Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations
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Research Article Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations Yu Wu,1 Xiaopei Li,2 and Shengfu Deng3 1
Yibin University, Yibin, Sichuan 644007, China Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524048, China 3 Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing 100088, China 2
Correspondence should be addressed to Shengfu Deng, sf [email protected] Received 7 September 2009; Accepted 14 January 2010 Academic Editor: Abdelkader Boucherif Copyright q 2010 Yu Wu et al. 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. Delay discrete inequalities with more than one nonlinear term are discussed, which generalize some known results and can be used in the analysis of various problems in the theory of certain classes of discrete equations. Application examples to show boundedness and uniqueness of solutions of a Volterra type difference equation are also given.
1. Introduction Gronwall-Bellman inequalities and their various linear and nonlinear generalizations play very important roles in the discussion of existence, uniqueness, continuation, boundedness, and stability properties of solutions of differential equations and difference equations. The literature on such inequalities and their applications is vast. For example, see 1–12 for continuous cases, and 13–20 for discrete cases. In particular, the book 21 written by Pachpatte considered three types of discrete inequalities: un ≤ an
n−1 fswus, s0
n−1 u2 n ≤ an 2 fsus, s0
u2 n ≤ an
n−1 s0
fswus.
1.1
2
Advances in Difference Equations
In this paper, we consider a delay discrete inequality un ≤ an
n−1 m bi fi n, swi us,
n ∈ N0
1.2
i1 sbi 0
which has m nonlinear terms where N0 {0, 1, 2, . . .}. We will show that many discrete inequalities like 1.1 can be reduced to this form. Our main result can be applied to analyze properties of solutions of discrete equations. We also give examples to show boundedness and uniqueness of solutions of a Volterra type difference equation.
2. Main Results Assume that C1 an is nonnegative for n ∈ N0 and a0 > 0; C2 bi n i 1, . . . , m are nondecreasing for n ∈ N0 , the range of each bi belongs to N0 , and bi n ≤ n; C3 all fi n, j i 1, . . . , m are nonnegative for n, j ∈ N0 ; C4 all wi i 1, . . . , m are continuous and nondecreasing functions on 0, ∞ and are positive on 0, ∞. They satisfy the relationship w1 ∝ w2 ∝ · · · ∝ wm where wi ∝ wi1 means that wi1 /wi is nondecreasing on 0, ∞ see 10. u Let Wi u ui dz/wi z for u ≥ ui where ui > 0 is a given constant. Then, Wi is strictly increasing so its inverse Wi−1 is well defined, continuous, and increasing in its corresponding domain. Define bi −1 −1, Δun un 1 − un a
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