- Research Article
- Open Access

# Volterra Discrete Inequalities of Bernoulli Type

- SungKyu Choi
^{1}and - Namjip Koo
^{1}Email author

**2010**:546423

https://doi.org/10.1155/2010/546423

© K. Choi and N. Koo 2010

**Received: **27 April 2010

**Accepted: **12 December 2010

**Published: **27 January 2011

## 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.

## Keywords

## 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]).

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

Corollary 2.2.

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.

Proof.

Since and , this implies that our inequality holds.

Remark 2.4.

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.

Proof.

where . Hence, the proof is complete.

Remark 2.6.

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.

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.

Proof.

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.

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.

where and are matrices for each and .

Lemma 3.1 (see [18, Theorem 2.9.1]).

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

Lemma 3.2.

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

with initial condition , where .

Theorem 3.3.

Suppose that the following conditions hold for , :

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

Proof.

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

## Declarations

### Acknowledgments

This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2010-0008835). The authors are thankful to the referee for giving valuable comments for the improvement of this paper.

## Authors’ Affiliations

## References

- Sugiyama S: On the stability problems of difference equations.
*Bulletin of Science and Engineering Research Laboratory*1969, 45: 140–144.MathSciNetGoogle Scholar - Bellman R: The stability of solutions of linear differential equations.
*Duke Mathematical Journal*1943, 10: 643–647. 10.1215/S0012-7094-43-01059-2MathSciNetView ArticleMATHGoogle Scholar - Bellman R:
*Stability Theory of Differential Equations*. McGraw-Hill, New York, NY, USA; 1953:xiii+166.MATHGoogle Scholar - Bellman R, Cooke KL:
*Differential-Difference Equations*. Academic Press, New York, NY, USA; 1963:xvi+462.MATHGoogle Scholar - Gronwall TH: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations.
*Annals of Mathematics. Second Series*1919, 20(4):292–296. 10.2307/1967124MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP:
*Difference Equations and Inequalities. Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 228*. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar - Baĭnov D, Simeonov P:
*Integral Inequalities and Applications, Mathematics and Its Applications (East European Series)*.*Volume 57*. Kluwer Academic Publishers, Dordrecht, UK; 1992:xii+245.Google Scholar - Choi SK, Deng S, Koo NJ, Zhang W: Nonlinear integral inequalities of Bihari-type without class .
*Mathematical Inequalities & Applications*2005, 8(4):643–654.MathSciNetView ArticleMATHGoogle Scholar - Lakshmikantham V, Trigiante D:
*Theory of Difference Equations. Numerical Methods and Applications, Mathematics in Science and Engineering*.*Volume 181*. Academic Press, Boston, Mass, USA; 1988:x+242.Google Scholar - Lakshmikantham V, Leela S, Martynyuk AA:
*Stability Analysis of Nonlinear Systems, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 125*. Marcel Dekker, New York, NY, USA; 1989:xii+315.Google Scholar - Pachpatte BG:
*Inequalities for Finite Difference Equations, Monographs and Textbooks in Pure and Applied Mathematics*.*Volume 247*. Marcel Dekker, New York, NY, USA; 2002:x+514.Google Scholar - Willett D, Wong JSW: On the discrete analogues of some generalizations of Gronwall's inequality.
*Monatshefte für Mathematik*1965, 69(4):362–367. 10.1007/BF01297622MathSciNetView ArticleMATHGoogle Scholar - Bihari I: A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations.
*Acta Mathematica Academiae Scientiarum Hungaricae*1956, 7: 81–94. 10.1007/BF02022967MathSciNetView ArticleMATHGoogle Scholar - Hull TE, Luxemburg WAJ: Numerical methods and existence theorems for ordinary differential equations.
*Numerische Mathematik*1960, 2: 30–41. 10.1007/BF01386206MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG: A note on Gronwall-Bellman inequality.
*Journal of Mathematical Analysis and Applications*1973, 44: 758–762. 10.1016/0022-247X(73)90014-0MathSciNetView ArticleMATHGoogle Scholar - Oguntuase JA: On an inequality of Gronwall.
*Journal of Inequalities in Pure and Applied Mathematics*2001, 2(1, article 9):6.MathSciNetGoogle Scholar - Choi SK, Kang B, Koo N: On inequalities of Gronwall type.
*Journal of Chungcheong Mathematical Society*2007, 20: 561–568.Google Scholar - Lakshmikantham V, Rama Mohana Rao M:
*Theory of Integro-Differential Equations, Stability and Control: Theory, Methods and Applications*.*Volume 1*. Gordon and Breach Science, Lausanne, Switzerland; 1995:x+362.Google Scholar

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