Periodic solutions for discrete convex Hamiltonian systems via Clarke duality. (English) Zbl 1121.39012

Yu, Jianshe; Bin, Honghua; Guo, Zhiming

Discrete Contin. Dyn. Syst. 15, No. 3, 939-950 (2006).

Applying Clarke duality and the perturbation technique, the authors prove the existence of a periodic solution for a certain class of periodic Hamiltonian systems, namely: \[ J\Delta u(n)+\nabla H(n, Lu(n))= f(n), \] where \[ u(n)= \begin{pmatrix} u_1(n)\\ u_2(n)\end{pmatrix},\quad Lu(n)= \begin{pmatrix} u_1(n+1)\\ u_2(n)\end{pmatrix},\quad f(n)= \begin{pmatrix} f_1(n)\\ f_2(n)\end{pmatrix} \] are in \(\mathbb{R}^{2N}\) with \(N\) a given positive integer, \(\Delta u(n)= u(n+ 1)- u(n)\) and \(u(n+ T)= u(n)\) for \(n\in\mathbb{Z}\) and \(T\) a fixed positive integer.

Reviewer: Costică Moroşanu (Iaşi)

MSC:

39A11	Stability of difference equations (MSC2000)
39A12	Discrete version of topics in analysis

Keywords:

Clarke duality; perturbation technique; dual least action principle; discrete Hamiltonian systems; periodic solution; convex