Multiplicity of kT-periodic solutions near a given T-periodic solution for nonlinear Hamiltonian systems (Q803625)

The paper considers the question of existence and multiplicity of subharmonic solutions of nonlinear Hamiltonian systems near an equilibrium point. Under certain conditions on the linearized part of the equation at 0 and on the nonlinear part the author proves a theorem on the existence and multiplicity of subharmonic solutions having prescribed period in a prescribed neighborhood of the origin, i.e., solutions with period kT, where T is the period of the Hamiltonian function. The hypotheses of the problem are more general than the ones of a previous work of the same author. The existence is proved by a well-known variational principle whereas the proof of the multiplicity follows by an application of an abstract critical point theorem due to Tarantello, taking advantage of a \({\mathbb{Z}}_ k\)-symmetry intrinsic in the problem of the search for k- periodic solutions.