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 Solutions of 2nth-Order Boundary Value Problem for Difference Equation via Variational Method | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 730484 10 pages doi 2009 730484 Research Article Solutions of 2nth-Order Boundary Value Problem for Difference Equation via Variational Method Qingrong Zou and Peixuan Weng School of Mathematics South China Normal University Guangzhou 510631 China Correspondence should be addressed to Peixuan Weng wengpx@ Received 7 July 2009 Accepted 15 October 2009 Recommended by Kanishka Perera The variational method and critical point theory are employed to investigate the existence of solutions for 2nth-order difference equation Nn pk-n nyk-n 1 n 1f fk yk 0 for k e 1 N with boundary value condition y1-n y2-n y0 0 yN 1 yN n 0 by constructing a functional which transforms the existence of solutions of the boundary value problem BVP to the existence of critical points for the functional. Some criteria for the existence of at least one solution and two solutions are established which is the generalization for BVP of the even-order difference equations. Copyright 2009 Q. Zou and P. Weng. 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 Difference equations have been applied as models in vast areas such as finance insurance biological populations disease control genetic study physical field and computer application technology. Because of their importance many literature deals with its existence and uniqueness problems. For example see 1-10 . We notice that the existing results are usually obtained by various analytical techniques for example the conical shell fixed point theorem 1 6 Banach contraction map method 7 Leray-Schauder fixed point theorem 2 10 and the upper and lower solution method 3 . It seems that the variational technique combining with the critical point theory 11 developed in the recent .
