A Note on the Minimal Period Problem for Second Order Hamiltonian Systems

Xiao, Huafeng

doi:10.1155/2014/385381

Cited by 2 publications

(3 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Together (23) with (24) we conclude that the claim holds. Arguing similarly to the proof of Theorem 1.1, we can prove that y 0 has T as its minimal period.…”

Section: Remark 42mentioning

confidence: 53%

“…Next, we will show that x 0 has T as its minimal period. Arguing as in [18,24,25], suppose that x 0 has minimal period T/k, where k ≥ 2 is an integer. Denote y 0 (t) = x 0 (t/k).…”

Section: Consequently |Smentioning

confidence: 99%

“…[19,20]). For more related results under the strongly global AR condition, we refer to [1,14,24] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Periodic solutions with prescribed minimal period to Hamiltonian systems

Xiao

Shen

2020

J Inequal Appl

Self Cite

View full text Add to dashboard Cite

In this article, we study the existence of periodic solutions to second order Hamiltonian systems. Our goal is twofold. When the nonlinear term satisfies a strictly monotone condition, we show that, for any $T>0$ T > 0 , there exists a T-periodic solution with minimal period T. When the nonlinear term satisfies a non-decreasing condition, using a perturbation technique, we prove a similar result. In the latter case, the periodic solution corresponds to a critical point which minimizes the variational functional on the Nehari manifold which is not homeomorphic to the unit sphere.

show abstract

“…Together (23) with (24) we conclude that the claim holds. Arguing similarly to the proof of Theorem 1.1, we can prove that y 0 has T as its minimal period.…”

Section: Remark 42mentioning

confidence: 53%

“…Next, we will show that x 0 has T as its minimal period. Arguing as in [18,24,25], suppose that x 0 has minimal period T/k, where k ≥ 2 is an integer. Denote y 0 (t) = x 0 (t/k).…”

Section: Consequently |Smentioning

confidence: 99%

See 1 more Smart Citation

Periodic solutions with prescribed minimal period to Hamiltonian systems

Xiao

Shen

2020

J Inequal Appl

Self Cite

View full text Add to dashboard Cite

show abstract

The Rabinowitz minimal periodic solution conjecture

2021

Int. J. Math.

View full text Add to dashboard Cite

In 1978, Rabinowitz proved the existence of a non-constant [Formula: see text]-periodic solution for nonlinear Hamiltonian systems on [Formula: see text] with Hamiltonian function being super-quadratic at the infinity and zero for any given [Formula: see text]. Since the minimal period of this solution may be [Formula: see text] for some positive integer [Formula: see text], he proposed the question whether there exists a solution with [Formula: see text] as its minimal period for such a Hamiltonian system. This is the so-called Rabinowitz minimal periodic solution conjecture. In the last more than 40 years, this conjecture has been deeply studied by many mathematicians. But under the original structural conditions of Rabinowitz, the conjecture is still open when [Formula: see text]. In this paper, I give a brief survey on the studies of this conjecture and hope to lead to more interests on it.

show abstract

A Note on the Minimal Period Problem for Second Order Hamiltonian Systems

Cited by 2 publications

References 17 publications

Periodic solutions with prescribed minimal period to Hamiltonian systems

Periodic solutions with prescribed minimal period to Hamiltonian systems

The Rabinowitz minimal periodic solution conjecture

Contact Info

Product

Resources

About