Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations
 Yu Wu^{1},
 Xiaopei Li^{2} and
 Shengfu Deng^{3}Email author
DOI: 10.1155/2010/795145
© Yu Wu et al. 2010
Received: 7 September 2009
Accepted: 14 January 2010
Published: 21 January 2010
Abstract
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
GronwallBellman 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:
In this paper, we consider a delay discrete inequality
which has nonlinear terms where . 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

(C_{1}) is nonnegative for and ;

(C_{2}) are nondecreasing for , the range of each belongs to , and ;

(C_{3}) all are nonnegative for ;

(C_{4}) all are continuous and nondecreasing functions on and are positive on . They satisfy the relationship where means that is nondecreasing on (see [10]).
Let for where is a given constant. Then, is strictly increasing so its inverse is well defined, continuous, and increasing in its corresponding domain. Define , and .
Theorem 2.1.
 (1)
is defined by (2.3) and when all satisfy . Different choices of in do not affect our results (see [2]).
 (2)
If for , then (2.1) gives the estimate of the following inequality:
by replacing , , , , and with , , , and , respectively. Especially, if and , then (1.2) for becomes the first inequality of (1.1). Equation (2.1) shows the same estimate given by ( ) of Theorem 4.2.3 in the book [21].
Lemma 2.3.
is nonnegative and nondecreasing in , and is nonnegative and nondecreasing in and for .
Proof.
which implies that is nondecreasing in . By induction, are nondecreasing in . Similarly, we can prove that they are nonnegative by induction again. Then are nonnegative and nondecreasing in and .
Proof of Theorem 2.1.
Before we prove (2.8), notice that . In fact, , , and are nondecreasing in by Lemma 2.3. Thus, satisfying (2.9) gets smaller as is chosen larger. In particular, satisfies the same (2.3) as for if is applied.
We divide the proof of (2.8) into two steps by using induction.
Step 1 ( ).
for ; that is, (2.8) is true for .
Step 2 ( ).
for since .
In the following, we prove that by induction again.
It is clear that for . Suppose that for . We have
where is applied. It implies that it is true for . Thus, for .
Equation (2.26) becomes
for . It shows that (2.8) is true for . Thus, the claim is proved.
Now we prove (2.1). Replacing by in (2.8), we have
This is exactly (2.1) since . This proves Theorem 2.1.
Remark 2.4.
which is the same as (2.15) with a complementary definition that . From ( ) of Remark 2.2, the estimate of (2.35) is independent of . Then we similarly obtain (2.1) and all are defined by the same formula (2.2) where we define for .
3. Some Corollaries
In this section, we apply Theorem 2.1 and obtain some corollaries.
Assume that is a strictly increasing function with where . Consider the inequality
Corollary 3.1.
for where , is the inverse of , , , , and other related functions are defined as in Theorem 2.1 by replacing with .
Proof.
Note that satisfy ( ) for . Using Theorem 2.1, we obtain the estimate about by replacing with . Then use the fact that and we get Corollary 3.1.
If where , then (3.1) reads
Directly using Corollary 3.1, we have the following result.
Corollary 3.2.
for where , is the inverse of , , , , and other related functions are defined as in Theorem 2.1 by replacing with .
If , , , (3.4) becomes the second inequality of (1.1) with and , and the third inequality of (1.1) with and , which are discussed in the book [21]. Equation (3.5) yields the same estimates of Theorem 4.2.4 in the book [21].
4. Applications to Volterra Type Difference Equations
In this section, we apply Theorem 2.1 to study boundedness and uniqueness of solutions of a nonlinear delay difference equation of the form
where is an unknown function, maps from to , and map from to , and satisfies the assumption ( ) for .
Theorem.
Proof.
which implies (4.3).
Theorem.
where . Then (4.1) has at most one solution on .
Proof.
where , , and . Appling Theorem 2.1, Remark 2.4, and the notation for , we obtain that which implies that the solution is unique.
Declarations
Acknowledgments
This paper was supported by Guangdong Provincial natural science Foundation (07301595). The authors would like to thank Professor Boling Guo for his great help.
Authors’ Affiliations
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