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Báo cáo hóa học: " Research Article Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems"

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 Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems | Hindawi Publishing Corporation Boundary Value Problems Volume 2o11 Article ID 172818 19 pages doi 10.1155 2011 172818 Research Article Existence of Positive Negative and Sign-Changing Solutions to Discrete Boundary Value Problems Bo Zheng 1 Huafeng Xiao 1 and Haiping Shi2 1 School of Mathematics and Information Sciences Guangzhou University Guangzhou Guangdong 510006 China 2 Department of Basic Courses Guangdong Baiyun Institute Guangzhou Guangdong 510450 China Correspondence should be addressed to Bo Zheng zhengbo611@yahoo.com.cn Received 11 November 2010 Accepted 15 February 2011 Academic Editor Zhitao Zhang Copyright 2011 Bo Zheng et al. 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 critical point theory Lyapunov-Schmidt reduction method and characterization of the Brouwer degree of critical points sufficient conditions to guarantee the existence of five or six solutions together with their sign properties to discrete second-order two-point boundary value problem are obtained. An example is also given to demonstrate our main result. 1. Introduction Let N z and 1 denote the sets of all natural numbers integers and real numbers respectively. For a b e z define z a b a a 1 . b when a b. A is the forward difference operator defined by Au n ufn 1 - ufri A2u n A Au n . Consider the following discrete second-order two-point boundary value problem BVP for short A2u n- 1 V u n 0 n el 1 T 1.1 u 0 0 u T 1 where V e C2 R R T 1 is a given integer. By a solution u to the BVP 1.1 we mean a real sequence u n T 0 u 0 u 1 . u T 1 satisfying 1.1 . For u u n T 0 with u 0 0 u T 1 we say that u f 0 if there exists at least one n el 1 T such that ufn 0. We say that u is positive and write u 0 if for all n el 1 T u n 0 and n el 1 T u nf 0 0 and similarly 2 Boundary Value Problems u is negative u 0 if for all n l 1