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Discrete Dynamics in Nature and Society
Volume 2009, Article ID 785068, 28 pages
Research Article

Existence of Solutions for a Nonlinear Algebraic System

1School of Science, Tianjin University of Commerce, Tianjin 300134, China
2School of Science, Qingdao Technological University, Fushun Road 11, Qingdao 266033, China

Received 30 April 2009; Accepted 15 August 2009

Academic Editor: Binggen Zhang

Copyright © 2009 Guang Zhang and Liang Bai. 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.


As well known, the existence and nonexistence of solutions for nonlinear algebraic systems are very important since they can provide the necessary information on limiting behaviors of many dynamic systems, such as the discrete reaction-diffusion equations, the coupled map lattices, the compartmental systems, the strongly damped lattice systems, the complex dynamical networks, the discrete-time recurrent neural networks, and the discrete Turing models. In this paper, both the existence of nonzero solution pairs and the nonexistence of nontrivial or nonzero solutions for a nonlinear algebraic system will be considered by using the critical point theory and Lusternik-Schnirelmann category theory. The process of proofs on the obtained results is simple, the conditions of theorems are also easy to be verified, however, some of them improve the known ones even if the system is reduced to the precial cases, in particular, others of them are still new.