Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations
© Yu Wu et al. 2010
Received: 7 September 2009
Accepted: 14 January 2010
Published: 21 January 2010
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.
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  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
(C4) all are continuous and nondecreasing functions on and are positive on . They satisfy the relationship where means that is nondecreasing on (see ).
is defined by (2.3) and when all satisfy . Different choices of in do not affect our results (see ).
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 .
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.
Equation (2.26) becomes
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.
Directly using Corollary 3.1, we have the following result.
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 . Equation (3.5) yields the same estimates of Theorem 4.2.4 in the book .
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
which implies (4.3).
This paper was supported by Guangdong Provincial natural science Foundation (07301595). The authors would like to thank Professor Boling Guo for his great help.
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