Open Access

Gronwall-OuIang-Type Integral Inequalities on Time Scales

Journal of Inequalities and Applications20102010:275826

Received: 20 April 2010

Accepted: 3 August 2010

Published: 11 August 2010


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.

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
implies that
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

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
where is a positive constant, then


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

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


Multiplying both sides of (2.16) by , we have
Integrating (2.17) from to , we obtain that

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
and define the differential operators , , by
For and a nonnegative function defined on , we set

Theorem 3.1.

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


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

This completes the proof.

Remark 3.2.

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

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
where and , then for all ,



Multiplying (3.25) by yields
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
Now, Theorem 3.1 yields

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 ,
then, for all ,

where , with for all .


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

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

Theorem 4.1.

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

then all nonoscillatory solutions of (4.1) are bounded.


Let be a nonoscillatory solution of (4.1). Without loss of generality, we assume there exists such that
Hence, is strictly decreasing on . Thus, either
or there exists such that
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
Hence, there exists such that

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
From the integration by parts in [8, Theorem ],
Thus, with , we have
Theorem 2.2 gives that
Applying Gronwall's inequality from Lemma 2.1 yields

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.


For , we have

and hence the statement follows from Theorem 4.1.

Example 4.3.

Consider the nonlinear one-dimensional integral equation of the form

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
where , are nonnegative constants and is an rd-continuous function. From (4.16) and using (4.17), it is easy to observe that
Now an application of Theorem 3.1 with gives

which gives the bound on .

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

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



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

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


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© Ailian Liu and Martin Bohner. 2010

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