Gronwall-OuIang-Type Integral Inequalities on Time Scales
© Ailian Liu and Martin Bohner. 2010
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.
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 .
The calculus on time scales has been introduced by Hilger  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 .
Now we will give the OuIang inequality on time scales.
Combining (2.4) and (2.11) yields (2.5) and completes the proof.
Combining (2.12) and (2.18), and using [8, Theorems and ] yields (2.13) and completes the proof.
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.
This completes the proof.
which gives the desired inequality (3.32). This concludes the proof.
4. Some Applications
then all nonoscillatory solutions of (4.1) are bounded.
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.
and hence the statement follows from Theorem 4.1.
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 .
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.
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