Abstract and Applied Analysis

Volume 2013 (2013), Article ID 534083, 5 pages

http://dx.doi.org/10.1155/2013/534083

Research Article

An Opial-Type Inequality on Time Scales

Qiao-Luan Li¹ and Wing-Sum Cheung²

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

Academic Editor: Allan Peterson

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., [2–7]). 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.

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