Advances in Difference Equations
Volume 2009 (2009), Article ID 486895, 17 pages
Research Article

An Extension to Nonlinear Sum-Difference Inequality and Applications

1Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China
2Department of Mathematics, Guangdong Ocean University, Zhanjiang 524088, China

Received 31 March 2009; Revised 31 March 2009; Accepted 17 May 2009

Academic Editor: Martin J. Bohner

Copyright © 2009 Wu-Sheng Wang and Xiaoliang 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.


We establish a general form of sum-difference inequality in two variables, which includes both more than two distinct nonlinear sums without an assumption of monotonicity and a nonconstant term outside the sums. We employ a technique of monotonization and use a property of stronger monotonicity to give an estimate for the unknown function. Our result enables us to solve those discrete inequalities considered in the work of W.-S. Cheung (2006). Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.