# Gronwall-OuIang-Type Integral Inequalities on Time Scales

- Ailian Liu
^{1, 2}and - Martin Bohner
^{2}Email author

**2010**:275826

https://doi.org/10.1155/2010/275826

© Ailian Liu and Martin Bohner. 2010

**Received: **20 April 2010

**Accepted: **3 August 2010

**Published: **11 August 2010

## Abstract

We present several Gronwall-OuIang-type integral inequalities on time scales. Firstly, an OuIang inequality on time scales is discussed. Then we extend the Gronwall-type inequalities to multiple integrals. Some special cases of our results contain continuous Gronwall-type inequalities and their discrete analogues. Several examples are included to illustrate our results at the end.

## Keywords

## 1. Introduction

OuIang inequalities and their generalizations have proved to be useful tools in oscillation theory, boundedness theory, stability theory, and other applications of differential and difference equations. A nice introduction to continuous and discrete OuIang inequalities can be found in [1, 2], and studies in [3–5] give some of their generalizations to multiple integrals and higher-dimensional spaces. Like Gronwall's inequality, OuIang's inequality is also used to obtain a priori bounds on unknown functions. Therefore, integral inequalities of this type are usually known as Gronwall-OuIang-type inequalities [6].

The calculus on time scales has been introduced by Hilger [7] in order to unify discrete and continuous analysis. For the general basic ideas and background, we refer to [8, 9]. In this paper, we are concerned with Gronwall-OuIang-type integral inequalities on time scales, which unify and extend the corresponding continuous inequalities and their discrete analogues. We also provide a more useful and explicit bound than that in [10–12].

## 2. OuIang Inequality

We first give Gronwall's inequality on time scales which could be found in [8, Corollary ]. Throughout this section, we fix and let .

Lemma 2.1.

and is the exponential function on time scales; for more details on time scales, see [8, 9].

Now we will give the OuIang inequality on time scales.

Theorem 2.2.

Proof.

Combining (2.4) and (2.11) yields (2.5) and completes the proof.

In 1979, Dafermos [13] published a so-called Gronwall-type inequality (see also [3]). In the same way as Theorem 2.2, we now extend this result to general time scales.

Theorem 2.3.

Proof.

Combining (2.12) and (2.18), and using [8, Theorems and ] yields (2.13) and completes the proof.

Remark 2.4.

If and , then Theorem 2.3 reduces to Theorem 2.2.

Remark 2.5.

If we multiply inequality (2.16) by another exponential function on time scales, for example, , we could get another kind of inequality, which is a special case of Theorem 3.4.

## 3. Gronwall-OuIang-Type Inequality

Pachpatte discussed several integral inequalities arising in the theory of differential equations and difference equations [3, 4]. Now, we extend some of these results to time scales. First, we give some notations and definitions which are used in our subsequent discussion.

Theorem 3.1.

Proof.

This completes the proof.

Remark 3.2.

Theorem 3.3.

Proof.

Noticing that (3.29) implies and , the bounds in (3.27) follow, which concludes the proof.

Theorem 3.4.

Proof.

which gives the desired inequality (3.32). This concludes the proof.

Remark 3.5.

As in the discussion in Remark 3.2, Theorem 3.4 also holds true for .

## 4. Some Applications

Theorem 4.1.

then all nonoscillatory solutions of (4.1) are bounded.

Proof.

contradicting for all . Similarly, we can prove that if , then and for .

which completes the proof.

The proof in Theorem 4.1 corrects an inaccuracy in the proof of [1, Theorem ]. We can also obtain the following results.

Corollary 4.2.

Let . If is a continuously differentiable positive function such that is nonnegative, then all nonoscillatory solutions of (4.1) are bounded.

Proof.

and hence the statement follows from Theorem 4.1.

Example 4.3.

where , , are rd-continuous functions, and is a constant. When , its physical meaning is to model the water percolation phenomena, and Okrasiński has studied the existence and uniqueness of solutions [14].

From (4.24), we see that the solution of (4.16) approaches zero as

## Declarations

### Acknowledgments

This work is supported by Grants 60673151 and 10571183 from NNSF of China, and by Grant 08JA910003 from Humanities and Social Sciences in Chinese Universities.

## Authors’ Affiliations

## References

- Yang-Liang O: The boundedness of solutions of linear differential equations .
*Advances in Mathematics*1957, 3: 409–415.MathSciNetGoogle Scholar - Yang E: On some nonlinear integral and discrete inequalities related to Ou-Iang's inequality.
*Acta Mathematica Sinica*1998, 14(3):353–360. 10.1007/BF02580438MathSciNetView ArticleMATHGoogle Scholar - Pachpatte BG: On a certain inequality arising in the theory of differential equations.
*Journal of Mathematical Analysis and Applications*1994, 182(1):143–157. 10.1006/jmaa.1994.1072MathSciNetView ArticleMATHGoogle Scholar - Pang PYH, Agarwal RP: On an integral inequality and its discrete analogue.
*Journal of Mathematical Analysis and Applications*1995, 194(2):569–577. 10.1006/jmaa.1995.1318MathSciNetView ArticleMATHGoogle Scholar - Cho YJ, Kim Y-H, Pečarić J: New Gronwall-Ou-Iang type integral inequalities and their applications.
*The ANZIAM Journal*2008, 50(1):111–127. 10.1017/S1446181108000266MathSciNetView ArticleMATHGoogle Scholar - Cheung W-S, Ma Q-H: On certain new Gronwall-Ou-Iang type integral inequalities in two variables and their applications. Journal of Inequalities and Applications 2005, (4):347–361.Google Scholar
- Hilger S: Analysis on measure chains—a unified approach to continuous and discrete calculus.
*Results in Mathematics*1990, 18(1–2):18–56.MathSciNetView ArticleMATHGoogle Scholar - Bohner M, Peterson A:
*Dynamic Equations on Time Scales: An Introduction with Application*. Birkhäuser, Boston, Mass, USA; 2001:x+358.View ArticleMATHGoogle Scholar - Bohner M, Peterson A (Eds):
*Advances in Dynamic Equations on Time Scales*. Birkhäuser, Boston, Mass, USA; 2003:xii+348.MATHGoogle Scholar - Li WN, Sheng W: Some nonlinear dynamic inequalities on time scales.
*Proceedings of Indian Academy of Sciences. Mathematical Sciences*2007, 117(4):545–554. 10.1007/s12044-007-0044-7MathSciNetView ArticleMATHGoogle Scholar - Akin-Bohner E, Bohner M, Akin F: Pachpatte inequalities on time scales. Journal of Inequalities in Pure and Applied Mathematics 2005, 6(1, article 6):-23.Google Scholar
- Li WN: Bounds for certain new integral inequalities on time scales.
*Advances in Difference Equations*2009, 2009:-16.Google Scholar - Dafermos CM: The second law of thermodynamics and stability.
*Archive for Rational Mechanics and Analysis*1979, 70(2):167–179.MathSciNetView ArticleMATHGoogle Scholar - Okrasiński W: On a nonlinear convolution equation occurring in the theory of water percolation.
*Annales Polonici Mathematici*1980, 37(3):223–229.MathSciNetMATHGoogle Scholar

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