Research Article | Open Access

# Nonlinear Dynamical Integral Inequalities in Two Independent Variables and Their Applications

**Academic Editor:**Hassan A. El-Morshedy

#### Abstract

In this paper, we investigate some nonlinear dynamical integral inequalities involving the forward jump operator in two independent variables. These inequalities provide explicit bounds on unknown functions, which can be used as handy tools to study the qualitative properties of solutions of certain partial dynamical systems on time scales pairs.

#### 1. Introduction

Theory of dynamical equations on time scales, which goes back to Hilger's landmark paper [1], has received considerable attention in recent years. For example, see the monographs [2, 3] and the references cited therein. Since dynamical integral inequalities usually can be used as handy tools to study the qualitative theory of dynamical equations on time scales, many researchers devoted to the study of different types of integral inequalities on time scales. We refer the readers to [4–19].

To the best of our knowledge, the theory of partial dynamic equations on time scales has received less attention [20–24]. The main purpose of this paper is to investigate several nonlinear integral inequalities in two independent variables on time scale pairs, which can be used to estimate explicit bounds of solutions of certain partial dynamical equations on time scales. Unlike some existing results in the literature (e.g., [12]), the integral inequalities considered in this paper involve the forward jump operator and on a pair of time scales and , which results in difficulties in the estimation on the explicit bounds of unknown functions for and . As an application, we study the qualitative property of certain partial dynamical equations on time scales.

Throughout this paper, a knowledge and understanding of time scales and time scale notations is assumed. In what follows, and are two unbounded time scales, and . is the set of right-dense continuous functions on . For an excellent introduction to the calculus on time scales, we refer the reader to monographs [2, 3].

#### 2. Problem Statements

Before establishing the main results of this paper, we first present two useful lemmas as follows.

Lemma 2.1. *Let , , and . Then, for any ,
**
holds, where .*

*Proof. *Set . It is not difficult to see that obtains its maximum at and
This completes the proof of Lemma 2.1.

Lemma 2.2. *Let with for . Then
**
implies
**
where and .*

*Proof. *Note that , we have
that is,
By Theorem 6.1 [2, page 255], we get that Lemma 2.2 holds.

Consider the following nonlinear integral inequalities in two independent variables on time scales : where , and ( are nonnegative right-dense continuous functions on , () are constants.

The reason for studying inequalities of type (2.7)–(2.9) is that sometimes we may need to estimate the solutions of the following partial dynamical equation in the form with boundary conditions , , and , where is right-dense continuous, , and is a constant. Integrating (2.10) yields Therefore, the study on the integral inequalities of type (2.7)–(2.9) can provide explicit bounds of solutions of system (2.10) in some cases.

#### 3. Main Results

Now, let us present the main results of this paper.

Theorem 3.1. *If there exists a positive function , such that
**
then inequality (2.7) implies
**
where
*

*Proof. *Define a function by
Then, for , is nondecreasing with respect to and , and
A delta derivative of with respect to yields
By Lemma 2.1, we have
It follows from (3.5), (3.6), and (3.7) that
Notice the definitions of , , and , we have
Since , by Lemma 2.2 we get
Then, (3.5) and (3.10) imply (3.2).

Theorem 3.2. *If there exist positive functions , such that is Δ-differentiable with respect to , , and
**
then inequality (2.8) implies
**
where
*

*Proof. *Set
Then, is nonnegative and nondecreasing with respect to and on , and
By Lemma 2.1, we have
Substituting (3.15) into (3.16), we get
Note that
Integrating by parts, we have
Therefore, it follows from (3.17) and (3.19) that
This together with Lemma 2.2 and (3.15) yields (3.12).

Theorem 3.3. *If there exist positive functions , such that are -differentiable with respect to , and
**
then inequality (2.9) implies
**
where
**
and .*

*Proof. *Let the nonnegative and nondecreasing function be defined by
Then,
Based on the same arguments as in Theorem 3.2, we have
Notice that
By (3.26) and (3.27), we have
Using the fact , Lemma 2.2 and (3.25), we get that (3.22) holds.

It is worthy to mention that although some additional assumptions such as and are imposed in Theorems 3.1–3.3, they are easy to be satisfied by choosing appropriate adjusting functions and .

#### 4. Applications

We now consider some applications of the main results in the partial dynamical system (2.10) under the boundary condition Denote . We have the following corollaries.

Corollary 4.1. *Let , , and
**
Then, the solution of system (2.10) under the boundary condition (4.1) satisfies
**
for .*

*Proof. *For , it follows from (2.11) and (4.2) that
Let be a constant. A straightforward computation yields
Since , we get (4.3) by Theorem 3.1.

Corollary 4.2. *Let , and
**
Then, the solution of system (2.10) under the boundary condition (4.1) satisfies
**
where is defined as in Corollary 4.1 for .*

*Proof. *For , it follows from (2.11) and (4.6) that
holds for . Let and . A straightforward computation yields
Hence, . By Theorem 3.2, we have that (4.7) holds.

For the case when satisfies
on , the solution of system (2.10) under the boundary condition (4.1) can be similarly estimated by Theorem 3.3. We omit it here.

#### Acknowledgment

This paper was supported by the Natural Science Foundations of Shandong Province (ZR2010AL002, JQ201119) and the National Natural Science Foundation of China (61174217).

#### References

- S. Hilger, “Analysis on measure chains-a unified approach to continuous and discrete calculus,”
*Results in Mathematics*, vol. 18, no. 1-2, pp. 18–56, 1990. View at: Google Scholar | Zentralblatt MATH - M. Bohner and A. Peterson,
*Dynamic Equations on Time Scales, An Introduction with Application*, Birkhäuser, Boston, Mass, USA, 2001. - M. Bohner and A. Peterson, Eds.,
*Advances in Dynamic Equations on Time Scales*, Birkhäuser, Boston, Mass, USA, 2003. - E. Akin-Bohner, M. Bohner, and F. Akin, “Pachpatte inequalities on time scales,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 6, no. 1, article 6, 23 pages, 2005. View at: Google Scholar | Zentralblatt MATH - D. R. Anderson, “Nonlinear dynamic integral inequalities in two independent variables on time scale pairs,”
*Advances in Dynamical Systems and Applications*, vol. 3, no. 1, pp. 1–13, 2008. View at: Google Scholar - D. R. Anderson, “Dynamic double integral inequalities in two independent variables on time scales,”
*Journal of Mathematical Inequalities*, vol. 2, no. 2, pp. 163–184, 2008. View at: Google Scholar | Zentralblatt MATH - W. N. Li, “Some new dynamic inequalities on time scales,”
*Journal of Mathematical Analysis and Applications*, vol. 319, no. 2, pp. 802–814, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. N. Li and W. Sheng, “Some nonlinear integral inequalities on time scales,”
*Journal of Inequalities and Applications*, Article ID 70465, 15 pages, 2007. View at: Google Scholar | Zentralblatt MATH - W. N. Li and W. Sheng, “Some nonlinear dynamic inequalities on time scales,”
*Proceedings of the Indian Academy of Sciences—Mathematical Sciences*, vol. 117, no. 4, pp. 545–554, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. N. Li, “Some Pachpatte type inequalities on time scales,”
*Computers & Mathematics with Applications*, vol. 57, no. 2, pp. 275–282, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. N. Li, “Some delay integral inequalities on time scales,”
*Computers & Mathematics with Applications*, vol. 59, no. 6, pp. 1929–1936, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - W. N. Li, “Some integral inequalities useful in the theory of certain partial dynamic equations on time scales,”
*Computers & Mathematics with Applications*, vol. 61, no. 7, pp. 1754–1759, 2011. View at: Publisher Site | Google Scholar - D. B. Pachpatte, “Explicit estimates on integral inequalities with time scale,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 7, no. 4, article 143, 8 pages, 2006. View at: Google Scholar | Zentralblatt MATH - B. G. Pachpatte, “On some new inequalities related to a certain inequality arising in the theory of differential equations,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 2, pp. 736–751, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. W. Meng and W. N. Li, “On some new nonlinear discrete inequalities and their applications,”
*Journal of Computational and Applied Mathematics*, vol. 158, no. 2, pp. 407–417, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. W. Meng and D. Ji, “On some new nonlinear discrete inequalities and their applications,”
*Journal of Computational and Applied Mathematics*, vol. 208, no. 2, pp. 425–433, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - Y. Sun, “Some sublinear dynamic integral inequalities on time scales,”
*Journal of Inequalities and Applications*, vol. 2010, Article ID 983052, 10 pages, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. H. Wong, C. C. Yeh, S. L. Yu, and C. H. Hong, “Young's inequality and related results on time scales,”
*Applied Mathematics Letters*, vol. 18, no. 9, pp. 983–988, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH - F. H. Wong, C. C. Yeh, and C. H. Hong, “Gronwall inequalities on time scales,”
*Mathematical Inequalities & Applications*, vol. 9, no. 1, pp. 75–86, 2006. View at: Google Scholar | Zentralblatt MATH - C. D. Ahlbrandt and C. Morian, “Partial differential equations on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 141, no. 1-2, pp. 35–55, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - M. Bohner and G. S. Guseinov, “Partial differentiation on time scales,”
*Dynamic Systems and Applications*, vol. 13, no. 3-4, pp. 351–379, 2004. View at: Google Scholar | Zentralblatt MATH - M. Bohner and G. S. Guseinov, “Double integral calculus of variations on time scales,”
*Computers & Mathematics with Applications*, vol. 54, no. 1, pp. 45–57, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH - J. Hoffacker, “Basic partial dynamic equations on time scales,”
*Journal of Difference Equations and Applications*, vol. 8, no. 4, pp. 307–319, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH - B. Jackson, “Partial dynamic equations on time scales,”
*Journal of Computational and Applied Mathematics*, vol. 186, no. 2, pp. 391–415, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH

#### Copyright

Copyright © 2011 Yuangong Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.