Periodic And Subharmonic Solutions Of A Class Of Superquadratic Hamiltonian Systems

Consider the Hamiltonianwhere H : 2n - K is a C1 function, and T-periodic in the second variable, i.e.(0.1) can be written more concisely aswhereis the standard symplectic matrix, In is the indentify matrix inis the gradient of H(z, t) with respect to z = (x1, ..., x2n).Periodic solutions and infinite distinct subharmonic solutions(i.e. kT periodic solutions, ) are obtained for nonconvex and nonautonomous Hamiltonian system (0.1). we haveTheorem 1 Supposing that H satisfies(H0) H is a class C1, andH(z,t + T) = H(z,t) (H1) H(z,t) 0 'Supported 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.C(H2) < 1, and 0 >0 such that(H3) > 0, > 0 such that(H4) 3 such thatthen (0.1) has a nontrivial T-periodic solution.Theorem 2 Supposing that H satisfies (H0)-(H2) and (H3)(H5) > 0, 1 > 0 such thatthen (0.1) possesses infinite distinct subharmonic solutions.