Báo cáo hóa học: "Research Article On the Nonexistence and Existence of Solutions for a Fourth-Order Discrete Boundary Value Problem"

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 On the Nonexistence and Existence of Solutions for a Fourth-Order Discrete Boundary Value Problem | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 389624 18 pages doi 2009 389624 Research Article On the Nonexistence and Existence of Solutions for a Fourth-Order Discrete Boundary Value Problem Shenghuai Huang and Zhan Zhou School of Mathematics and Information Science Guangzhou University Guangzhou Guangdong 510006 China Correspondence should be addressed to Zhan Zhou zzhou0321@ Received 16 July 2009 Accepted 16 October 2009 Recommended by Patricia J. Y. Wong By using the critical point theory we establish various sets of sufficient conditions on the nonexistence and existence of solutions for the boundary value problems of a class of fourth-order difference equations. Copyright 2009 S. Huang and Z. Zhou. 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 In this paper 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 following boundary value problem BVP A2 p n - 1 A2u n - 2 1 A q n Au n - 1 f n ufnỴ n e Z 1 k u -1 u 0 0 u k 1 u k 2 . Here k e N p n is nonzero and real valued for each n e Z 0 k 1 q ri is real valued for each n e Z 1 k 1 . f n u is real-valued for each n u e Z 1 k X R and continuous in the second variable u. A is the forward difference operator defined by Au n ufn 1 - u n and A2u n A Au n . We may think of as being a discrete analogue of the following boundary value problem p t x t q t x t f t x t t e a b x a x a 0 x b x b 0 2 Advances in Difference Equations which are used to describe the bending of an elastic beam see for example 1-10 and references therein. Owing to its importance in physics many methods are applied to study fourth-order boundary value problems by many authors. For example fixed

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