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Gronwall-OuIang-Type Integral Inequalities on Time Scales

Journal of Inequalities and Applications20102010:275826

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

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

Chain RuleIntegral InequalityDiscrete AnalogueNonoscillatory SolutionNonnegative Constant

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 [35] 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 [1012].

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.

Let , , , for all , and . Then
(2.1)
implies that
(2.2)
Above, is defined as the set of all regressive and rd-continuous functions, is the positive regressive part of , the "circle minus" subtraction on is defined by
(2.3)

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.

Let and be real-valued nonnegative rd-continuous functions defined on . If
(2.4)
where is a positive constant, then
(2.5)

Proof.

Let
(2.6)
From (2.4), we have
(2.7)
The definition of gives
(2.8)
Dividing both sides of (2.8) by and integrating from to , we have
(2.9)
According to the chain rule [8, Theorem ] and since is increasing,
(2.10)
so
(2.11)

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.

Let and be nonnegative rd-continuous functions on . Let , , be nonnegative constants and . If
(2.12)
then
(2.13)

Proof.

Let
(2.14)
Then,
(2.15)
Hence,
(2.16)
Multiplying both sides of (2.16) by , we have
(2.17)
Integrating (2.17) from to , we obtain that
(2.18)

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.

To simplify the expression, we let , choose rd-continuous functions such that
(3.1)
and define the differential operators , , by
(3.2)
For and a nonnegative function defined on , we set
(3.3)

Theorem 3.1.

Let and be real-valued nonnegative rd-continuous functions on , and let be a constant. If
(3.4)
where is a constant, then
(3.5)

Proof.

Let
(3.6)
From (3.6), it is easy to observe that
(3.7)
From (3.7) and using the facts that and are nonnegative, and
(3.8)
we have
(3.9)
that is,
(3.10)
Integrating (3.10) with respect to from to and using the fact that , we obtain that
(3.11)
which implies that
(3.12)
Again as above, from (3.12), we observe that
(3.13)
that is,
(3.14)
By setting in (3.14) and integrating with respect to from to and using the fact that , we get
(3.15)
Continuing this way, we obtain that
(3.16)
that is,
(3.17)
For , from the chain rule in [8, Theorem ],
(3.18)
Letting in (3.17) and integrating with respect to from to , we have
(3.19)
which means that
(3.20)

This completes the proof.

Remark 3.2.

Theorem 3.1 also holds for . To show this, assume (3.4) holds for , that is,
(3.21)
Now, let be arbitrary. Then
(3.22)
that is, (3.4) holds for . By Theorem 3.1, (3.5) also holds for , that is,
(3.23)
Since (3.23) holds for arbitrary , we may let in (3.23) to arrive at
(3.24)

that is, (3.5) holds for .

Theorem 3.3.

Let , , and for be real-valued nonnegative rd-continuous functions on and let be a constant. If , , and are nonnegative constants such that
(3.25)
(3.26)
where and , then for all ,
(3.27)

where

Proof.

Multiplying (3.25) by yields
(3.28)
Define
(3.29)
By taking the th power on both sides of (3.29) and using the elementary inequality , where are nonnegative reals, and also noticing (3.26) and , we get
(3.30)
Now, Theorem 3.1 yields
(3.31)

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

Theorem 3.4.

Let and be the set of all nonnegative real-valued rd-continuous functions defined on . Let and be monotone increasing linear operators on . If there exists a positive constant such that, for ,
(3.32)
then, for all ,
(3.33)

where , with for all .

Proof.

Let
(3.34)
Hence, for all , so that on , and thus
(3.35)
Hence , and therefore Similarly, . Using this and (3.32), we obtain that
(3.36)
By the product rule [8, Theorem ], we have
(3.37)
In summary,
(3.38)
Obviously
(3.39)
so that the chain rule [9, Theorem ] yields
(3.40)
Dividing both sides of (3.38) by provides that
(3.41)
Integrating both sides of (3.41) from to and noticing (3.40), we find that
(3.42)
Substitute the expression of , we have
(3.43)

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

In this section, we indicate some applications of our results to obtain the estimates of the solutions of certain integral equations for which inequalities obtained in the literature thus far do not apply directly. As an application of Theorem 2.2, we consider the second-order dynamic equation
(4.1)

Theorem 4.1.

Assume that is a differentiable positive function such that is rd-continuous. If there exist and such that
(4.2)

then all nonoscillatory solutions of (4.1) are bounded.

Proof.

Let be a nonoscillatory solution of (4.1). Without loss of generality, we assume there exists such that
(4.3)
Then
(4.4)
Hence, is strictly decreasing on . Thus, either
(4.5)
or there exists such that
(4.6)
We now claim that (4.6) is impossible to hold. To show this, let us assume that (4.6) is true. Then is strictly decreasing on and
(4.7)
Hence, there exists such that
(4.8)

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

Multiplying (4.1) on both sides by and taking integral from to , we have
(4.9)
From the integration by parts in [8, Theorem ],
(4.10)
Thus, with , we have
(4.11)
Theorem 2.2 gives that
(4.12)
Applying Gronwall's inequality from Lemma 2.1 yields
(4.13)
Hence,
(4.14)

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.

For , we have
(4.15)

and hence the statement follows from Theorem 4.1.

Example 4.3.

Consider the nonlinear one-dimensional integral equation of the form
(4.16)

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].

Here, we assume that every solution of (4.16) exists on the interval . We suppose that the functions , , in (4.16) satisfy the conditions
(4.17)
where , are nonnegative constants and is an rd-continuous function. From (4.16) and using (4.17), it is easy to observe that
(4.18)
Now an application of Theorem 3.1 with gives
(4.19)

which gives the bound on .

Now, we consider (4.16) under the conditions
(4.20)
where and are as above, is a constant, is an rd-continuous function, and
(4.21)
From (4.16) and (4.20), it is easy to observe that
(4.22)
Applying Theorem 3.1 with yields
(4.23)
So,
(4.24)

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

(1)
School of Statistics and Mathematics, Shandong Economic University, Jinan, China
(2)
Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, USA

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Copyright

© Ailian Liu and Martin Bohner. 2010

This article is published under license to BioMed Central Ltd. 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.

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