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 Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2010 Article ID 470375 19 pages doi 2010 470375 Research Article Existence of Homoclinic Solutions for a Class of Nonlinear Difference Equations Peng Chen and X. H. Tang School of Mathematical Sciences and Computing Technology Central South University Changsha Hunan 410083 China Correspondence should be addressed to X. H. Tang tangxh@ Received 5 May 2010 Accepted 2 August 2010 Academic Editor Jianshe Yu Copyright 2010 P. Chen and X. H. Tang. 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. By using the critical point theory we establish some existence criteria to guarantee that the nonlinear difference equation A p n Ax n - 1 fi - q n x nf S f n x n has at least one homoclinic solution where n e Z x n e R and f Z X R R is non periodic in n. Our conditions on the nonlinear term f n x n are rather relaxed and we generalize some existing results in the literature. 1. Introduction Consider the nonlinear difference equation of the form A p n Au n - 1 5j - q n x n 6 f n u n n e Z where A is the forward difference operator defined by Au n u n 1 - u n A2u n A Au n Ỗ 0 is the ratio of odd positive integers p n j and q n j are real sequences p n j 0. f Z X R R. As usual we say that a solution u n of is homoclinic to 0 if u n 0 as n x. In addition if u n f 0 then ufn is called a nontrivial homoclinic solution. Difference equations have attracted the interest of many researchers in the past twenty years since they provided a natural description of several discrete models. Such discrete models are often investigated in various fields of science and technology such as computer science economics neural network ecology cybernetics biological systems optimal control and population dynamics. These studies cover many of the
