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`Abstract and Applied AnalysisVolume 2013 (2013), Article ID 534083, 5 pageshttp://dx.doi.org/10.1155/2013/534083`
Research Article

An Opial-Type Inequality on Time Scales

1College of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050024, China
2Department of Mathematics, The University of Hong Kong, Hong Kong

Received 4 January 2013; Accepted 16 January 2013

Copyright © 2013 Qiao-Luan Li and Wing-Sum Cheung. 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 Opial-type inequalities involving higher order delta derivatives on time scales. These extend some known results in the continuous case in the literature and provide new estimates in the setting of time scales.

1. Introduction

Opial's inequality appeared for the first time in 1960 in [1] and has been receiving continual attention throughout the years (cf., e.g., [27]). The inequality together with its numerous generalizations, extensions, and discretizations has been playing a fundamental role in the study of the existence and uniqueness properties of solutions of initial and boundary value problems for differential equations as well as difference equations [8, 9]. Two excellent surveys on these inequalities can be found in [10, 11].

In 1960, Opial established the following integral inequality.

Theorem A (see  [1]). If satisfies and for all , then

Shortly after the publication of Opial’s paper, Olech provided a modified version of Theorem A. His result is stated in the following.

Theorem B (see  [12]). If is absolutely continuous on with , then
The equality in (2) holds if and only if , where is a constant.

The first natural extension of Opial’s inequality (1) involving higher order derivatives is embodied in the following.

Theorem C (see  [10]). Let be such that . Then the following inequality holds:

In 1997, Alzer [13] considered Opial-type inequalities which involve higher-order derivatives of two functions. These generalize earlier results of Agarwal and Pang [14].

In this paper, we consider the Opial-type inequality which involves higher-order delta derivatives of two functions on time scales. Our results in special cases yield some of the recent results on Opial’s inequality and provide some new estimates on such types of inequalities in this general setting.

2. Main Results

Let be a time scale; that is, is an arbitrary nonempty closed subset of real numbers. Let , . We suppose that the reader is familiar with the basic features of calculus on time scales for dynamic equations. Otherwise one can consult Bohner and Peterson’s book [15] for most of the materials needed.

We first quote the following elementary lemma and the delta time scales Taylor formula.

Lemma 1 (see  [16]). Let be real constants. Then for any .

Lemma 2 (see [17]). Let the set of functions that are times differentiable with rd-continuous derivatives on . Then for any and , where .

Our main results are given in the following theorems.

Theorem 3. Let , , be real numbers, and let , be integers with . Let and be measurable functions on . Further, let with , and let be absolutely continuous on such that the integrals and exist. Then one has where , ,

Proof. Since , we obtain from Taylor's theorem that for all , and hence
From (9) and Hölder's inequality we get where ,  .
Let
Then we have
So (10) together with (12) implies where . Integrating both sides of (13) over and making use of Hölder's inequality, we obtain
Similarly, we get
Recall the elementary inequalities where
Let . Since and is nondecreasing, from (14)–(16), we have
By Lemma 1, we get
From (18) and (19), we conclude
The proof is complete.

Theorem 4. Let , , , be real numbers, and let , be integers with . Let , and be measurable functions on . Further, let ,  with let , and be absolutely continuous on such that the integrals and exist. Then one has where , , , , , and

Proof. Following the proof of Theorem 3, we obtain
Using (16),
The proof is complete.

Remark 5. In the special case where , Theorem 4 reduces to Theorem of [13].

Theorem 6. Let , be such that , , let be absolutely continuous on , and let . Then

Proof. From the hypotheses, we have
Multiplying (26) by and using Cauchy-Schwarz inequality, we obtain
Integrating both sides over from to and using Cauchy-Schwarz inequality, we observe
The proof is complete.

Theorem 7. Let , be nonnegative and measurable on , and let be such that , . If is absolutely continuous on , then for , , and any , where , ,.

Proof. Following the hypotheses, it is easy to see that (26) holds. By using Hölder's inequality with indices and , we obtain where , . So we get and hence for any ,
Thus for ,
Integrating both sides of (33) from to and applying Hölder's inequality with indices and , we obtain where . The proof is complete.

Remark 8. In the special case where , Theorems 6 and 7 reduce to Theorems and of [18].

Acknowledgments

The first author's research was supported by NSF of China (11071054), Natural Science Foundation of Hebei Province (A2011205012). The second author's research was partially supported by an HKU URG grant.

References

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