Advances in Difference Equations
Volume 2008 (2008), Article ID 695495, 12 pages
A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations
Department of Mathematics, Hechi College, Guangxi, Yizhou 546300, China
Received 2 October 2007; Accepted 29 January 2008
Academic Editor: Rigoberto Medina
Copyright © 2008 Wu-Sheng Wang. 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 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 by Cheung and Ren (2006). Furthermore, we apply our result to a boundary value problem of a partial difference equation for boundedness, uniqueness, and continuous dependence.