Volterra Discrete Inequalities of Bernoulli Type

Abstract

We obtain the discrete versions of integral inequalities of Bernoulli type obtained in Choi (2007) and give an application to study the boundedness of solutions of nonlinear Volterra difference equations.

1. Introduction

Integral inequalities of Gronwall type have been very useful in the study of ordinary differential equations. Sugiyama [1] proved the discrete analogue of the well-known Gronwall-Bellman inequality [2–5] which find numerous applications in the theory of finite difference equations. See [6–11] for differential inequalities and difference inequalities.

Willett and Wong [12] established some discrete generalizations of the results of Gronwall [5]. The discrete analogue of the result of Bihari [13] was partially given by Hull and Luxemburg [14] and was used by them for the numerical treatment of ordinary differential equations. Pachpatte [15] obtained some general versions of Gronwall-Bellman inequality. Oguntuase [16] established some generalizations of the inequalities obtained in [15]. However, there were some defects in the proofs of Theorems  2.1 and 2.7 in [16]. Choi et al. [17] improved the results of [16] and gave an application to boundedness of the solutions of nonlinear integrodifferential equations.

In this paper, we establish the discrete analogues of integral inequalities of Bernoulli type in [17] and give an application to study the boundedness of solutions of nonlinear Volterra difference equations.

2. Main Results

Pachpatte [11] proved the following useful discrete inequality which can be used in the proof of various discrete inequalities. Let and for fixed nonnegative integers and .

Lemma 2.1 (see [11, Theorem  2.3.4]).

Let be a positive sequence defined on , and let and be nonnegative sequences defined on . Suppose that

(2.1)

where , is a constant. Then, one has

(2.2)

where and

(2.3)

If we set in Lemma 2.1, then we can obtain the following corollary.

Corollary 2.2.

Suppose that

(2.4)

Then,

(2.5)

where

(2.6)

Willet and Wong [12, Theorem  4] proved the nonlinear difference inequality by using the mean value theorem. We obtain the following result which is slightly different from Willet and Wong's Theorem  4.

Theorem 2.3.

Let be nonnegative sequences defined on and for each . Suppose that

(2.7)

where , and is a positive constant. Then one has

(2.8)

where

(2.9)

Proof.

Let the right hand of (2.7) denote by

(2.10)

Then, we have

(2.11)

since is nondecreasing. Multiplying (2.11) by the factor , we obtain

(2.12)

since is a positive sequence on . From Corollary 2.2, we obtain

(2.13)

where

(2.14)

Since and , this implies that our inequality holds.

Remark 2.4.

Note that (2.7) with in Theorem 2.3 implies

(2.15)

Hence, we can obtain a comparison result for linear difference inequalities.

The following theorem can be regarded as an extension of the inequality given by Willett and Wong in [12] which is the discrete analogue of the inequality given by Choi et al. in [17, Theorem  2.7].

Theorem 2.5.

Let and be nonnegative sequences defined on , and let be a nonnegative function for with . Suppose that

(2.16)

where is a positive constant and , is a constant. Then, one has

(2.17)

where , , , and

(2.18)

Proof.

Define by the right member of (2.16). Then,

(2.19)

by and for . Letting

(2.20)

we obtain

(2.21)

for each . By Theorem 2.3, we have

(2.22)

where and

(2.23)

Substituting (2.22) into (2.19) and then summing it from to , we have

(2.24)

where . Hence, the proof is complete.

Remark 2.6.

We suppose further that is a nonnegative function for with in Theorem 2.5. Then, we have

(2.25)

where

(2.26)

If we set in Theorem 2.5, then we obtain the following corollary from Theorem 2.5. This is an analogue of the nonlinear difference inequality in [17, Corollary  2.8].

Corollary 2.7.

Let be nonnegative sequences defined on , and let be a positive constant. Suppose that

(2.27)

where , is a constant. Then, one has

(2.28)

where , , and

(2.29)

If we use Lemma 2.1 in the proof of Theorem 2.5, then we obtain the following bound of which contains double fold summations.

Corollary 2.8.

Let and be nonnegative sequences defined on , and let be a nonnegative function for with . Suppose that

(2.30)

where is a positive constant, and , is a constant. Then, one has

(2.31)

where and

(2.32)

Proof.

Define by the right member of (2.30). Then, we have

(2.33)

since and is nondecreasing in . From Lemma 2.1, we obtain

(2.34)

where and

(2.35)

Since , the proof is complete.

If we set in Theorem 2.5, then we can obtain the following discrete analogue of Theorem  2.2 in [17] which improve in [16, Theorem  2.1].

Corollary 2.9.

Let and be nonnegative sequences defined on and be a nonnegative function for each with . Suppose that

(2.36)

where is a positive constant. Then, one has

(2.37)

where .

The proof of this corollary follows by the similar argument as in the proof of Theorem 2.5. We omit the details.

3. An Application

In this section, we present an application of nonlinear difference inequalities established in Theorem 2.5 to study the boundedness of the solutions of nonlinear Volterra difference equations.

Consider the difference equation of Volterra type

(3.1)

where and are matrices for each and .

Lemma 3.1 (see [18, Theorem  2.9.1]).

Assume that there exists a matrix defined on and satisfying

(3.2)

where .

Then, (3.1) is equivalent to the ordinary linear difference equation

(3.3)

where and

(3.4)

Consider the linear nonhomogeneous difference equation

(3.5)

where is a matrix over and .  We present the variation of constants formula of difference equations.

Lemma 3.2.

The solution of (3.5) is given by the variation of constants formula

(3.6)

where is a fundamental matrix solution of the difference equation such that is the identity matrix.

Now, we give an application of our results. We consider the perturbation of linear Volterra difference equation (3.1) with

(3.7)

with initial condition , where .

Theorem 3.3.

Suppose that the following conditions hold for , :

(i),

(ii),

(iii), ,

where and are some positive constants and is a nonnegative sequence defined on with . Then, all solutions of (3.7) are bounded in .

Proof.

By Lemma 3.1, (3.7) is equivalent to

(3.8)

with , where and is a solution of (3.2). It follow from Lemma 3.2 that the solution of (3.8) is given by

(3.9)

By using the conditions (i)–(iii), we obtain

(3.10)

Letting , and

(3.11)

and employing the above estimate by Corollary 2.9, then we have

(3.12)

where , because

(3.13)

and . Hence, the solutions of (3.7) are bounded in , and the proof is complete.