bounded Fredholm operator on ˆ E and dim ker ˆ A = 0, the Fredholm index of ˆ A iszero. ˆ B is a compact operator on ˆ E .For any positive integer m , we defineˆ E m = Σ mj =1 E τ (2 j − . For m ≥ 1, let ˆ P m be the orthogonal projection from ˆ E to ˆ E m . Then { ˆ P m } is a Galerkin approxi-mation scheme w.r.t. ˆ A . 19 heorem 3.1. For any B ( t ) ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1) and < d ≤ || ( A τ − B τ ) || − , there exists m ∗ > such that for m ≥ m ∗ there hold dim M + d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn − i L √− ( B ) − ν L √− ( B ) , (3.5)dim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn + i L √− ( B ) , (3.6)dim M d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = ν L √− ( B ) . (3.7) Proof. The method of the proof here is similar as that of Theorem 2.1 in [51].For any positive integer m , we define˜ E m = m X j =0 E τ (2 j ) . For m ≥ 1, let ˜ P m be the orthogonal projection from ˜ E to ˜ E m . Then { ˜ P m } is a Galerkin approxi-mation scheme w.r.t. ˜ A .For any y ∈ ˆ E m and z ∈ ˜ E m , it is easy to check that h ( P τ,m ( A τ − B τ ) P τ,m y, z ) i = 0 . (3.8)So we have the following P τ,m ( A τ − B τ ) P τ,m orthogonal decomposition E τ, m = ˆ E m ⊕ ˜ E m . (3.9)Similarly, we have the following A τ − B τ orthogonal decomposition E τ = ˆ E ⊕ ˜ E. (3.10)Hence, under above decomposition we have( A τ − B τ ) = ( ˆ A − ˆ B ) ⊕ ( ˜ A − ˜ B ) . (3.11)Thus || ( A τ − B τ ) || − ≤ || ( ˆ A − ˆ B ) || − (3.12) || ( A τ − B τ ) || − ≤ || ( ˜ A − ˜ B ) || − (3.13)By the definitions of M ∗ d ( · ) for P τ, m ( A τ − B τ ) P τ, m , ˆ P m ( ˆ A − ˆ B ) ˆ P m , and ˜ P m ( ˜ A − ˜ B ) ˜ P m with ∗ = + , − , 0. So for ∗ ∈ { + , − , } we havedim M ∗ d ( P τ, m ( A τ − B τ ) P τ, m ) = dim M ∗ d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) + dim M ∗ d ( ˜ P m ( ˜ A − ˜ B ) ˜ P m ) . (3.14)20ote 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 that ˜ E is the τ -periodic extending of E τ from S τ to S τ , and ˜ E m is the τ -periodic extending of E τ, m from S τ to S τ too.Thus we have || ( A τ − B τ ) || − = || ( ˜ A − ˜ B ) || − . (3.15)By (3.13) and (3.15) we have || ( A τ − B τ ) || − ≤ || ( A τ − B τ ) || − . (3.16)For ∗ ∈ { + , − , } we havedim M ∗ d ( P τ,m ( A τ − B τ ) P τ,m ) = M ∗ d ( ˜ P m ( ˜ A − ˜ B ) ˜ P m ) . (3.17)Then for 0 < d ≤ || ( A τ − B τ ) || − , by Theorem 2.1 in [31] there exists m > m ≥ m we have dim M + d ( P τ, m ( A τ − B τ ) P τ, m ) = 2 mn − i L ( γ B ) − ν L ( γ B ) , (3.18)dim M − d ( P τ, m ( A τ − B τ ) P τ, m ) = 2 mn + n + i L ( γ B ) , (3.19)dim M d ( P τ, m ( A τ − B τ ) P τ, m ) = ν L ( γ B ) . (3.20)By (3.16), we have 0 < d ≤ || ( A τ − B τ ) || − . By Theorem 2.1 in [31] again there exists m > 0, such that for m ≥ m we havedim M + d ( P τ,m ( A τ − B τ ) P τ,m ) = mn − i L ( γ B ) − ν L ( γ B )) , (3.21)dim M − d ( P τ,m ( A τ − B τ ) P τ,m ) = mn + n + i L ( γ B )) , (3.22)dim M d ( P τ,m ( A τ − B τ ) P τ,m ) = ν L ( γ B )) . (3.23)Let m ∗ = max { m , m } . Then for m ≥ m ∗ , all of (3.18)-(3.23) hold.So by (3.14), (3.17), and (3.18)-(3.23) we havedim M + d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn − ( i L ( γ B ) − i L ( γ B )) − ( ν L ( γ B ) − ν L ( γ B )) , (3.24)dim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn + i L ( γ B ) − i L ( γ B ) , (3.25)dim M d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = ν L ( γ B ) − ν L ( γ B ) . (3.26)Thus (3.5)-(3.7) hold from (3.24)-(3.26), Definition 2.3, and Theorem 2.2. The proof of Theorem3.1 is complete. 21 emark 3.1. Let any B ∈ C ([0 , τ ] , L s ( R n )) be a constant matrix path satisfying condition (B1).By Theorem 5.1 of [42], for d = 0 the same conclusions of Theorem 2.1 of [31] still holds . Hencefor d = 0 the same conclusions of Theorem 3.1 still hold, i.e., there exists m ∗ > such that for m ≥ m ∗ there hold dim M + ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn − i L √− ( B ) − ν L √− ( B ) , dim M − ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn + i L √− ( B ) , dim M ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = ν L √− ( B ) . In the following, we study some monotonicity of the the Maslov-type i L √− index. In thispaper, for any two symmetric matrices B and B , we say B > B if B − B is positive definiteand we say B ≥ B if B − B is semipositive. Similarly for two symmetric matrix paths B , B ∈ C ([0 , τ ] , L s ( R n )), we say B > B if B ( t ) − B ( t ) is positive definite for all t ∈ [0 , τ ] and wesay B ≥ B if B ( t ) − B ( t ) is semipositive definite for all t ∈ [0 , τ ]. Lemma 3.1. For any τ > and B , B ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1). If B ≥ B , then there hold i L √− ( B ) ≥ i L √− ( B ) (3.27) and i L √− ( B ) + ν L √− ( B ) ≥ i L √− ( B ) + ν L √− ( B ) . (3.28) Moreover, if Z τ ( B ( t ) − B ( t )) dt > , (3.29) then there holds i L √− ( B ) ≥ i L √− ( B ) + ν L √− ( B ) . (3.30) Proof. Let the space ˆ E and the orthogonal projection operator ˆ P m be the ones defined inSection 2. Correspondingly we define the compact operators ˆ B and ˆ B . By Theorem 3.1, for d > m ∗ > M + d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn − i L √− ( B ) − ν L √− ( B ) , (3.31)dim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn + i L √− ( B ) , (3.32)dim M d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = ν L √− ( B ) . (3.33)22nd dim M + d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn − i L √− ( B ) − ν L √− ( B ) , (3.34)dim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = mn + i L √− ( B ) , (3.35)dim M d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) = ν L √− ( B ) . (3.36)If B ≥ B , we have ˆ P m ( ˆ A − ˆ B ) ˆ P m ≤ ˆ P m ( ˆ A − ˆ B ) ˆ P m , Sodim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) ≥ dim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) . (3.37)Then by (3.32) and (3.35), (3.27) holds. Also we havedim M + d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) ≤ dim M + d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) . (3.38)Then by (3.31) and (3.34), (3.28) holds.If R τ ( B ( t ) − B ( t )) dt > 0, thenˆ P m ( ˆ A − ˆ B ) ˆ P m < ˆ P m ( ˆ A − ˆ B ) ˆ P m . (3.39)So we havedim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) ≥ dim M − d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) + M d ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) . (3.40)Then by (3.32), (3.35) and (3.36), (3.30) holds and the proof of Lemma 3.1 is complete. Corollary 3.1. For any τ > and B ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1) and B ≥ ,there holds i L √− ( B ) ≥ . (3.41) proof. By Lemma 3.1, we have i L √− ( B ) ≥ i L √− (0) . (3.42)Then the conclusion holds from the fact that i L √− (0) = i L √− ( γ ) = 0 , (3.43)Where γ is the identity symplectic path.By Theorem 2.1 of [31] and the Remark below Theorem 2.1 in [31] and the similar proof ofLemma 3.1 we have the following lemma. Lemma 3.2. If τ > and B , B ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1) and B ≥ B ,then for j = 0 , there hold i L j ( B ) ≥ i L j ( B ) (3.44)23 nd i L j ( B ) + ν L j ( B ) ≥ i L j ( B ) + ν L j ( B ) . (3.45) Moreover, if R τ ( B ( t ) − B ( t )) dt > , then there holds i L j ( B ) ≥ i L j ( B ) + ν L j ( B ) . (3.46)Since i L j (0) = − n and ν L j (0) = n for j = 0 , 1, a direct consequence of Lemma 3.2 is thefollowing Corollary 3.2. If τ > and B ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1) and B ≥ , then for j = 0 , there hold i L j ( B ) + ν L j ( B ) ≥ , i L j ( B ) ≥ − n. (3.47) Moreover if R τ B ( t ) dt > , there holds i L j ( B ) ≥ . (3.48)Moreover we can give a stronger version of Corollary 3.2, i.e., the following Lemma 3.3. Lemma 3.3. Let τ > and B ∈ C ([0 , τ ] , L s ( R n )) with the n × n matrix square block form B ( t ) = B ( t ) B ( t ) B ( t ) B ( t ) satisfying condition (B1) and B ≥ .If R τ B ( t ) dt > , there holds i L ( B ) ≥ . (3.49) If R τ B ( t ) dt > , there holds i L ( B ) ≥ . (3.50) Proof. Without loss of generality, assume λ > Z τ B ( t ) ≥ λI n . (3.51)Also we can extend B to [0 , τ ] by B ( τ t ) = N B ( τ − t ) N, ∀ t ∈ [0 , τ . (3.52)Then since B ( τ ) = B (0), we can extend it τ -periodically to R , so we can see B as an element in C ( S τ , L s ( R n )). Then we have Z τ B ( t ) ≥ λI n . (3.53)24or any m ∈ N , we define two subspaces of E as follows E − τ,m = z ∈ E τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z ( t ) = m X j =1 exp( − jπtτ J ) b j , ∀ t ∈ R ; ∀ b j ∈ L ,E τ (0) = { z ∈ E τ | z ( t ) ≡ b, b ∈ L } . Then for any z = αx + βy ∈ E τ (0) ⊕ E − τ,m with α + β = 1 and || x || = || y || = 1, we have h ( A τ − B τ ) z, z i = h ( A τ − B τ )( αx + βy ) , αx + βy i = − β h A τ y, y i − h B τ ( αx + βy ) , αx + βy i≤ −|| A τ || − β − h B τ ( αx + βy ) , αx + βy i . (3.54)Since B ≥ 0, note that x ( t ) ≡ b = (0 , b ) ∈ L for all t ∈ S τ with τ | b | = 1, we have h B τ ( αx + βy ) , αx + βy i = Z τ ( α Bx · x + β By · y + 2 αβBx · y ) dt ≥ α Z τ Bx · x dt + β Z τ By · y dt − | α || β | ( Z τ Bx · x dt ) / ( Z τ By · y dt ) / ≥ α Z τ Bx · x dt + β Z τ By · y dt − 11 + ε α Z τ Bx · x dt − (1 + ε ) β Z τ By · y dt = εα ε Z τ Bx · x dt − εβ Z τ By · y dt = εα ε (cid:18)Z τ B ( t ) dt (cid:19) b · b − εβ Z τ By · y dt = εα ε (cid:18)Z τ B ( t ) dt (cid:19) b · b − εβ Z τ By · y dt ≥ εα ε λ | b | − εβ || B τ || || y || = 2 ελα (1 + ε ) τ − εβ || B τ || (3.55)for any ε > ε = min { , || A τ || − || B τ || − } . By (3.54) and (3.55), we have h ( A τ − B τ ) z, z i ≤ −|| A τ || − β − ελα (1 + ε ) τ + εβ || B τ ||≤ − || A τ || − β − ελα τ ≤ − d ( α + β )= − d , (3.56)25here d = min { || A τ || − , ελτ } = min { || A τ || − , λτ , λ || A τ || − || B τ || − τ } . Note that d is independent of m , so for 0 < d ≤ min { d , || ( A τ − B τ ) || − } , by Theorem 2.1 of [31] there exists m ∗ > m ≥ m ∗ , we have dim M − d ( P τ,m ( A τ − B τ ) P τ,m ) = mn + n + i L ( B ) . (3.57)By (3.56) we havedim M − d ( P τ,m ( A τ − B τ ) P τ,m ) ≥ dim( E τ (0) ⊕ E − τ,m ) = mn + n. (3.58)Then by (3.57) and (3.58) we have i L ( B ) ≥ R τ B ( t ) dt > 0, by similar proof we have i L ( B ) ≥ 0. The proof of Lemma 3.3 is complete.Now we give the following Theorem 3.2 which will play a important role in the proof of our mainresults in Section 5. This results implies that the corresponding Maslov-type index of a periodicsymmetric solution of a first order even semipositive Hamilton increases with the increasing of theiteration time of the solution. Theorem 3.2. If τ > and B ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1) and B ≥ , then forany two positive integers p > q there holds i L √− ( γ pB ) ≥ i L √− ( γ qB ) . (3.59) Proof. Extend γ B ( t ) to [0 , pτ ] as γ pB , we still denote it by γ B . By definition of i L o √− and the Pathadditivity and Symplectic invariance property of µ CLMF in [11], we have i L √− ( γ pB ) − i L √− ( γ qB )= µ CLMF ( L × J L , Gr( γ B ) , [0 , pτ − µ CLMF ( L × J L , Gr( γ B ) , [0 , qτ µ CLMF ( L × J L , Gr( γ B ) , [ qτ , pτ µ CLMF ( L × L , Gr( − J γ B ) , [ qτ , pτ . (3.60)By the first geometrical definition of the index µ CLMF in section 4 of [11], there is a ε > e − ε J Gr( − J γ B ( pτ ∩ ( L × L ) = { } = ( e − εJ Gr( γ B ( qτ ∩ ( L × L ) (3.61)26nd µ CLMF ( L × L , Gr( − J γ B ) , [ qτ , pτ µ CLMF ( L × L , e − ε J Gr( − J γ B ) , [ qτ , pτ µ CLMF ( L × L , Gr( − e − εJ J γ B e − εJ ) , [ qτ , pτ , (3.62)where in the second equality we have used Symplectic invariance property of µ CLMF index in [11].Choose a C path γ ∈ P pτ such that γ ( t ) = − e − εJ J γ B e − εJ for all t ∈ [ qτ, pτ ]. Denote by D ( t ) = − J ˙ γ ( t ) γ ( t ) − for t ∈ [0 , pτ ]. For t ∈ [ qτ, pτ ], by direct computation we have D ( t ) = − J ddt ( − e − εJ J γe − εJ )( − e − εJ J γe − εJ ) − = − J e − εJ B ( t ) e εJ J. (3.63)Since B ≥ D ( t ) ≥ t ∈ [ qτ, pτ ] and D ∈ C ([0 , pτ ] , L s ( R n )). For s ≥ 0, we define D s ( t ) = D ( t ) + sI n and symplectic path γ s ( t ) by ddt γ s ( t ) = J D s ( t ) γ s ( t ) , t ∈ [0 , pτ γ s (0) = I n . It is clear that γ = γ. (3.64)By the same argument of step2 of the proof of Theorem 5.1 in [42], we have − J dds γ s ( t )( γ s ( t )) − > , for t = pτ , qτ . (3.65)By (3.61) and definition of γ s we have ν L ( γ ( pτ ν L ( γ ( qτ . (3.66)So by (3.65), there is a σ > ν L ( γ s ( pτ ν L ( γ s ( qτ , ∀ s ∈ [0 , σ ] . (3.67)So we have µ CLMF ( L × L , Gr( γ s ( pτ , s ∈ [0 , σ ]) = 0 ,µ CLMF ( L × L , Gr( γ s ( qτ , s ∈ [0 , σ ]) = 0 . (3.68)By the Homotopy invariance with respect to end points and Path additivity properties of µ CLMF index in [11], we have µ CLMF ( L × L , Gr( γ s ( pτ , s ∈ [0 , σ ]) + µ CLMF ( L × L a, Gr( γ σ ( t )) , t ∈ [ qτ , pτ µ CLMF ( L × L , Gr( γ ( t )) , t ∈ [ qτ , pτ µ CLMF ( L × L , Gr( γ s ( pτ , s ∈ [0 , σ ]) . (3.69)27o by (3.60), (3.62), (3.64),(3.68) and (3.69), we have i L √− ( γ pB ) − i L √− ( γ qB ) = µ CLMF ( L × L , Gr( γ σ ( t )) , t ∈ [ qτ , pτ . (3.70)Since D ( t ) ≥ t ∈ [ qτ , pτ ], we have D σ ( t ) > , ∀ t ∈ [ qτ , pτ . (3.71)So by the proof of Lemma 3.1 of [42] and Lemma 2.6 of [42], we have µ CLMF ( L × L , Gr( γ σ ( t )) , t ∈ [ qτ , pτ X t ∈ [ qτ , pτ ) ν L ( γ σ ( t )) ≥ . (3.72)Thus by (3.70) and (3.72), (3.59) holds. The proof of Theorem 3.1 is complete.By similar proof of Theorem 3.2 we have the following Theorem 3.3. Theorem 3.3. If τ > B ∈ C ([0 , τ ] , L s ( R n )) satisfying condition (B1) and B ≥ 0, then for j = 0 , p ≥ q there holds i L j ( γ pB ) ≥ i L j ( γ qB ) . (3.73) In this section we study the minimal period problem for brake orbits of the reversible Hamiltoniansystem (1.1) and complete the proof of Theorems 1.1-1.3 and Corollary 1.2.For T > 0, we set E = W / , ( S T , R n ) with the usual norm and inner product denoted by || · || and h·i respectively, and two subspaces of E by E T = { x ∈ W / , ( S τ , R n ) | x ( − t ) = N x ( t ) a.e. t ∈ R } and ˇ E T = { x ∈ W / , ( S τ , R n ) | x ( − t ) = − N x ( t ) a.e. t ∈ R } . Then we have E = E T ⊕ ˇ E T . (4.1)As in Section 3, we define two selfadjoint operators A T on E T by the same way as (3.3). Wealso define two selfadjoint operators ˇ A T on ˇ E T by the following bilinear form: h ˇ A T x, y i = Z T − J ˙ x · y dt. (4.2)Then A T is a bounded operator on E T and dim ker A T = n , the Fredholm index of A T is zero, andˇ A T is a bounded operator on ˇ E T and dim ker ˇ A T = n , the Fredholm index of ˇ A T is zero.28et E T ( j ) = (cid:26) z ∈ E T (cid:12)(cid:12)(cid:12)(cid:12) z ( t ) = exp( 2 jπtT J ) a + exp( − jπtT J ) b, ∀ t ∈ R ; ∀ a, b ∈ L (cid:27) ,E T,m = E T (0) + E T (1) + · · · + E T ( m )and ˇ E T ( j ) = (cid:26) z ∈ E T (cid:12)(cid:12)(cid:12)(cid:12) z ( t ) = exp( 2 jπtT J ) a + exp( − jπtT J ) b, ∀ t ∈ R ; ∀ a, b ∈ L (cid:27) , ˇ E T,m = ˇ E T (0) + ˇ E T (1) + · · · + ˇ E T ( m ) . Let P T,m be the orthogonal projection from E T to E T,m and ˇ P T,m be the orthogonal projectionfrom ˇ E T to ˇ E T,m for m = 0 , , , ... , then Γ T = { P T,m : m = 0 , , , ... } and ˇΓ T = { ˇ P T,m : m =0 , , , ... } are the usual Galerkin approximation schemes w.r.t. A T and ˇ A T respectively.For z ∈ E T , we define f ( z ) = 12 h A T z, z i − Z T H ( z ) dt. (4.3)It is well known that f ∈ C ( E T , R ) whenever, H ∈ C ( R n ) and | H ′′ ( x ) | ≤ a | x | s + a (4.4)for some s ∈ (1 , + ∞ ) and all x ∈ R n .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 theoremwhich was proved in [24, 28, 48]. Theorem 4.1. Let W be a real Hilbert space with orthogonal decomposition E = X ⊕ Y , where dim X < + ∞ . Suppose f ∈ C ( W, R ) satisfies (PS) condition and the following conditions:(i) There exist ρ, δ > such that f ( w ) ≥ δ for any w ∈ W ;(ii) There exist e ∈ ∂B (0) ∩ Y and r > ρ > such that for any w ∈ ∂Q , f ( w ) < δ where Q = ( B r (0) ∩ X ) ⊕ { re : 0 ≤ r ≤ r } , B r (0) = { w ∈ W : || w || ≤ r } .Then (1) f possesses a critical value c ≥ δ , which is given by c = inf h ∈ Γ max w ∈ Q f ( h ( w )) , where Γ = { h ∈ C ( Q, E ) : h = id on ∂Q } ;(2) There exists w ∈ K c ≡ { w ∈ E : f ′ ( w ) = 0 , f ( w ) = c } such that the Morse index m − ( w ) of f at w satisfies m − ( w ) ≤ dim X + 1 . roof of Theorem 1.3. For any given T > 0, we prove the existence of T -periodic brakesolution 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 function ˆ H suitably and evenly such that it satisfies the growthcondition (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 K > χ ∈ C ∞ ( R , R ) such that χ ≡ y ≤ K , χ ≡ y ≥ K and χ ′ ( y ) < y ∈ ( K, K + 1),Where K will be determined later. Setˆ H K ( z ) = χ ( | z | ) ˆ H ( z ) + (1 − χ ( | z | )) R K | z | (4.5)and H K ( z ) = 12 B x · x + ˆ H K ( z ) , (4.6)where the constant R K satisfies R K ≥ max K ≤| z |≤ K +1 H ( z ) | z | . (4.7)Then H K ∈ C ( R n , R ). Since ˆ H satisfies (H3), ∀ ε > 0, there is a δ > H K ( z ) ≤ ε | z | for | z | ≤ δ . It is easy to see that H K ( z ) | z | is uniformly bounded as | z | → + ∞ , there is an M = M ( ε, K ) such that ˆ H K ( z ) ≤ M | z | for | z | ≥ δ . Soˆ H K ( z ) ≤ ε | z | + M | z | , ∀ z ∈ R n . (4.8)Set f K ( z ) = 12 h A T z, z i − Z T H K ( z ) dt, ∀ z ∈ ˆ E. Then f K ∈ C ( E T , R ) and f K ( z ) = 12 h ( A T − B T ) z, z i − Z T ˆ H K ( z ) dt, ∀ z ∈ ˆ E, where B T is the selfadjoint linear compact operator on E T defined by h B T z, z i = Z T B z ( t ) · z ( t ) dt. 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 X m = M − ( P T,m ( A T − B T ) P T,m ) ⊕ M ( P T,m ( A T − B T ) P T,m ) ,Y m = M + ( P T,m ( A T − B T ) P T,m ) . z ∈ Y m , by (4.8), (3.2), and the fact that P T,j B T = P T,j B T for j > 0, we have f Km ( z ) = 12 h ( A T − B T ) z, z i − Z T ˆ H K ( z ) dt ≥ || ( A T − B T ) || − || z || − ( ε || z || L + M || z || L ) ≥ || ( A T − B T ) || − || z || − ( εC + M C || z || ) || z || , (4.9)where C and C are constants for s = 2 , m and K .So if choose ε > εC < || ( A T − B T ) || − , then there exists ρ = ρ ( K ) > δ = δ ( K ) > 0, which are independent of m , such that f m ( z ) ≥ δ, ∀ z ∈ ∂B ρ (0) ∩ Y m . (4.10)Let e ∈ B (0) ∩ Y m and set Q m = { re : 0 ≤ r ≤ r } ⊕ ( B r (0) ∩ X m ) , where r will be determined later. Let z = z − + z ∈ B r (0) ∩ X m , we have f Km ( z + re ) = 12 h ( A T − B T ) z , z i + 12 r h ( A T − B T ) e, e i − Z T ˆ H K ( z + re ) dt ≤ || A T − B T || r − || ( A T − B T ) || − || z − || − Z T ˆ H K ( z + re ) dt. (4.11)Since ˆ H satisfies (H2) we haveˆ H K ( x ) ≥ a | x | α − a , ∀ x ∈ R n , where α = min { µ, } , a > a are two constants independent of K and m . Then there holds Z T ˆ H K ( z + re ) dt ≥ a Z T | z + re | α − T a ≥ a ( || z || αL α + r α ) − a , (4.12)where a and a are constants independent of K and m . By (4.11) and (4.12) we have f Km ( z + re ) ≤ || A T − ˆ B || r − || ( A − B ) || − || z − || − a ( || z || αL α + r α ) + a . Since α > r > ρ > 0, which are independent of K and m , such that f Km ≤ , ∀ z ∈ ∂Q m . (4.13)31hen by Theorem 4.1, f Km has a critical value c Km , which is given by c Km = inf g ∈ Γ m max z ∈ Q m f Km ( g ( z )) , (4.14)where Γ m = { g ∈ C ( Q m , ˆ E m | g = id ; on ∂Q m } . Moreover there is a critical point x Km of f Km whichsatisfies m − ( x Km ) ≤ dim X m + 1 . (4.15) Step 3. We prove that there exists a T -periodic brake orbit solution x T of (1.1) which satisfies i L ( x T ) ≤ i L ( B ) + ν L ( B ) + 1.Note that id ∈ Γ m , by (4.11) and condition (H4), we have c Km ≤ sup z ∈ Q m f Km ( z ) ≤ || A T − B T || r . 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 ) → f Km ( z m ) → c possesses a convergentsubsequence in E T . Hence in the sense of subsequence we have x Km → x K , f K ( x K ) = c K , f ′ K ( x K ) = 0 . (4.16)By similar argument in [44], x K is a classical nonconstant symmetric T -periodic solution of˙ x = J H ′ K ( x ) , x ∈ R n . Set B K ( t ) = H ′′ K ( x K ( t )), Then B K ∈ C ([0 , T / , L s ( R n )) and satisfies condition (B1). Let B K T be the operator defined by the same way of the definition of B T . It is easy to show that || f ′′ ( z ) − ( A T − B K T ) || → || z − x K || → . So for 0 < d ≤ || ( A T − B K T ) || − , there exists r > || f ′′ Km ( z ) − P T,m ( A T − B K T ) P T,m || ≤ || f ′′ ( z ) − ( A T − B K T ) || ≤ d, ∀ z ∈ { z ∈ E T : || z − x K || ≤ r } . Then for z ∈ { z ∈ E T : || z − x K || ≤ r } ∩ E T,m , ∀ u ∈ M − d ( P T,m ( A T − B K T ) P T,m ) \ { } , we have h f ′′ Km ( z ) u, u i ≤ h P T,m ( A T − B K T ) P T,m u, u i + k f ′′ Km ( z ) − P T,m ( A T − B K T ) P T,m kk u k ≤ − d k u k . 32o we have m − ( f ′′ Km ( z )) ≥ dim M − d ( P T,m ( A T − B K T ) P T,m ) . (4.17)By Theorem 2.1 of [31] and Remark 3.1, there is m ∗ > m ≥ m ∗ we havedim X m = mn + n + i L ( B ) + ν L ( B ) , (4.18)dim M − d ( P T,m ( A T − B K T ) P T,m ) = mn + n + i L ( B K ) . (4.19)Then by (4.15), (4.16), and (4.17)-(4.19), we have i L ( B K ) ≤ i L ( B ) + ν L ( B ) + 1 . By the similar argument as in the section 6 of [44], there is a constant M independent of K such that || x K || ∞ ≤ M . Choose K > M . Then x K is a non-constant symmetric T -periodicsolution of the problem (1.1). From now on in the proof of Theorem 1.3, we write B = B K and x T = x K . Then x T is a non-constant symmetric T -periodic solution of the problem (1.1), and B satisfies i L ( x T ) = i L ( B ) ≤ i L ( B ) + ν L ( B ) + 1 . (4.20)Since x T obtained in Step 3 is a nonconstant and symmetric T -period solution, its minimalperiod τ = Tk 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 kτ being the k times iteration of x τ . As in Section 1, let γ x T and γ x τ be the symplectic path as-sociated to ( τ, x ) and ( T, x T ) respectively. Then γ x τ ∈ C ([0 , τ ] , Sp(2 n )) and γ x T ∈ C ([0 , T ] , Sp(2 n )).Also we have γ x T = γ kx τ . Step 4. We prove that i L ( γ x τ ) + ν L ( γ x τ ) ≥ . We follow the way of the proof of Theorem 1.2 of [18]. By the same way as ˇ E T and ˇ A T we candefine the space ˇ E τ and the operator ˇ A τ on it. Also we can define the orthogonal projection ˇ P τ , m and the subspaces ˇ E τ,m for m = 0 , , , ... . Let ˇ B τ be the selfadjoint linear compact operator onˇ E T defined by: h ˇ B τ z, z i = Z τ B ( t ) z ( t ) · z ( t ) dt, ∀ z ∈ ˇ E τ . For z ∈ ˇ E τ , set f τ ( z ) = 12 h ( ˇ A τ − ˇ B τ ) z, z i = 12 h ˇ A τ z, z i − Z τ H ′′ ( x τ ( t )) z · z dt f τm ( w ) = f τ ( w ) , ∀ w ∈ ˇ E τ,m . Let X = { z ∈ L | B z = 0 and ˆ H ′′ ( x τ ( t )) z = 0 , ∀ t ∈ R } and Y be the orthogonal complement of X in L , i.e., L = X ⊕ Y . Since H ′′ ( x τ ( t )) = B +ˆ H ′′ ( x τ ( t )), by (H4) it is easy to see that there exists λ > Z τ H ′′ ( x τ ( t )) z · z dt ≥ λ || z || , ∀ z ∈ Y. Thus for any z = z − + z ∈ ˇ P τ,m M − ( ˇ A τ ) ⊕ Y with || z || = 1, we have f τm ( z ) = 12 h ( ˇ A τ − ˇ B τ ) z, z i = 12 h ˇ A τ z − , z − i − Z τ H ′′ ( x τ ( t )) z · z dt (4.21) ≤ − || ˇ A τ || − || z − || − Z τ H ′′ ( x τ ( t )) z · z dt − Z τ H ′′ ( x τ ( t )) z − · z dt ≤ − || ˇ A τ || − || z − || − λ || z || + max t ∈ [0 ,τ ] || H ′′ ( x τ ( t )) || || z − || || z || . (4.22)Since || z − || || z || ≤ ε || z − || + 1 ε || z || , ∀ ε > . By choosing ε suitably one can see that there exists 0 < c < | − c | small enough suchthat if || z || ≤ c , f τm ( z ) ≤ − λ c . (4.23)When || z || ≤ c , we have || z − || ≥ − c . By (4.21) and (H4) f τm ( z ) ≤ − || ˇ A τ || − || z − || ≤ − || ˇ A τ || − (1 − c ) . Hence we always have f τm ( z ) ≤ − c || z || , ∀ z ∈ ˇ P τ,m M − ( ˇ A τ ) ⊕ Y, (4.24)where c = max { λ c , || ˇ A τ || − (1 − c ) } is independent of m . Let d = min { || ( ˇ A τ − ˇ B τ ) || − , c } . 34y (4.24) and Theorem 2.1 of [31] and Remark 3.1 and the definition of i L ( γ ( x τ )), for m largeenough, we have mn + n + i L ( γ ( x τ )) = dim M − d ( ˇ P τ,m ( ˇ A τ − ˇ B τ ) ˇ P τ,m ) ≥ dim( ˇ P τ,m M − ( ˇ A τ ) ⊕ Y )= mn + n − dim X, (4.25)which implies that i L ( γ ( x τ )) ≥ − dim X. (4.26)Since x τ is a nonconstant brake solution of (1.1), by the definition of X we have ν L ( γ ( x τ )) ≥ dim X + 1 . (4.27)Hence by (4.26) and (4.27) we have i L ( γ ( x τ )) + ν L ( γ ( x τ )) ≥ . (4.28) Step 5. Finish the proof of Theorem 1.3.By Theorem 2.1 and Theorem 6.2 below (also Theorem 2.6 of [32]) we have i L ( γ kx τ ) ≥ i L ( γ x τ ) + k − 12 ( i ( γ ) + ν ( γ ) − n ) , if k ∈ N − , (4.29) i L ( γ kx τ ) ≥ i L ( γ x τ ) + i L √− ( γ x τ ) + ( k − i ( γ ) + ν ( γ ) − n ) , if k ∈ N . (4.30)Since B is semipositive and ˆ H satisfies (H4), by Corollary 3.2, we have i L ( γ x τ ) + ν L ( γ x τ ) ≥ . (4.31)By Proposition C of [42] and the definitions of i L and i L we have i ( γ ) = i L ( γ ) + i L ( γ ) + n,ν ( γ ) = ν L ( γ ) + ν L ( γ ) . So by (4.28) and (4.31) we have i ( γ ) + ν ( γ ) − n ≥ . (4.32)So by (4.29), (4.30) and (4.32) we have 35 L ( γ kx τ ) ≥ i L ( γ x τ ) + k − , if k ∈ N − , (4.33) i L ( γ kx τ ) ≥ i L ( γ x τ ) + k − , if k ∈ N . (4.34)By (4.20) and the definition of γ x τ we have i L ( γ x τ ) k ) ≤ i L ( B ) + ν L ( B ) + 1 . (4.35)By Corollary 3.2, we have i L ( γ x τ ) ≥ − n. (4.36)So by (4.33)-(4.36) we have k ≤ i L ( B ) + ν L ( B )) + 2 n + 4 . (4.37) Claim 2. k can not be 2( i L ( B ) + ν L ( B )) + 2 n + 3 and 2( i L ( B ) + ν L ( B )) + 2 n + 4.Hence by Claim 2, k ≤ i L ( B ) + ν L ( B )) + 2 n + 2, and Theorem 1.3 holds. Proof of Claim 2. We first show that k can not be 2( i L ( B ) + ν L ( B )) + 2 n + 3. Otherwise,we have k = 2( i L ( B ) + ν L ( B )) + 2 n + 3 . (4.38)The equality in (4.29) holds, then by (4.32), in this case there must hold that i ( γ ) + ν ( γ ) − n = 1 (4.39)and i L ( γ x τ ) = − n. (4.40)By Corollary 3.2 again we have that ν L ( γ x τ ) = n. (4.41)Also by (4.39) we have i L ( γ ( x τ ) + ν L ( γ ( x τ )) = 1 . Denote by ν L ( γ ( x τ )) = r . Then we have i L ( γ ( x τ )) = 1 − r, (4.42) ν L ( γ ( x τ )) ≥ . (4.43)36y (4.40) and (4.42) we have i L ( γ x τ ) − i L ( γ x τ ) = r − n − . (4.44)So we can write γ x τ ( τ ) = A C D with A, C, D to be n × n real matrices. Hence by (4.2)of [35] we have γ x τ ( τ ) = N γ x τ ( τ − N γ x τ ( τ D T A C T A A T D . Since γ x τ ( τ ) is a symplectic matrix we have A T D = D T A = I n , C T A = A T C. So we have γ x τ ( τ ) = I n C T A I n . Note that here C T A is a symmetric matrix and A is invertible. So by (4.43) there exists a orthogonalmatrix Q such that Q ( C T A ) Q T = diag(0 , , ..., , λ , λ , ..., λ p , λ p +1 , ..., λ n − p − r ) (4.45)with λ j > j = 1 , , ..., p and λ j < 0, for j = p + 1 , p + 2 , ..., n − p − r , where 1 ≤ p ≤ n − r .Then it is easy to check that ( I ) ⋄ r ⋄ N (1 , − ⋄ p ⋄ N (1 , ⋄ ( n − p − r ) ∈ Ω ( γ x τ ) with Ω ( γ xτ ) to bedefined in Section 6 below. Then by Theorem 6.2 below or Theorem 2.6 of [32], when the equalityin (4.29) holds, there must hold p = n − r . Hence we have Q ( C T A ) Q T = diag(0 , , ..., , λ , λ , ..., λ n − r ) , (4.46) λ j > , for j = 1 , , ..., n − r. (4.47) Case 1. If det A > 0, then there exists a invertible matrix path ρ ( s ) for s ∈ [0 , τ ] connecting itand I n such that ρ (0) = I n and ρ (1) = A .We define a symplectic path φ by φ ( s ) = ρ ( s ) − ρ ( s ) T A C D , ∀ s ∈ [0 , τ . ν L j ( φ ( s ) = constant for j = 0 , s ∈ [0 , τ ]. So by Definition 2.5 and Lemma 2.8 andProposition 2.11 of [42], for j = 1 , µ CLMF ( V j , Gr( φ ) , [0 , τ . (4.48)Also we have φ (0) = A C D and φ (0) = I n A T C I n .Note that we can always choose the orthogonal matrix Q in (4.46) such that det Q = 1 (otherwisewe replace it by diag( − , , ..., Q ). Then there exists a invertible matrix path ρ ( s ) for s ∈ [0 , τ ]connecting it and I n such that ρ (0) = I n and ρ ( τ ) = Q . We define a symplectic path φ by φ ( s ) = I n ρ ( s ) A T Cρ ( s ) T I n , ∀ s ∈ [0 , τ . Then ν L j ( φ ( s ) = constant and for j = 0 , s ∈ [0 , τ ]. So by Definition 2.5 and Lemma 2.8and Proposition 2.11 of [42] again, for j = 1 , µ CLMF ( V j , Gr( φ ) , [0 , τ . (4.49)Also we have φ (0) = I n A T C I n φ ( τ I n QA T CQ T I n = ( I ) ⋄ r ⋄ N (1 , λ ) ⋄ · · · ⋄ N (1 , λ n − r ) . (4.50)By the Reparametrization invariance and Path additivity of the Maslov index µ CLMF in [11] and(4.48) and (4.49), for j = 1 , µ CLMF ( V j , Gr( γ x τ ) , [0 , τ µ CLMF ( V j , Gr( φ ∗ ( φ ∗ γ x τ )) , [0 , τ , where the joint path φ ∗ ( φ ∗ γ x τ ) is defined by (6.1). So by definition for j = 0 , i L j ( γ x τ ) = i L j ( φ ∗ ( φ ∗ γ x τ )) . (4.51)Then by Theorem 2.3 and (4.50) we have i L ( γ x τ ) − i L ( γ x τ ) = 12 sgn M ε (( I ) ⋄ r ⋄ N (1 , λ ) T ⋄ · · · ⋄ N (1 , λ n − r ) T ) . (4.52)By Remark 2.1 and the computations (2.68)-(2.71) at the end of Section 2, for ε > M ε (( I ) ⋄ r ⋄ N (1 , λ ) T ⋄ · · · ⋄ N (1 , λ n − r ) T ) = 2( r − n ) . (4.53)38o we have i L ( γ x τ ) − i L ( γ x τ ) = r − n, (4.54)which contradicts to (4.44). Case 2. If det A < 0, then there exists a invertible matrix path ρ ( s ) for s ∈ [0 , τ ] such that ρ (0) = diag( − , , , ..., 1) and ρ (1) = A . by similar arguments we can show that i L ( γ x τ ) − i L ( γ x τ ) = 12 sgn M ε (( − I ) ⋄ ( I ) ⋄ ( r − ⋄ N (1 , λ ) ⋄ · · · ⋄ N (1 , λ n − r )) = r − n, (4.55)which still contradicts to (4.44).Hence we have proved that k can not be 2( i L ( B ) + ν L ( B )) + 2 n + 3. By the same argumentwe can prove that k can not be 2( i L ( B ) + ν L ( B )) + 2 n + 4. Thus Claim 2 is proved and theproof of Theorem 1.3 is complete. Proof of Theorem 1.1. Note that this is the case B = 0 of Theorem 1.3. Then by Theorem1.3 and the fact that i L (0) = − n and ν L (0) = n , the minimal period of x T is no less than T n +2 .In the following we prove that if (1.9) holds then the minimal period of x T belongs to { T, T } .Let x T is the k -time iteration of x τ with τ being the minimal period of x τ and τ = Tk . Thenby the proof of Theorem 1.3 with B = 0 we have (4.28), (4.29) and (4.30) hold. Since (1.9) holds,by Lemma 3.3 we have i L ( γ xτ ) ≥ . (4.56)So by (4.29) if k is odd, we have 1 ≥ k − . (4.57)Hence k ≤ 3. Now we prove that k can not be 3, other wise we have i L ( γ x τ ) = 0 , (4.58) ν L ( γ x τ ) = 0 , (4.59) i L ( γ x τ ) + ν L ( γ x τ ) = 1 . (4.60)And by Theorem 2.1 and Theorem 6.2 we have1 ≥ i L ( γ x τ ) = i L ( γ x τ ) + i e π/ ( γ x τ ) ≥ ( i ( γ x τ ) − ν ( γ x τ ) − n ) ≥ . (4.61)Then all the equalities of (4.61) hold. By Lemma 6.2 and 2 of Theorem 6.2 again, there exist p ≥ q ≥ p + q ≤ n and 0 < θ ≤ θ ≤ ... ≤ θ n − ( p + q ) ≤ π/ I ) ⋄ p ⋄ N (1 , − ⋄ q ⋄ R ( θ ) ⋄ R ( θ ) ⋄ ... ⋄ R ( θ n − p − q ) ∈ Ω (( γ x τ )( τ )) , (4.62)39here Ω ( M ) for a symplectic matrix M is defined in Section 6. By (4.62) we have − / ∈ σ (( γ x τ )( τ )) . (4.63)Now we denote by γ x τ ( τ ) = A BC D with A, B, C, D are all n × n matrices. Claim 1. Both D and A are invertible.We first prove D is invertible. Otherwise, there exists a n × n invertible matrix P such that P − DP = R and R is a ( n − r ) × ( n − r ) matrix with r ≥ 1. So we have P T P − A BC D ( P − ) T P := ˜ A ˜ B ˜ C ˜ D with ˜ D = R . Since ˜ A ˜ B ˜ C ˜ D is a symplectic matrix, we have˜ A T D − ˜ C T ˜ B = I n . (4.64)Since ˜ D = R , ˜ B T ˜ D and ˜ A T ˜ D both have form˜ B T ˜ D = ∗ ∗ , ˜ A T ˜ D = ∗ ∗ . (4.65)So by (4.64) and (4.65) we have˜ A T ˜ D + ˜ C T ˜ B = 2 ˜ A T ˜ D − I n = − I r ∗ ∗ . (4.66)By direct computation and (4.65) and (4.66) we have N ˜ A ˜ B ˜ C ˜ D − N ˜ A ˜ B ˜ C ˜ D = P T P − N A BC D − N A BC D ( P − ) T P (4.67)= ˜ D T ˜ A + ˜ B T ˜ C B T ˜ D A T ˜ C ˜ A T ˜ D + ˜ C T ˜ B = ∗ ∗ ∗∗ ∗ ∗∗ ∗ − I r ∗∗ ∗ ∗ (4.68)40ince by (4.2) o f[35] we have γ x τ ( τ ) = N A BC D − N A BC D , by (4.67) and (4.68) we have − ∈ σ ( γ x τ ( τ )) , which contradicts to (4.63). Thus we have proved that D is invertible. Similarly we can prove A isinvertible, and Claim 1 is proved. Claim 2. There exists a invertible n × n real matrix Q with det Q > Q − ( B T C ) Q = diag(0 , , ..., , λ , λ , ...λ n − r ) (4.69)with r = ν L ( γ x τ ) and λ i ∈ ( − , 0) for i = 1 , , ..., n − r .In fact γ x τ ( τ ) = N A BC D − N A BC D = D T B T C T A T A BC D = I + 2 B T C B T D A T C I + 2 C T B . (4.70)Since B and D are both invertible, for any ω ∈ C , we have I n − ( I n + 2 C T B − ωI n ) D − ( B T ) − I n I + 2 B T C − ωI n B T D A T C I + 2 C T B − ωI n = I + 2 B T C − ωI n B T D − ( I n + 2 C T B − ωI n ) D − ( B T ) − ( I + 2 B T C − ωI n ) + 2 A T C . So we havedet( γ x τ ( τ ) − ωI n ) = det( B T D )det(( I n + 2 C T B − ωI n ) D − ( B T ) − ( I + 2 B T C − ωI n ) − A T C )= det( D [( I n + 2 C T B − ωI n ) D − ( B T ) − ( I + 2 B T C − ωI n ) − A T C ] B T )= det( D [ I n + 2 C T B − ωI n ) D − ( B T ) − ( I + 2 B T C − ωI n ] B T − DA T CB T )= det(( I + 2 B T C − ωI n ) − CB T ) CB T )= det( ω I n − ω ( I + 2 CB T ) + I ) . (4.71)41y (4.62) we have σ ( γ x τ ( τ )) ⊂ U . (4.72)So for ω ∈ U by (4.71) we havedet( γ x τ ( τ ) − ωI n ) = ( − n ω n det( CB T − 12 (Re ω − . (4.73)Hence by (4.62) again we have σ ( CB T ) ⊂ ( − , n × n matrix S such that S − CB T S = diag(0 , , ..., , λ , λ , ..., λ n − r ) . (4.74)with r = ν L ( γ x τ ) and λ i ∈ ( − , 0) for i = 1 , , ..., n − r . Since S − CB T S = ( B T S ) − B T C ( B T S ),let Q = B T S , if det Q < B T S diag( − , , , ..., Continue the proof of Theorem 1.1. If det B > 0, there is a continuous symplectic matrix path joint B − B T and I n . Since B − B T A BC D = B − A I n B T C B T D . By Lemma 2.2, for ε > M ε A BC D = sgn M ε B − A I n B T C B T D . (4.75)If det B < 0, there is a continuous symplectic path joint B − B T and ( − I ) ⋄ I n − . Bydirect computation we havesgn M ε A BC D = sgn M ε (cid:0) ( − I ) ⋄ I n − (cid:1) A BC D . So by Lemma 2.2 again we have (4.75) holds. So whenever det( B ) > P T P − B − A I n B T C B T D P 00 ( P − ) T = ˜ A I n ˜ C ˜ D . By Claim 2,we have ˜ C = diag(0 , , .., , λ , λ , ..., λ n − r ) . (4.76)Since ˜ A I n ˜ C ˜ D is a symplectic matrix, we have ˜ A and ˜ D are both symmetric and have thefollow forms: ˜ A = A A , ˜ D = D D , A and D are r × r invertible matrices, A and D are ( n − r ) × ( n − r ) invertiblematrices. So we have ˜ A I n ˜ C ˜ D = A I r D ⋄ A I n − r Λ D , (4.77)where Λ = diag( λ , λ , ..., λ n − r ).Since N A I r D − N A I r D = I r D I r , by (4.62) D is negative defi-nite. So we can joint it to − I r by a invertible symmetric matrix path. Then by Lemma 2.2, Remark2.1, and computations below Remark 2.1 in Section 2, we havesgn M ε A I r D = sgn M ε − I r I r − I r = r sgn M ε ( N ( − , r. (4.78)Since M ε A I n − r Λ D is invertible for ε = 0, for ε > M ε A I n − r Λ D = sgn M A I n − r Λ D = sgn A Λ I n − r D − I n − r − I n − r A I n − r Λ D + I n − r I n − r = sgn − A Λ − Λ − Λ − D = sgn − A Λ − Λ − Λ − D . (4.79)Since A I n − r Λ D is a symplectic matrix, we have A D − Λ = I n − r ,A Λ = Λ A . (4.80)Hence A − Λ − D = A − (Λ − A D ) = − A − . 43o we have I n − r − A − I n − r − A Λ − Λ − Λ − D I n − r − A − I n − r = − A Λ 00 A − Λ − D = − A Λ 00 − A − . (4.81)By (4.80), there exist invertible matrix R such that R − A R = diag( α , α , ..., α n − r ) , α i ∈ R \ { } , i = 1 , , ..., n − r, (4.82) R − Λ R = diag( λ i , λ i , ..., λ i n − r ) , { i , i , ..., i n − r } = { , , ..., n − r } . (4.83)So we have R − ( − A Λ) R = diag( − λ i α , − λ i α , ..., − λ i n − r α n − r ) , (4.84) R − ( − A − ) R = diag( − α , − α , ..., α n − r ) . (4.85)Since λ i ∈ ( − , 0) for i = 1 , , ..., n − r , by (4.82)-(4.85) we havesgn( − A Λ) + sgn( − A − ) = 0 . (4.86)Hence by (4.79), (4.81) and (4.86) we havesgn M ε A I n − r Λ D = sgn( − A Λ) + sgn( − A − ) = 0 . (4.87)Since det Q > I n by a invertible matrix path. Hence by Lemma 2.2 andRemark 2.1, (4.77), (4.78) and (4.87), we havesgn M ε B − A I n B T C B T D = sgn M ε A I r D + sgn M ε A I n − r Λ D = 2 r + 0= 2 r. (4.88)Then by Theorem 2.3, (4.75) and (4.88) we have i L ( γ x τ ) − i L ( γ x τ ) = r. (4.89)44owever by (4.58), (4.60) and ν L ( γ x τ ) = r we have i L ( γ x τ ) − i L ( γ x τ ) = r − , which contradicts to (4.89).Thus we have prove that k can not be 3. So if k is odd, it must be 1. By the same proof wehave if k is even, it must be 2. Then τ ∈ { T, T } . The proof of Theorem 1.1 is complete. Proof of Corollary 1.2. Since 0 < T < π || B || , there is ε > ≤ B ≤ || B || I n < ( πT − ε ) I n . It is easy to see that γ ( πT − ε ) I n ( t ) = exp(( πT − ε ) tJ ) ∀ t ∈ [0 , T . So we have ν L ( γ ( πT − ε ) I n ) = 0 ,i L (( πT − ε ) I n ) = 0 . Then by (5.40) and Lemma 3.1 and Corollary 3.1 we have0 ≤ i − ( B ) + ν − ( B ) ≤ i − (( πT − ε ) I n ) = 0 . So we have i − ( B ) + ν − ( B ) = 0 . Hence by the same proof of Theorem 1.1, the conclusions of Corollary 1.2 holds. Remark 4.1. Under the same conditions of Theorem 1.3, if R T H ′′ ( x T ( t )) dt > 0, by the sameproof of Theorem 1.1, we have τ ≥ T i L ( B ) + ν L ( B )) + 2 . Moreover, if 0 < T < π || B || or i L ( B ) + ν L ( B ) = 0, we have τ ∈ { T, T } . Proof of Theorem 1.2. This is the case n = 1 and B = 0 of Theorem 1.3, by the proofTheorem 1.3, for any T > x T satisfies i L ( γ x T ) ≤ . (4.90)45f it’s minimal period is τ = T /k for some k ∈ N , we denote x τ = x T | [0 ,τ ] . Then by the proof ofTheorem 1.3 we have i ( γ xτ ) + ν ( γ x τ ) ≥ . (4.91)In the following we prove Theorem 1.2 in 2 steps. Step 1. For k = 2 p + 1 for some p ≥ 0, we prove that p = 0.Firstly by the proof of Theorem 1.3 we have1 ≥ i L ( γ p +1 x τ ) ≥ p ( i ( γ x τ ) + ν ( γ x τ ) − 1) + i L ( γ ) . (4.92)We divide the argument into three cases. Case 1. i ( γ xτ ) + ν ( γ x τ ) = 2. If ν ( γ x τ ) = 1, then i ( γ x τ ) = 1 ∈ Z + 1. By Lemma 6.3, wehave N (1 , ∈ Ω ( γ x τ ( τ )). Since1 = i ( γ x τ ) = i L ( γ x τ ) + i L ( γ xτ ) + 1 . (4.93)By Corollary 2.1 we have | i L ( γ x τ ) − i L ( γ x τ ) | ≤ . (4.94)Then by (4.93) and (4.94) we have i L ( γ x τ ) = i L ( γ x τ ) = 0 . (4.95)So by Theorem 2.1, Lemma 6.2, and (6.13), we have i L ( γ x τ ) = i L ( γ x τ ) + i e π √− / ( γ xτ )= i L ( γ x τ ) + i ( γ x τ ) + S N (1 , (1)= 0 + 1 + 1= 2 > ≥ i L ( γ p +1 xτ ) . (4.96)Then by Theorem 3.3 we have 2 p + 1 < . Hence p = 0.If ν ( γ x τ ) = 2, then i ( γ x τ ) = 0. But now γ x τ ( τ ) = I , by Lemma 6.3 i ( γ x τ ) ∈ Z + 1, whichyields a contradiction. So this case can not happen. So in Case 1, we have proved p = 0. Case 2. i ( γ xτ ) + ν ( γ x τ ) = 3. 46f ν ( γ x τ ) = 1, then i ( γ xτ ) = 2 ∈ Z . (4.97)By Lemma 6.3 we have N (1 , − ∈ Ω (( γ x τ )( τ )). So if p ≥ 1, by Theorem 3.3, Theorem 2.1,Lemma 6.2 and (6.13), we have we have1 ≥ i L ( γ p +1 x τ ) ≥ i L ( γ x τ )= i L ( γ x τ ) + i e π √− / ( γ x τ )= i L ( γ x τ ) + i ( γ xτ ) + S N (1 , − (1) ≥ − . (4.98)So there must hold i L ( γ xτ ) = − . Then by Corollary 2.1 we have i L ( γ x τ ) ≤ . So we have i ( γ x τ ) = i L ( γ x τ + i L ( γ x τ ) + 1 ≤ − , which contradicts (4.97). Thus we have p = 0.If ν ( γ x τ ) = 2, then i ( γ x τ ) = 1 , γ x τ ( τ ) = I . (4.99)If p ≥ 1, by Theorem 3.3, Theorem 2.1, Corollary 3.2, Lemma 6.2 and (6.13), we have we have1 ≥ i L ( γ p +1 x τ ) ≥ i L ( γ x τ )= i L ( γ x τ ) + i e π √− / ( γ x τ )= i L ( γ x τ ) + i ( γ x τ ) + S I (1) ≥ − . So there must hold i L ( γ xτ ) = − . i L ( γ x τ ) ≤ . So we have i ( γ xτ ) = i L ( γ x τ + i L ( γ x τ ) + 1 ≤ − , which contradicts (4.99). Thus we have p = 0. Case 3. i ( γ x τ ) + ν ( γ x τ ) ≥ i ( γ x τ ) + ν ( γ x τ ) − ≥ 3. By Corollary 3.2 we have i L ≥ − 1. So by (4.92) we have p ≤ / , (4.100)which yields p = 0. So we finish Step 1. Step 2. For k = 2 p + 2 for some p ≥ 0, we prove that p = 0.In fact, apply Bott-type iteration formula of Theorem 2.1 to the the case of the iteration timeequals to 4 and note that by Corollary 3.1 i √− ( γ x τ ) ≥ 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 thefollowing remark. Remark 4.2. Only use the Maslov-type index theory to estimate the iteration time of the T -periodic brake solution x T obtained by the first 4 steps in the proof of Theorem 1.3 with B = 0,we can not hope to prove T is the minimal period of x T . Even H ′′ ( z ) > z ∈ R n \ { } .For n = 1 and T = 4 π , we can not exclude the following case: x T ( t ) = sin t cos t ,H ′ ( x T ( t )) = x T ( t ) ,H ′′ ( x T ( t )) ≡ I n . It is easy to check that γ x T ( t ) = R ( t ) for t ∈ [0 , π ]. Hence by Lemma 5.1 of [30] or the proof ofLemma 3.1 of [42] we have i L ( γ x T ) = X In this section we study the minimal period problem for symmeytric brake orbit solutions of theeven reversible Hamiltonian system (1.1) and complete the proof of Theorems 1.4-1.5 and Corollary1.4.For T > 0, let E T = { x ∈ W / , ( S τ , R n ) | x ( − t ) = N x ( t ) a.e. t ∈ R } with the usual W / , norm and inner product. Correspondingly ˆ E and ˜ E are defined to be the symmetric ones and the T -periodic ones in E T respectively. Also { P T,m } and { ˆ P m } are the Galerkin approximation schemew.r.t. A T and ˆ A respectively, where { P T,m } , { ˆ P m } , A T , and ˆ A are defined by the same way as inSection 2, we only need to replace τ by T .For z ∈ E T , we define f ( z ) = 12 h A T z, z i − Z T H ( z ) dt. (5.1)For z ∈ ˆ E , we define ˆ f ( z ) = 12 h ˆ Az, z i − Z T H ( z ) dt. (5.2)We have the following lemma. Lemma 5.1. Let z ∈ ˆ E . If ˆ f ′ ( z ) = 0 , then f ′ ( z ) = 0 . Proof. Let z ∈ ˆ E and ˆ f ′ ( z ) = 0. So for any y ∈ ˆ E we have h ˆ f ′ ( z ) , y i = Z T J ˙ z ( t ) · y ( t ) dt − Z T H ′ ( z ( t )) · y ( t ) dt = 0 , ∀ y ∈ ˆ E. (5.3)Since H is even and z ∈ ˆ E , we have H ′ ( z ( t + T H ′ ( − z ( t )) = − H ′ ( z ( t )) . (5.4)So H ′ ( z ) ∈ ˆ E and h f ′ ( z ) , y i = Z T J ˙ z ( t ) · y ( t ) dt − Z T H ′ ( z ( t )) · y ( t ) dt = 0 , ∀ y ∈ ˜ E. (5.5)By (5.4) and (5.5), we have h f ′ ( z ) , y i = Z T J ˙ z ( t ) · y ( t ) dt − Z T H ′ ( z ( t )) · y ( t ) dt = 0 , ∀ y ∈ E T . (5.6)Hence f ′ ( z ) = 0By Lemma 5.1 and arguments in the proof of Theorem 1.3 in Section 4, to look for the T -periodsymmetric solutions of (1.1) is equivalent to look for critical points of ˆ f . Proof of Theorem 1.5. For any given T > 0, we prove the existence of T -periodic symmetricbrake orbit solution of (1.1) whose minimal period satisfies the inequalities in the conclusion of49heorem 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 proofinto several steps. Step 1. Similarly as Step 1 in the proof of Theorem 1.3, for any K > H suitably and evenly to ˆ H K such that it satisfies the growth condition (4.4). Correspondinglywe obtain a new even and reversible function H K satisfies condition (4.4).Set ˆ f K ( z ) = 12 h ˆ Az, z i − Z T H K ( z ) dt, ∀ z ∈ ˆ E. (5.7)Then ˆ f K ∈ C ( ˆ E, R ) andˆ f K ( z ) = 12 h ( ˆ A − ˆ B ) z, z i − Z T ˆ H K ( z ) dt, ∀ z ∈ ˆ E, (5.8)where ˆ B is the selfadjoint linear compact operator on ˆ E defined by h ˆ B z, z i = Z T B z ( t ) · z ( t ) dt. (5.9) Step 2. For m > 0, let ˆ f Km = ˆ f | ˆ E m , where ˆ E m = ˆ P m ˆ E . Set X m = M − ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) ⊕ M ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) ,Y m = M + ( ˆ P m ( ˆ A − ˆ B ) ˆ P m ) . By the same argument of Step 2 in the proof of Theorem 1.3, we can show that ˆ f K m satisfiesthe hypotheses of Theorem 4.1. Moreover, we obtain a critical point x Km of ˆ f Km with critical value C K m which satisfies m − ( x Km ) ≤ dim X m + 1 . (5.10)and δ ≤ C Km ≤ || ˆ A − ˆ B || r , (5.11)where δ is a positive number depending on K and r > K and m . Step 3. We prove that there exists a symmetric T -periodic brake orbit solution x T of (1.1)which satisfies i L √− ( γ x T ) ≤ i L √− ( B ) + ν L √− ( B ) + 1 . (5.12)From the proof of Theorem 1.3 we have f K satisfies ( P S ) ∗ c condition for c ∈ R , by the sameproof of Lemma 5.1, we have ˆ f K satisfies ( P S ) ∗ c condition for c ∈ R , i.e., any sequence z m such50hat z m ∈ ˆ E m , ˆ f ′ Km ( z m ) → f Km ( z m ) → c possesses a convergent subsequence in ˆ E . Hencein the sense of subsequence we have x Km → x K , ˆ f K ( x K ) = c K , ˆ f ′ K ( x K ) = 0 . (5.13)By similar argument as in [44], x K is a classical nonconstant symmetric T -periodic solution of˙ x = J H ′ K ( x ) , x ∈ R n . (5.14)Set B K ( t ) = H ′′ K ( x K ( t )), Then B K ∈ C ( S T/ , L s ( R n )). Let ˆ B K be the operator defined by thesame way of the definition of ˆ B . It is easy to show that || ˆ f ′′ ( z ) − ( ˆ A − ˆ B K ) || → || z − x K || → . (5.15)So for 0 < d < || ( A T − B K T ) || − , there exists r > || ˆ f ′′ Km ( z ) − ˆ P m ( ˆ A − ˆ B K ) ˆ P m || ≤ || ˆ f ′′ ( z ) − ( ˆ A − ˆ B K ) || < d, ∀ z ∈ { z ∈ ˆ E : || z − x K || ≤ r } . (5.16)Then for z ∈ { z ∈ ˆ E : || z − x K || ≤ r } ∩ ˆ E m , ∀ u ∈ M − d ( ˆ P m ( ˆ A − ˆ B T ) ˆ P m ) \ { } , we have h ˆ f ′′ Km ( z ) u, u i ≤ h ˆ P m ( ˆ A − ˆ B K ) ˆ P m u, u i + k ˆ f ′′ Km ( z ) − ˆ P m ( ˆ A − ˆ B K ) ˆ P m kk u k ≤ − d k u k . So we have m − ( ˆ f ′′ Km ( z )) ≥ dim M − d ( ˆ P m ( ˆ A − ˆ B K ) ˆ P m ) . (5.17)By Theorem 3.1, Remark 3.1, there is m ∗ > m ≥ m ∗ we havedim X m = mn + i L √− ( B ) + ν L √− ( B ) , (5.18)dim M − d ( ˆ P m ( ˆ A − ˆ B K ) ˆ P m ) = mn + i L √− ( B K ) . (5.19)Then by (5.10), (5.13), and (5.17)-(5.19), we have i L √− ( B K ) ≤ i L √− ( B ) + ν L √− ( B ) + 1 . (5.20)By the similar argument as in the section 6 of [44], there is a constant M independent of K such that || x K || ∞ ≤ M . Choose K > M . Then x K is a non-constant symmetric T -periodic brakeorbit solution of the problem (1.1). From now on in the proof of Theorem 1.2, we write B = B K and x T = x K . Then x T is a non-constant symmetric T -periodic solution of the problem (1.1), and B satisfies i L √− ( γ x T ) = L √− ( B ) ≤ i L √− ( B ) + ν L √− ( B ) + 1 . (5.21)51 tep 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, itsminimal period τ = T r + 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 r + sτ being the 4 r + s times iteration of x τ . As in Section 1, let γ x T and γ x τ thesymplectic path associated to ( τ, x ) and ( T, x T ) respectively. Then γ x τ ∈ C ([0 , τ ] , Sp(2 n )) and γ x T ∈ C ([0 , T ] , Sp(2 n )). Also we have γ x T = γ r + sx τ , which is the 4 r + s times iteration of γ x τ .By (5.21) we have i L √− ( γ r + sx τ ) ≤ i L √− ( B ) + ν L √− ( B ) + 1 . (5.22)Since x τ is also a nonconstant symmetric periodic solution of (1.1). It is clear that ν − ( x τ ) ≥ . (5.23)Since ˆ H satisfies condition (H5) and B is semipositive, by Corollary 3.1 of [51] (also by Theorem6.2) we have i − ( γ x τ ) ≥ . (5.24)By Corollary 3.2 of [51] (cf. aslo [29]), we have i ( γ x τ ) + ν ( γ x τ ) ≥ n. (5.25)It is easy to see that γ x τ ( τ t ) = γ x τ ( t ) γ x τ ( τ , ∀ t ∈ [0 , τ . (5.26)So by Theorem 6.1 of Bott-type iteration formula we have i ( γ x τ ) + ν ( γ x τ ) = i ( γ x τ ) + ν ( γ x τ ) + i − ( γ x τ ) + ν − ( γ x τ ) ≥ n + 0 + 1= n + 1 . (5.27)If r ≥ 1, then by Theorems 2.2 and 6.2 and (5.27) we have i − ( γ rx τ ) = i − (( γ x τ ) p )= r X j =1 i ω j − r ( γ x τ ) ≥ r X j =1 ( i ( γ x τ ) + ν ( γ x τ ) − n ) (5.28)= r ( i ( γ x τ ) + ν ( γ x τ ) − n ) ≥ r, (5.29)52here ω r = e π √− / (2 r ) as defined in Theorem 2.2.By Theorem 3.2, we have i L √− ( γ r + sx τ ) ≥ i L √− ( γ rx τ ) . (5.30)Then (5.22), (5.29) and (5.30) yield r ≤ i L √− ( B ) + ν L √− ( B ) + 1 . (5.31)Thus for i L √− ( B ) + ν L √− ( B ) is odd, by (5.31) we have4 r + s ≤ r + 3 ≤ i L √− ( B ) + ν L √− ( B )) + 7 . (5.32) Claim 3. For i L √− ( B ) + ν L √− ( B ) is even, the equality in (5.31) can not hold.Otherwise, r ≥ i ω j − r ( γ x τ ) = i ( γ x τ ) + ν ( γ x τ ) − n = 1 , j = 1 , , ..., r. (5.33)By the definition of ω r , we have ω j − r = − j = 1 , , ..., r . So by (5.33) and 2 of Theorem6.2, we have I p ⋄ N (1 , − ⋄ q ⋄ K ∈ Ω ( γ x τ ( τ )) for some non-negative integers p and q satisfying0 ≤ p + q ≤ n and K ∈ Sp(2( n − p − q )) with σ ( K ) ∈ U \ { } satisfying the condition that alleigenvalues of K located with the arc between 1 and ω r in U + \ {± } possess total multiplicity n − p − q . So there are no eigenvalues of K on the arc between ω j − r and − ω r − r with r = 1. However, whether ω r − r ∈ σ ( γ x τ ( τ )) or not, we always have S + γ xτ ( τ ) ( ω r − r ) = 0 , (5.34) i ω r − r ( γ x τ ) = 1 . (5.35)So (6.13) and Lemma 6.2, we have i − ( γ x τ ) = i ω r − r ( γ x τ ) + S + γ xτ ( τ ) ( ω r − r )= 1 + 0 = 1 . (5.36)But by (5.26), Lemma 6.1, and Theorem 6.1, we have i − ( γ rx τ ) = i − (( γ rx τ ) )= i √− ( γ rx τ ) + i −√− ( γ rx τ )= 2 i √− ( γ rx τ ) . i − ( γ rx τ ) is an even integer, which yields a contradiction to (5.36). So Claim 3 holds, and wehave r ≤ i L √− ( B ) + ν L √− ( B ) . (5.37)Hence 4 r + s ≤ r + 3 ≤ i L √− ( B ) + ν L √− ( B )) + 3 . (5.38)Theorem 1.5 holds from (5.32) and (5.38). Proof of Theorem 1.4. This is the case B ≡ i L √− (0) = 0 , ν L √− (0) = 0 . (5.39)Then i L √− (0) + ν L √− (0) = 0 and is also even. So Theorem 1.4 holds from Theorem 1.5. Proof of Corollary 1.2. Since 0 < T < π || B || , there is ε > ≤ B ≤ || B || I n < ( πT − ε ) I n . (5.40)It is easy to see that γ ( πT − ε ) I n ( t ) = exp(( πT − ε ) tJ ) ∀ t ∈ [0 , T . (5.41)Since ν L (exp(( πT − ε ) tJ )) = 0 , ∀ t ∈ [0 , T . (5.42)We have i L ( γ πT − ε ) I n ) = 0 , i L ( γ ( πT − ε ) I n ) = 0 . (5.43)So by Theorem 2.1 we have i L √− (( πT − ε ) I n ) = i L ( γ πT − ε ) I n ) − i L ( γ ( πT − ε ) I n ) = 0 . (5.44)Then by (5.40) and Lemma 3.1 and Corollary 3.1 we have0 ≤ i − ( B , T ν − ( B , T ≤ i − (( πT − ε ) I n , T . (5.45)So we have i − ( B , T ν − ( B , T . (5.46)Hence by Theorem 1.1 or Corollary 1.1, the conclusion of Corollary 1.2 holds.Also a natural question is that can we prove the minimal period is T in this way? We have thefollowing remark. 54 emark 5.1. Only use the Maslov-type index theory to estimate the iteration time of the sym-metric T -periodic brake solution x T obtained in the proof of Theorem 1.5 with B = 0, we can nothope to prove T is the minimal period of x T . Even H ′′ ( z ) > z ∈ R n \ { } . For n = 1 and T = 6 π , we can not exclude the following case: x T ( t ) = sin t cos t ,H ′ ( x T ( t )) = x T ( t ) ,H ′′ ( x T ( t )) ≡ I n . (5.47)It is easy to check that γ x T ( t ) = R ( t ) for t ∈ [0 , π ]. Hence by Theorem 2.1 and Lemma 5.1 of [30]or the proof of Lemma 3.1 of [42] we have i L √− ( γ x T ) = X π/ ≤ s< π ν L ( γ x T )( s ) = 1 . (5.48)In this case the minimal period of x T is T . Similarly for n > ( 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].For any ω ∈ U , the following codimension 1 hypersuface in Sp(2 n ) is defined by:Sp(2 n ) ω = { M ∈ Sp(2 n ) | det( M − ωI n ) = 0 } . For any two continuous path ξ and η : [0 , τ ] → Sp(2 n ) with ξ ( τ ) = η (0), their joint path is definedby η ∗ ξ ( t ) = ξ (2 t ) if 0 ≤ t ≤ τ ,η (2 t − τ ) if τ ≤ t ≤ τ. (6.1)Given any two (2 m k × m k )- matrices of square block form M k = A k B k C k D k for k = 1 , 2, as in[41], the ⋄ -product of M and M is defined by the following (2( m + m ) × m + m ))-matrix M ⋄ M : M ⋄ M = A B A B C D C D . 55 special path ξ n is defined by ξ n ( t ) = − tτ 00 (2 − tτ ) − ⋄ n , ∀ t ∈ [0 , τ ] . Definition 6.1. For any ω ∈ U and M ∈ Sp(2 n ), define ν ω ( M ) = dim C ker( M − ωI n ) . (6.2)For any γ ∈ P τ (2 n ), define ν ω ( γ ) = ν ω ( γ ( τ )) . (6.3)If γ ( τ ) / ∈ Sp(2 n ) ω , we define i ω ( γ ) = [Sp(2 n ) ω : γ ∗ ξ n ] , (6.4)where the right-hand side of (6.4) is the usual homotopy intersection number and the orientationof γ ∗ ξ n is its positive time direction under homotopy with fixed endpoints.If γ ( τ ) ∈ Sp(2 n ) ω , we let F ( γ ) be the set of all open neighborhoods of γ in P τ (2 n ), and define i ω ( γ ) = sup U ∈F ( γ ) inf { i ω ( β ) | β ( τ ) ∈ U and β ( τ ) / ∈ Sp(2 n ) ω } . (6.5)Then ( i ω ( γ ) , ν ω ( γ )) ∈ Z × { , , ..., n } , is called the index function of γ at ω . Lemma 6.1. (Lemma 5.3.1 of [41]) For any γ ∈ P τ (2 n ) and ω ∈ U , there hold i ω ( γ ) = i ¯ ω ( γ ) , ν ω ( γ ) = ν ¯ ω ( γ ) . (6.6)As in [38], for any M ∈ Sp(2 n ) we defineΩ( M ) = { P ∈ Sp(2 n ) | σ ( P ) ∩ U = σ ( M ) ∩ U and ν λ ( P ) = ν λ ( M ) , ∀ λ ∈ σ ( M ) ∩ U } . (6.7)We denote by Ω ( M ) the path connected component of Ω( M ) containing M , and call it the homo-topy component of M in Sp(2 n ).The following symplectic matrices were introduced as basic normal forms in [41]: D ( λ ) = λ λ − , λ = ± , (6.8) N ( λ, b ) = λ b λ , λ = ± , b = ± , , (6.9)56 ( θ ) = cos( θ ) − sin( θ )sin( θ ) cos( θ ) , θ ∈ (0 , π ) ∪ ( π, π ) , (6.10) N ( ω, b ) = R ( θ ) b R ( θ ) , θ ∈ (0 , π ) ∪ ( π, π ) , (6.11)where b = b b b b with b i ∈ R and b = b .For any M ∈ Sp(2 n ) and ω ∈ U , splitting number of M at ω is defined by S ± M = lim ǫ → + i ω exp( ±√− ǫ ) ( γ ) − i ω ( γ ) (6.12)for any path γ ∈ P τ (2 n ) satisfying γ ( τ ) = M .Splitting numbers possesses the following properties. Lemma 6.2. (cf. [40], Lemma 9.1.5 and List 9.1.12 of [41]) Splitting number S ± M ( ω ) are welldefined; that is they are independent of the choice of the path γ ∈ P τ (2 n ) satisfying γ ( τ ) = M . For ω ∈ U and M ∈ Sp(2 n ) , S ± N ( ω ) are constant for all N ∈ Ω ( M ) . Moreover we have(1) ( S + M ( ± , S − M ( ± , for M = ± N (1 , b ) with b = 1 or ;(2) ( S + M ( ± , S − M ( ± , for M = ± N (1 , b ) with b = − ;(3) ( S + M ( e √− θ ) , S − M ( e √− θ )) = (0 , for M = R ( θ ) with θ ∈ (0 , π ) ∪ ( π, π ) ;(4) ( S + M ( ω ) , S − M ( ω )) = (0 , for ω ∈ U \ R and M = N ( ω, b ) is trivial i.e., forsufficiently small α > , M R (( t − α ) ⋄ n possesses no eigenvalues on U for t ∈ [0 , .(5) ( S + M ( ω ) , S − M ( ω ) = (1 , for ω ∈ U \ R and M = N ( ω, b ) is non-trivial .(6) ( S + M ( ω ) , S − M ( ω ) = (0 , for any ω ∈ U and M ∈ Sp(2 n ) with σ ( M ) ∩ U = ∅ .(7) S ± M ⋄ M ( ω ) = S ± M ( ω ) + S ± M ( ω ) , for any M j ∈ Sp(2 n j ) with j = 1 , and ω ∈ U . By the definition of splitting numbers and Lemma 6.2, for 0 ≤ θ < θ < π and γ ∈ P τ (2 n )with τ > 0, we have i exp( √− θ ) ( γ ) = i exp( √− θ ) + S + γ ( τ ) ( e √− θ )+ X θ ∈ ( θ ,θ ) (cid:16) S + γ ( τ ) ( e √− θ ) − S − γ ( τ ) ( e √− θ ) (cid:17) − S − γ ( τ ) ( e √− θ ) . (6.13)For any symplectic path γ ∈ P τ (2 n ) and m ∈ N , we define its m th iteration in the periodicboundary sense γ ( m ) : [0 , mτ ] → Sp(2 n ) by γ ( m )( t ) = γ ( t − jτ ) γ ( τ ) j for jτ ≤ t ≤ ( j + 1) τ, j = 0 , , ..., m − . (6.14)57 efinition 6.2. (cf.[40], [41]) For any γ ∈ P τ (2 n ) and ω ∈ U , we define( i ω ( γ, m ) , ν ω ( γ, m )) = ( i ω ( γ ( m )) , ν ω ( γ ( m ))) , ∀ m ∈ N . (6.15)We have the following Bott-type iteration formula. Theorem 6.1. (cf. [40], Theorem 9.2.1 of [41]) For any τ > , γ ∈ P τ (2 n ) , z ∈ U , and m ∈ N , i z ( γ, m ) = X ω k = z i ω ( γ ) , ν z ( γ, m ) = X ω m = z ν ω ( γ ) . (6.16)By Theorem 8.1.4 of [41], we have the following Lemma. Lemma 6.3. For γ ∈ P τ (2) with τ > , the following results hold.1. If N (1 , ∈ Ω ( γ ( τ )) , then i ( γ, m ) = m ( i ( γ ) + 1) − , ν ( γ, m ) = 1 , ∀ m ∈ N , (6.17) i ( γ ) ∈ Z + 1 . (6.18) 2. If N (1 , ∈ Ω ( γ ( τ )) , then i ( γ, m ) = m ( i ( γ ) + 1) − , ν ( γ, m ) = 2 , ∀ m ∈ N , (6.19) i ( γ ) ∈ Z + 1 . (6.20) 3. If N (1 , − ∈ Ω ( γ ( τ )) , then i ( γ, m ) = m ( i ( γ ) , ν ( γ, m ) = 1 , ∀ m ∈ N , (6.21) i ( γ ) ∈ Z . (6.22)Denote by U + = { ω ∈ U | Im ω ≥ } and U − = { ω ∈ U | Im ω ≤ } . The following theoremwas proved by Liu and Long in [33, 34], which plays a important role in the proof of our mainresults in Sections 4-5. Theorem 6.2. (Theorem 10.1.1 of [41]) 1. For any γ ∈ P τ (2 n ) and ω ∈ U \ { } , it always holds that i ( γ ) + ν ( γ ) − n ≤ i ω ( γ ) ≤ i ( γ ) + n − ν ω ( γ ) . (6.23) 2. The left equality in (6.23) holds for some ω ∈ U + \ { } (or U − \ { } ) if and only if I p ⋄ N (1 , − ⋄ q ⋄ K ∈ Ω ( γ ( τ )) for some non-negative integers p and q satisfying ≤ p + q ≤ n and K ∈ Sp(2( n − p − q )) with σ ( K ) ∈ U \ { } satisfying the condition that all eigenvalues of K located ith the arc between and ω including U + \ { } (or U − \ { } )possess total multiplicity n − p − q . If ω = − , all eigenvalues of K are in U \ R and those in U + \ R (or U − \ R ) are all Krein-negative(or Krein-positive) definite. If ω = − , it holds that ( − I s ) ⋄ N ( − , ⋄ t ⋄ H ∈ Ω ( γ ( τ )) for somenon-negative integers s and t satisfying ≤ s + t ≤ n − p − q , and some H ∈ Sp(2( n − p − q − s − t )) satisfying σ ( H ) ⊂ U \ R and that all elements in σ ( H ) ∩ U + (or σ ( H ) ∩ U − ) are all Krein-negative(or Krein-positive) definite.3. The left equality of (6.23) holds for all ω ∈ U \ { } if and only if I p ⋄ N (1 , − ⋄ ( n − p ) ∈ Ω ( γ ( τ )) for some integer p ∈ [0 , n ] . Especially in this case, all the eigenvalues of γ ( τ ) are equalto and ν γ = n + p ≥ n .4. The right equality in (6.23) holds for some ω ∈ U + \ { } (or U − \ { } ) if and only if I p ⋄ N (1 , ⋄ r ⋄ K ∈ Ω ( γ ( τ )) for some non-negative integers p and r satisfying ≤ p + r ≤ n and K ∈ Sp(2( n − p − r )) with σ ( K ) ∈ U \ { } satisfying the condition that all eigenvalues of K locatedwith the arc between and ω including U + \ { } (or U − \ { } )possess total multiplicity n − p − r . If ω = − , all eigenvalues of K are in U \ R and those in U + \ R (or U − \ R ) are all Krein-positive(or Krein-negative) definite. If ω = − , it holds that ( − I s ) ⋄ N ( − , ⋄ t ⋄ H ∈ Ω ( γ ( τ )) for somenon-negative integers s and t satisfying ≤ s + t ≤ n − p − r , and some H ∈ Sp(2( n − p − q − r − t )) satisfying σ ( H ) ⊂ U \ R and that all elements in σ ( H ) ∩ U + (or σ ( H ) ∩ U − ) are all Krein-positive(or Krein-negative) definite.5. The right equality of (6.23) holds for all ω ∈ U \ { } if and only if I p ⋄ N (1 , ⋄ ( n − p ) ∈ Ω ( γ ( τ )) for some integer p ∈ [0 , n ] . Especially in this case, all the eigenvalues of γ ( τ ) are equalto and ν γ = n + p ≥ n .6. Both equalities of (6.23) holds for all ω ∈ U \ { } if and only if γ ( τ ) = I n . Acknowledgements Part of the work was finished during the author’s visit at University ofMichgan, he sincerely thanks Professor Yongbin Ruan for his invitation and the Department ofMathematics of University of Michigan for its hospitality. References [1] A. Ambrosetti, V. Benci, Y. Long, A note on the existence of multiple brake orbits. NonlinearAnal. T. M. A. , 21 (1993) 643-649.[2] A. Ambrosetti and V. Coti Zelati, Solutions with minimal period for Hamiltonian systems ina potential well. Ann. I. H. P. Anal. non lin ´ e aire Math. Ann. 255 (1981), 405-421.[4] T. An and Y. Long, Index theories of second order Hamiltonian systems. Nonlinear Anal. Ann. I. H. P. Analyse Nonl. 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