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Báo cáo hóa học: " Research Article Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 41830 13 pages doi 10.1155 2007 41830 Research Article Multiple Periodic Solutions to Nonlinear Discrete Hamiltonian Systems Bo Zheng Received 15 April 2007 Revised 27 June 2007 Accepted 19 August 2007 Recommended by Ondrej Dosly An existence result of multiple periodic solutions to the asymptotically linear discrete Hamiltonian systems is obtained by using the Morse index theory. Copyright 2007 Bo Zheng. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let Z and R be the sets of all integers and real numbers respectively. For a b e Z define Z a a a 1 . and Z a b a a 1 . b when a b. Let A be an n X m matrix. AT denotes the transpose of A. When n m Ơ A and det A denote the set of eigenvalues and the determinant of A respectively. In this paper we study the existence of multiple p-periodic solutions to the following discrete Hamiltonian systems Ax n JVH Lx n n e Z 1.1 where p 2 is a prime integer Ax n x n 1 - x n x n xỉ n with xi n e Rd i 1 2 L is defined by Lx n Ợỉ n Ị J i -3d is the standard symplectic matrix with Id the identity matrix on Rd H e C1 R2d R and VH z denotes the gradient of H in z. We may think of systems 1.1 as being a discrete analog of the following Hamiltonian systems x JVH x t t e R 1.2 2 Advances in Difference Equations which has been studied extensively by many scholars. For example by using the critical point theory some significant results for the existence and multiplicity of periodic and subharmonic solutions to 1.2 were obtained in 1-5 . Some authors have also contributed to the study of 1.1 for the disconjugacy boundary value problems oscillations and asymptotic behavior see for example 6-9 . In recent years existence and multiplicity results of periodic solutions to discrete .