Nonlinear Delay Discrete Inequalities and Their Applications to Volterra Type Difference Equations

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

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:

(1.1)

In this paper, we consider a delay discrete inequality

(1.2)

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

Assume that

• (C₁) is nonnegative for and ;

• (C₂) are nondecreasing for , the range of each belongs to , and ;

• (C₃) all are nonnegative for ;

• (C₄) 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.

Suppose that ()–() hold and is a nonnegative function for satisfying (1.2). Then

(2.1)

where , , is determined recursively by

(2.2)

, , (Identity), and is the largest positive integer such that

(2.3)

Remark 2.2.

• (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:

(2.4)

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.

By the definitions of and , it is easy to check that they are nonnegative and nondecreasing in , and and for each fixed where . in () implies that for all . Clearly,

(2.5)

where is used, which yields that and are nondecreasing in . Assume that is nondecreasing in . Then

(2.6)

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.

Take any arbitrary positive integer and consider the auxiliary inequality

(2.7)

Claim that in (2.7) satisfies

(2.8)

for where is the largest positive integer such that

(2.9)

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 ().

Let for and . It is clear that is nonnegative and nondecreasing. Observe that (2.7) is equivalent to for and by assumptions () and () and Lemma 2.3,

(2.10)

Since is nondecreasing and , we have

(2.11)

Then

(2.12)

and so

(2.13)

The definition of in Theorem 2.1 and show

(2.14)

Equation (2.9) shows that the right side of (2.14) is in the domain of for all . Thus the monotonicity of implies

(2.15)

for ; that is, (2.8) is true for .

Step 2 ().

Assume that (2.8) is true for . Consider

(2.16)

Let and . Then is nonnegative and nondecreasing and satisfies for . Moreover, we have

(2.17)

Since and are nondecreasing in their arguments and , we have by the assumption

(2.18)

for where for , which gives

(2.19)

Therefore,

(2.20)

that is,

(2.21)

or equivalently

(2.22)

the same as (2.7) for where and

(2.23)

From the assumption (), each , , is continuous and nondecreasing on and is positive on since is continuous and nondecreasing on . Moreover, . By the inductive assumption, we have

(2.24)

for where , , (Identity), , is the inverse of , , ,

(2.25)

i and is the largest positive integer such that

(2.26)

Note that

(2.27)

Thus, we have from (2.24) that

(2.28)

for since .

In the following, we prove that by induction again.

It is clear that for . Suppose that for . We have

(2.29)

where is applied. It implies that it is true for . Thus, for .

Equation (2.26) becomes

(2.30)

for . It implies that . Thus, (2.28) becomes

(2.31)

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

(2.32)

Since (2.8) is true for any , we replace by and get

(2.33)

This is exactly (2.1) since . This proves Theorem 2.1.

Remark 2.4.

If for all , then . Let where is given in . Using the same arguments as in (2.11) where is replaced with the positive , we have and (2.14) becomes

(2.34)

that is,

(2.35)

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

(3.1)

Corollary 3.1.

Suppose that )–() hold. Ifin(3.1)is nonnegative for, then

(3.2)

for where , is the inverse of , , , , and other related functions are defined as in Theorem 2.1 by replacing with .

Proof.

Let . Then (3.1) becomes

(3.3)

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

(3.4)

Directly using Corollary 3.1, we have the following result.

Corollary 3.2.

Suppose that )–() hold. If in (3.4) is nonnegative for, then

(3.5)

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

(4.1)

where is an unknown function, maps from to , and map from to , and satisfies the assumption () for .

Theorem.

Suppose that and the functions and in (4.1) satisfy the conditions

(4.2)

where . If is a solution of (4.1) on , then

(4.3)

where

(4.4)

Proof.

Using (4.1) and (4.2), the solution satisfies

(4.5)

where

(4.6)

Clearly, for all since . For positive constants , we have

(4.7)

It is obvious that and satisfy (). Applying Theorem 2.1 gives

(4.8)

which implies (4.3).

Theorem.

Suppose that and the functions and in (4.1) satisfy the conditions

(4.9)

where . Then (4.1) has at most one solution on .

Proof.

Let and be two solutions of (4.1) on . From (4.9), we have

(4.10)

where , , and . Appling Theorem 2.1, Remark 2.4, and the notation for , we obtain that which implies that the solution is unique.