tailieunhanh - Báo cáo hóa học: " Research Article Unbounded Perturbations of Nonlinear Second-Order Difference Equations at Resonance Ruyun Ma"

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 Unbounded Perturbations of Nonlinear Second-Order Difference Equations at Resonance Ruyun Ma | Hindawi Publishing Corporation Advances in Difference Equations Volume 2007 Article ID 96415 12 pages doi 2007 96415 Research Article Unbounded Perturbations of Nonlinear Second-Order Difference Equations at Resonance Ruyun Ma Received 19 March 2007 Accepted 30 May 2007 Recommended by Johnny L. Henderson We study the existence of solutions of nonlinear discrete boundary value problems A2u t 1 p1u t g t u t h t t e T u a u b 2 0 where T a 1 . b 1 h T - R P1 is the first eigenvalue of the linear problem A2u t 1 pu t 0 t e T u a u b 2 0 g T X R - R satisfies some asymptotic nonuniform resonance conditions and g t u u 0 for u e R. Copyright 2007 Ruyun Ma. 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 Let a b e N be two integers with b a 2. Let T a 1 . b 1 and T a a 1 . b 1 b 2 . Definition . Suppose that a function y T - R. If y t 0 then t is a zero of y. If y t 0 and Ay t 0 then t is a simple zero of y. If y t y t 1 0 then y has a node at the point s ty t 1 t 1 y t y t 1 y t e t t 1 . The nodes and simple zeros of y are called the simple generalized zeros of y . Let p is a real parameter. It is well known that the linear eigenvalue problem A2y t 1 py t 0 t e T u a u b 2 0 has exactly N b a 1 eigenvalues P1 P2 PN 2 Advances in Difference Equations which are real and the eigenspace corresponding to any such eigenvalue is one dimensional. The following lemma is crucial to the study of nonlinear perturbations of the linear problem . The required results are somewhat scattered in 1 Chapters 6-7 . Lemma 1 . Let pi yi i e 1 . N denote eigenvalue pairs of with 6 1 y Vj t Vj t 1 j e 1 . N . t a 1 Then 1 yi has i - 1 simple generalized zeros in a 1 b 1 also if j k then 6 1 y Vj t yk t 0 t a 1 2 if h a 1 . b 1 R is given then the problem A2u t - 1 Ầ1 u t h t t e