Periodic And Homoclinic Solutions Of A Class Of Hamiltonian Systems

For the first order Hamiltonian systemsand the second order Hamiltonian systemswhere H, F satisfy the following conditWith the aids of minimax theorems, existence results for periodic solutions of Hamilto-nian systems (HS1) and (HS2) are obtained when they satisfy (A).For (HS1), A(t) is a given continuous T- periodic and symmetric 2N 2N matrix H C1([0, T] R2N,R) is T-periodic in t, J = ( ) is the standard symplecticmatrix. Suppose that H satisfies the following conditions: (H2) There exist constants > 1, > A, a1 > 0, a2 > 0 and L > 0 such that(H3) There exists 8 > 0 such thatTheorem 1. Suppose that H C1([0,T] R2N, R) satisfies (A), (H1) and (H2). If 0 is an eigenvalue of -J(d/dt) + A(t) (with periodic boundary conditions) assume also (H3).1Supported by National Natural Science Foundation of China, by Major Project of Science and Technology of MOE, P.R.C and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.CThen (HS1) has at least one nontrivial T-periodic solution.For (HS2), we have similar results, suppose that F Cl(R x RN, R) is T--periodic in t, the main results are the following:Theorem 2. Suppose that F satisfies the following conditions:(F4) There exists 6 > 0 such thatThen (HS1) has at least one nontrivial T-periodic solution.Theorem 3. Suppose that F satisfies (F1)-(F3) and the following conditi (F4') There exists 8 > 0 such thatThen (HSl) has at least one nontrivial T-periodic solution. For the second order Hamiltonian systemsu(t) - B(t)u(t) + F(f , u(t)) = 0 (HS )where B(t) Cl(R, RN2) is a symmetric matrix valued function, following, for two N x N symmetric matrices S1 and S2, by S1 < S2(|S1| < |S2|). we mean that for all RN and | | = 1. Suppose that B satisfies the following conditions:(B1) For the smallest eigenvalue of B(t), i.e. b(t) = inf| |=1 B(t) , there exists a < 2 such thatwhere B'(t) = (d/dt)B(t), B"(t) = (d2 / dt2) B (t). the main result is the following:Theorem 4. Suppose that B satisfies (Bl) and (B2), F satisfies the following conditions:(F8) There exist constants 0, d2 > 0 such thatandThen (HS3) has at least one nontrivial homoclinic solution.