Periodic Solutions of Second-Order Differential Equations in Hilbert Spaces

Abstract

We prove the existence of periodic solutions of some infinite-dimensional systems by the use of the lower/upper solutions method. Both the well-ordered and non-well-ordered cases are treated, thus generalizing to systems some well-established results for scalar equations.

On the Dirichlet problem associated with bounded perturbations of positively-(p, q)- homogeneous Hamiltonian systems

“…] , giving easily, passing to the complementary, Ω μ κ,1 \ Ω μ κ,4 = Ω μ κ,2 ∪ Ω μ κ,3 and consequently Ω μ κ,4 = Ω μ κ,1 \ Ω μ κ,2 ∪ Ω μ κ,3 . Arguing as in [11,, we can prove that deg I − L −1 N d , Ω μ is well defined for every μ ∈ {1, 2, 3, 4} N −M , and it is equal to (−1) m , where m is the number of times the number 4 appears in the multi-index μ. In particular, we have that deg I − L −1 N d , Ω (4,4,...,4,4) = (−1) N −M = 0 .…”

“…For the non-well-ordered case, we need to introduce the notion of strict lower and upper solutions. To this aim we will follow the ideas developed in [11], and distinguish the components which are well ordered from the others.…”

“…, N}, we will mainly follow the procedure developed in Section 5 if n ∈ J , and the one in Section 6 if n ∈ K, so that the couple of variables (x n , y n ) will overshadow the remaining ones, which will essentially act as parameters. We first need to modify problem (53) both in the kvariables, following the lines of the proof of Theorem 3, and in the j-variables, to apply a topological degree argument, as in the proof of [11,Theorem 10].…”

“…Remark 30. A generalization of Theorem 28 can be obtained removing the strictness assumption on one of the components κ ∈ K. Indeed, the above proof can be easily adapted following the ideas in the proof of [11,Theorem 19]. In such a situation, the conclusion (K) of the statement must be slightly changed concerning the estimates on the κth component: the solution will be such that α κ / x κ and x κ / β κ .…”

Unbounded solutions to systems of differential equations at resonance

“…For some recent advances on the topic of ODEs in Hilbert spaces, see [4,11,12] and the references therein.…”