Minimal P-cyclic periodic brake orbits in semi-positive Hamiltonian system

In this paper, the authors study the minimal period problems for brake orbits in two types of first order Hamiltonian systems in $R^{2n}$, namely the superquadratic type and asymptotically linear type. In both cases the Hamiltonian systems are assumed to be reversible, semi-positive, and symmetric with respect to certain orthogonal symplectic linear transformation P generating a p-order cyclic subgroup acting freely on $R^{2n}\setminus \left\{0\right\}$. The authors prove that if $P=R\left({\theta }_1\right)\diamond \dots \diamond R\left({\theta }_n\right)$ and ${\theta }_i\in [0,\pi ]$ for each $1\le i\le n$, then for each $T>0$ there exists a pT-periodic P-cyclic brake orbit with minimal period belonging to an finite set with the form$\{\frac{pT}{lp+q}:l,q\in Z,0\le l\le n+1,\gcd (q,p)=1,1\le q\le p-1\}$ for both cases, which is an generalization of the results in [10]. The main tools involved are the iteration inequalities for Maslov-type indices, the saddle point reduction method and the Galerkin approximation method under the corresponding Lagrangian boundary condition.