An Extension to Nonlinear Sum-Difference Inequality and Applications
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Research Article An Extension to Nonlinear Sum-Difference Inequality and Applications Wu-Sheng Wang1 and Xiaoliang Zhou2 1 2
Department of Mathematics, Hechi University, Yizhou, Guangxi 546300, China Department of Mathematics, Guangdong Ocean University, Zhanjiang 524088, China
Correspondence should be addressed to Xiaoliang Zhou, [email protected] Received 31 March 2009; Revised 31 March 2009; Accepted 17 May 2009 Recommended by Martin J. Bohner 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. Copyright q 2009 W.-S. Wang and X. 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.
1. Introduction Gronwall-Bellman inequality 1, 2 is a fundamental tool in the study of existence, uniqueness, boundedness, stability, invariant manifolds and other qualitative properties of solutions of differential equations and integral equation. There are a lot of papers investigating them such as 3–15. Along with the development of the theory of integral inequalities and the theory of difference equations, more attentions are paid to some discrete versions of Bellman-Gronwall type inequalities e.g., 16–18. Starting from the basic form un ≤ an
n−1
fsus,
1.1
s0
discussed in 19, an interesting direction is to consider the inequality n−1 αu2 s Qgsus , u2 n ≤ P 2 u2 0 2 s0
1.2
2
Advances in Difference Equations
a discrete version of Dafermos’ inequality 20, where α, P, Q are nonnegative constants and u, g are nonnegative functions defined on {1, 2, . . . , T } and {1, 2, . . . , T − 1}, respectively. Pang and Agarwal 21 proved for 1.2 that un ≤ 1 αn P u0 n−1 s0 Qgs for all 0 ≤ n ≤ T . Another form of sum-difference inequality n−1 u2 n ≤ c2 2 f1 suswus f2 sus
1.3
s0
n−1 estimated by Pachpatte 22 as un ≤ Ω−1 Ωc n−1 s0 f2 s s0 f1 s, where Ωu : was u ds/ws. Recently, Pachpatte 23, 24 discussed the inequalities of two variables u0 um, n ≤ c
m−1 n−1
us, t as, t log us, t bs, tg log us, t ,
s0 t0
um, n ≤ c
m−1 n−1
f1 s, tgus, t
s0 t0
s0 t0
κs, t, σ, τguσ, τ
σ0 τ0
1.4
⎞⎞ ⎛ τ−1 s−1 σ−1 t−1
⎝ ⎝ h s, t, σ, τ, ξ, η g u ξ, η ⎠⎠, ⎛
s−1 t−1
m−1 n−1
m−1 n−1 s0 t0
σ0 τ0
ξ0 η0
where g is nondecreasing. In 25 a
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