LOWER BOUND ON THE NUMBER OF PERIODIC SOLUTIONS FOR ASYMPTOTICALLY LINEAR PLANAR HAMILTONIAN SYSTEMS

In this work we prove the lower bound for the number of T-periodic solutions of an asymptotically linear planar Hamiltonian system. Precisely, we show that such a system, T-periodic in time, with T-Maslov indices i(0), i(infinity) at the origin and at infinity, has at least vertical bar i(infinity) - i(0) vertical bar periodic solutions, and an additional one if i(0) is even. Our argument combines the Poincare-Birkhoff Theorem with an application of topological degree. We illustrate the sharpness of our result, and extend it to the case of second orders ODEs with linear-like behaviour at zero and infinity.