Discrete Dynamics in Nature and Society
Volume 2010 (2010), Article ID 364079, 26 pages
Research Article

Existence and Multiplicity of Solutions to Discrete Conjugate Boundary Value Problems

College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, Guangdong 510006, China

Received 21 March 2010; Accepted 29 April 2010

Academic Editor: Yong Zhou

Copyright © 2010 Bo Zheng. 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.


We consider the existence and multiplicity of solutions to discrete conjugate boundary value problems. A generalized asymptotically linear condition on the nonlinearity is proposed, which includes the asymptotically linear as a special case. By classifying the linear systems, we define index functions and obtain some properties and the concrete computation formulae of index functions. Then, some new conditions on the existence and multiplicity of solutions are obtained by combining some nonlinear analysis methods, such as Leray-Schauder principle and Morse theory. Our results are new even for the case of asymptotically linear.