Abstract
We investigate the existence of periodic solutions of linear Hamiltonian systems with a nonlinear perturbation. Under generalized Ahmad-Lazer-Paul type coercive conditions for the nonlinearity on the kernel of the linear part, existence of periodic solutions is obtained by saddle point theorems. A note on a result of Rabinowitz is also given.
Highlights
For the second-order Hamiltonian system ut ∇F t, u t 0, u 0 − u T u 0 − u T 0, 1.1 the existence of periodic solutions is related to the coercive conditions of F t, u on u
For some recent developments of the secondorder systems 1.1, see 9. We use this kind of condition to consider the existence of periodic solutions of first-order linear Hamiltonian system with a nonlinear perturbation u JA t u J∇G u, t, 1.5 where A t is a symmetric 2π-periodic 2N × 2N continuous matrix function, G u, t ∈ C1 R2N × R, R is 2π-periodic for t, and J is the standard symplectic matrix
It is clear that conditions 1 and 2 in Theorem 2.1 hold
Summary
For the second-order Hamiltonian system ut ∇F t, u t 0, u 0 − u T u 0 − u T 0, 1.1 the existence of periodic solutions is related to the coercive conditions of F t, u on u. We use this kind of condition to consider the existence of periodic solutions of first-order linear Hamiltonian system with a nonlinear perturbation u JA t u J∇G u, t , 1.5 where A t is a symmetric 2π-periodic 2N × 2N continuous matrix function, G u, t ∈ C1 R2N × R, R is 2π-periodic for t, and J is the standard symplectic matrix. The 2π-periodic solutions of the problem correspond to the critical points of the functional. Assume that the linear problem 1.14 has only the trivial 2π-periodic solution u 0 and the condition G2 holds. For the use of Morse theory, more regularity restrictions than those in the above theorems about G t, u have to be used
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