Tham khảo luận văn - đề án 'báo cáo hóa học: " research article existence theorems of periodic solutions for second-order nonlinear difference equations"', luận văn - báo cáo phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Hindawi Publishing Corporation Advances in Difference Equations Volume 2008 Article ID 247071 11 pages doi 2008 247071 Research Article Existence Theorems of Periodic Solutions for Second-Order Nonlinear Difference Equations Xiaochun Cai1 and Jianshe Yu2 1 College of Statistics Hunan University Changsha Hunan 410079 China 2 College of Mathematics and Information Science Guangzhou University Guangzhou 510405 China Correspondence should be addressed to Xiaochun Cai cxchn8@ Received 14 August 2007 Accepted 14 November 2007 Recommended by Patricia J. Y. Wong The authors consider the second-order nonlinear difference equation of the type A pn Axn-i ổ qnx5n f n xn n e Z using critical point theory and they obtain some new results on the existence of periodic solutions. Copyright 2008 X. Cai and J. Yu. 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 We denote by N Z R the set of all natural numbers integers and real 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 nonlinear second-order difference equation A pn Axn-1 ổ q xf f n Xn n e Z where the forward difference operator A is defined by the equation Axn xn i - xn and A2xn-1 A Axn-1 Axn - Axn-1. In the given real sequences pn qn satisfy pn T pn 0 qn T qn for any n e Z f Z X R R is continuous in the second variable and f n T z f n z for a given positive integer T and for all n z e Z X R. -1 ổ -1 Ổ 0 and Ổ is the ratio of odd positive integers. By a solution of we mean a real sequence x xn n e Z satisfying . In 1 2 the qualitative behavior of linear difference equations of type A pnAxn qnxn 0 2 Advances in Difference Equations has been investigated. In 3 the nonlinear difference equation A pn AXn-1 qnXn f n Xn has been considered. However results on periodic .
