Journal of Applied Mathematics

Volume 2013 (2013), Article ID 625063, 9 pages

http://dx.doi.org/10.1155/2013/625063

Research Article

Some New Gronwall-Bellman-Type Inequalities on Time Scales and Their Applications

Bin Zheng,¹ Qinghua Feng,1,2 and Fanwei Meng²

¹School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

²School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China

Received 28 May 2013; Accepted 11 October 2013

Academic Editor: Samir H. Saker

Copyright © 2013 Bin Zheng et al. 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.

Abstract.

We establish some new Gronwall-Bellman-type inequalities on time scales. These inequalities are of new forms compared with other Gronwall-Bellman-type inequalities established so far in the literature. Based on them, new bounds for unknown functions are derived. For illustrating the validity of the inequalities established, we present some applications for them, in which the boundedness for solutions for some certain dynamic equations on time scales is researched.

1. Introduction

As is known, various integral and differential inequalities play an important role in the research of boundedness, global existence, and stability of solutions of differential and integral equations as well as difference equations. Among the investigations for inequalities, generalization of the Gronwall-Bellman-inequality [1, 2] is a hot topic, as such inequalities provide explicit bounds for unknown functions concerned. During the past decades, many Gronwall-Bellman-type inequalities have been discovered (e.g., see [3–18]). Recently, with the development of the theory of time scales [19], many integral inequalities on time scales have been established, for example, [20–32], which have proved to be very effective in the analysis of qualitative as well as quantitative analysis of solutions of dynamic equations. But for some certain dynamic equations, for example,

or

it is inadequate to research the boundedness of their solutions by use of the existing results in the literature. So it is necessary to seek new approach to fulfill such analysis for them.

Based on the analysis above, in this paper, we establish some new Gronwall-Bellman-type inequalities on time scales, which are designed so as to be used as a handy tool to research the boundedness of the solutions of the equations mentioned above. By use of the established inequalities, some new bounds for the solutions for the two equations are derived.

In the rest of the paper, denotes the set of real numbers, and . denotes an arbitrary time scale, and , where . On we define the forward and backward jump operators and such that and . The graininess is defined by . Obviously, if , while if . A point with is said to be left dense if , right dense if , left scattered if , and right scattered if . The set is defined to be if does not have a left scattered maximum; otherwise it is without the left scattered maximum.

The following definitions and theorems in the theory of time scales are known to us.

Definition 1. A function is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while is called regressive if . denotes the set of rd-continuous functions, while denotes the set of all regressive and rd-continuous functions, and .

Definition 2. For some and a function , the delta derivative of at is denoted by (provided that it exists) with the property such that, for every , there exists a neighborhood of satisfying

Definition 3. If and , then is called an antiderivative of , and the Cauchy integral of is defined by

where .

Definition 4. For , the exponential function is defined by

Definition 5. If and , we define

Theorem 6 (see [20, Theorem 2.1]). If and , then

(i)

(ii)If are delta differential at , then is also delta differential at , and

Theorem 7 (see [20, Theorem 2.2]). If , and , then one has the following:

(i); (ii); (iii); (iv); (v); (vi)if for all , then .

Theorem 8 (see [29, Theorem 5.2]). If , then the following conclusions hold:

(i) and ;(ii);(iii)if , then for all ;(iv)if , then ;(v),

where .

Remark 9. If , then Theorem 8 ((iii), (v)) still holds.

Theorem 10 (see [29, Theorem 5.1]). If and fix , then the exponential function   is the unique solution of the following initial value problem:

2. Main Results

Lemma 11. Suppose that , , and . Then

implies

Proof. Denote that . Then

Since , then from Theorem 8 (iv), we have , and furthermore from Theorem 8 (iii), we obtain for all .

According to Theorem 6 (ii),

On the other hand, from Theorem 10, we have

So combining (13), (14) and Theorem 8, it follows that

Substituting with , an integration for (15) with respect to from to yields

By and , it follows that

which is followed by

Since is arbitrary, after substituting with , we obtain the desired inequality.

Lemma 12. Under the conditions of Lemma 11, furthermore, if and is nonincreasing on , then

Proof. Since and is nondecreasing on , then and

From [33, Theorems 2.39 and 2.36 (i)], we have

Combining the above information, we can obtain the desired inequality.

Lemma 13 (see [4]). Assume that , and ; then for any , the following inequality holds:

First one will study the following Gronwall-Bellman-type inequality on time scales:

Theorem 14. Suppose that are nonincreasing, , and are constants, , and . If for , satisfies (23), then

Provided that , where

Proof. Let the right side of (23) be . Then

Let

Then we have

From Lemma 11, considering is nonincreasing on , we can obtain

where are defined in (25).

Let

Then

From Lemma 13, we have

A combination of (32), (33), (34), and (35) yields

where are defined in (26) and (27), respectively. By being nonincreasing on and , according to Lemma 12, we have

Combining (32), (34), and (37), we obtain

From (28), (38), we can obtain the desired inequality (13).

Based on Theorem 14, we will establish two Volterra-Fredholm type delay integral inequalities on time scales in the following two theorems.

Theorem 15. Suppose that , and are defined as in Theorem 14 with , is a constant, and is a fixed number. If for , satisfies the following inequality:

and furthermore, , then for ,

provided that , where

Proof. Let the right side of (39) be . Then

Considering , it follows that

We notice that the structure of (46) is just similar to that of (29). So following the same manner as the process of (29)–(38) in Theorem 14 (i.e., takes the place of in Theorem 14, and let in Theorem 14), considering , we can obtain

where , and are defined in (42), (43), and (44), respectively. Setting in (47), we obtain

As , it follows that

that is,

Combining (45), (47), and (50), we can obtain the desired inequality (39).

Theorem 16. Suppose that , and are defined as in Theorem 14 with , is a constant, is a fixed number, , and for , where . If for , satisfies the following inequality:

with the initial condition (11), and furthermore, , then

provided that , where

Proof. Let the right side of (51) be . Then

Furthermore, considering , we have

Let

Then

Considering is nonincreasing on , by Lemma 12 we obtain

Combining (57) and (59), it follows that

On the other hand, from Lemma 13 one can see that the following inequalities hold:

So combining (60) and (61), we have

where are defined in (53) and (54), respectively.

Considering is nonincreasing on , then by Lemma 12, we have

Combining (59) and (63), we obtain

Setting in (64), considering , we obtain

which is followed by

Then combining (55), (64), and (66), we can obtain the desired inequality (52).

3. Applications

In this section, we apply the results established above to analysis of boundedness of solutions for certain dynamic equations.

Example 17. Consider the following dynamic equation:

where , is a constant with , and is a constant.

Theorem 18. Suppose that is a solution of (66)-(67) and assume that and , where , and are defined the same as in Theorem 14; then the following inequality holds:

where

Proof. The equivalent integral equation of (56) can be denoted by

Then we have

A suitable application of Theorem 14 (i.e., takes the place of a() in Theorem 14, and in Theorem 14) yields (68).

Remark 19. Under the conditions of Theorem 18, considering , furthermore we have the following estimate:

Example 20. Consider the following dynamic equation:

where , , is a constant with , and is a constant.

Theorem 21. Suppose that is a solution of (73) and assume that and , where , and are defined the same as in Theorem 16. Then the following inequality holds:

provided that and , where are defined as in Theorem 16.

Proof. From (73), we have

Then under the condition , a suitable application of Theorem 16 yields (74).

Acknowledgments

This work was partially supported by Natural Science Foundation of Shandong Province (China) (Grant no. ZR2013AQ009) and Doctoral initializing Foundation of Shandong University of Technology (China) (Grant no. 4041-413030).

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