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Báo cáo hóa học: " Research Article Periodic Solutions for Subquadratic 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 Periodic Solutions for Subquadratic Discrete Hamiltonian Systems | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 13916 16 pages doi 10.1155 2007 13916 Research Article Periodic Solutions for Subquadratic Discrete Hamiltonian Systems Xiaoqing Deng Received 4 February 2007 Accepted 26 April 2007 Recommended by Ondrej Dosly Some existence conditions of periodic solutions are obtained for a class of nonautono-mous subquadratic first-order discrete Hamiltonian systems by the minimax methods in the critical point theory. Copyright 2007 Xiaoqing Deng. 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 and statement of main results We denote N Z R C by the set of all natural numbers integers real and complex numbers respectively. For a b e Z define Z a a a 1 . Z a b a a 1 . b when a b. Consider the nonautonomous first-order discrete Hamiltonian systems JAx n VH n Lx n 0 n e Z 1.1 where J N -0N x n X1 n xi n e RN i 1 2 N is a given positive integer and In denotes the N X N identity matrix Ax n x n 1 - x n Lx n xx n 1 for all n e Z and H e C1 Z X R2N R . For a given integer T 0 we suppose that H n T z H n z for all n e Z and z e R2N and VH n z denotes the gradient of H n z in z e R2N. Our purpose is to establish the existence of T-periodic solutions of 1.1 where H is subquadratic. 2 Advances in Difference Equations Let H n Lx n H n x1 n 1 x2 n 1 2 x1 n 1 2 V n x2 n with xfin 1 Ax2 n where V e C1 Z X RN R is T-periodic in n and VV n z denotes the gradient of V n z in z e RN. Then from 1.1 we obtain A2x2 n - 1 VV n x2 n 0 n e Z x2 n e RN. 1.2 As the author knows in the past two decades there has been a large number of papers devoted to the existence of periodic and subharmonic solutions for subquadratic first-order see 1-3 or second-order see 4-8 continuous Hamiltonian systems by using the critical point theory. On the other hand