Periodic solutions for a class of nonautonomous second order Hamiltonian systems

Abstract

In this paper, some existence theorems of periodic solutions of a class of the nonautonomous second order Hamiltonian systemsare obtained by a mountain pass theorem and a local link theorem.

Periodic solutions for Hamiltonian systems without Ambrosetti–Rabinowitz condition and spectrum 0

“…By using the linking theorem, Fei [6] proved an existence theorem of (1.1) with A(t) = 0 for all t ∈ [0, T ] and a superquadratic potential function H ∈ C 1 ([0, T ], R N ), where H does not satisfy the AR-condition. He and Wu [11], Luan and Mao [16] and Tao, Yan and Wu [26] studied (1.1) with A(t) ≡ 0 by the local linking theorem (see [15]), and employed additional restrictive conditions, such as the positivity of the potential function H , superquadratic behavior near the origin or additional growth restrictions near infinity (see, for example, hypothesis H 3 in [16]). We should also mention that some authors studied systems with nonsmooth, locally Lipschitz potentials (hemivariational inequalities), see [4,18,19].…”

New existence of periodic solutions for second order non-autonomous Hamiltonian systems

“…In 2008, He and Wu [16] had obtained some results of the nontrivial T -periodic solutions for systems (1.1) under much weaker assumptions, which greatly generalized the corresponding results in [5]. More precisely, they established the following two main theorems.…”

“…There are functions that satisfy our theorems but not satisfy those results in [5,16]. For example, let…”

“…Motivated by the ideas of [5,7,16,24], we will use the following conditions to generalize the results of [16] in another direction.…”

Multiple Periodic Solutions for Perturbed Second-Order Impulsive Hamiltonian Systems

“…Background information and applications of Hamiltonian systems can be found for example in [16,28,31,37]. The monographs [29,32] have inspired a great deal of work on the existence and multiplicity of periodic solutions for Hamiltonian systems using variational techniques; for example, see [9,10,11,13,14,15,18,19,24,25,26,36,38,40,42,43,45] and the references therein.…”