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Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 157301, 22 pages
http://dx.doi.org/10.1155/2012/157301
Research Article

Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received 25 April 2012; Accepted 16 August 2012

Academic Editor: Eric R. Kaufmann

Copyright © 2012 Samir H. Saker. 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 will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Hölder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases.

1. Introduction

In the past decade a number of Opial dynamic inequalities have been established by some authors which are motivated by some applications; we refer to the papers [13]. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function is a so-called time scale , which may be an arbitrary closed subset of the real numbers , to avoid proving results twice, once on a continuous time scale which leads to a differential inequality and once again on a discrete time scale which leads to a difference inequality. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [4]), that is, when , and where . A cover story article in New Scientist [5] discusses several possible applications of time scales. In this paper, we will assume that and define the time scale interval by . Since the continuous and discrete inequalities involving higher order derivatives are important in the analysis of qualitative properties of solutions of differential and difference equations [68], we also believe that the dynamic inequalities involving higher order derivatives on time scales will play the same effective act in the analysis of qualitative properties of solutions of dynamic equations [2, 3, 9]. To the best of the author’s knowledge there are few inequalities involving higher order derivatives established in the literature [1013]. In the following, we recall some of these results that serve and motivate the contents of this paper.

In [13] the authors proved that if is delta differentiable times with , for , and is a positive rd-continuous function on , then In [10] it is proved that if is delta differentiable times ( odd) with , for , then where and satisfy . Also in [10] it is proved that if is delta differentiable times with , for , and is increasing, then where and satisfy . As a generalization of (1.3) it is proved in [10] that if is delta differentiable times with , for , and is increasing, then where and satisfy . In [12] the authors proved that if and are positive rd-continuous functions on such that is nonincreasing, and is delta differentiable times with , for , then where and . For contributions of different types of dynamic inequalities on time scales, we refer the reader to the papers [1, 2, 1417] and the references cited therein.

Following this trend, to develop the qualitative theory of dynamic inequalities on time scales, we will prove some new inequalities of Opial’s type involving higher order derivatives by making use of the Hölder inequality (see, [18, Theorem 6.13]): where and and , the formula which is a simple consequence of Keller’s chain rule [18, Theorem 1.90], and the Taylor monomials on time scales. The results in this paper extend and improve the pervious results in the sense that our results contain two different weighted functions and do not require the monotonicity condition on (the results in [10] required that should be increasing). Some results on continuous and discrete spaces, which lead to differential and difference inequalities, will be derived from our results as special cases. This paper is a continuation of the papers [3, 1013, 16].

2. Main Results

In this section, we will prove the main results. For completeness, we recall the following concepts related to the notion of time scales. A time scale is an arbitrary nonempty closed subset of the real numbers . We assume throughout that has the topology that it inherits from the standard topology on the real numbers . The forward jump operator and the backward jump operator are defined by where . A point is said to be left-dense if and , is right-dense if , is left-scattered if and right-scattered if . The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [4]), that is, when and where . For more details of time scale analysis we refer the reader to the two books by Bohner and Peterson [18, 19] which summarize and organize much of the time scale calculus.

A function is said to be right-dense continuous (rd-continuous) provided is continuous at right-dense points and at left-dense points in ; left-hand limits exist and are finite. The set of all such rd-continuous functions is denoted by .

The graininess function for a time scale is defined by , and for any function the notation denotes . We will assume that and define the time scale interval by . Fix and let . Define to be the number (if it exists) with the property that given any there is a neighborhood of with In this case, we say is the (delta) derivative of at and that is (delta) differentiable at . We will frequently use the following results which are due to Hilger [20]. Assume that and let .(i) If is differentiable at , then is continuous at .(ii) If is continuous at and is right-scattered, then is differentiable at with (iii) If is differentiable and is right-dense, then (iv) If is differentiable at , then .

We will make use of the following product and quotient rules for the derivative of the product and the quotient (where , here ) of two differentiable functions and : In this paper, we will refer to the (delta) integral which we can define as follows: if , then the Cauchy (delta) integral of is defined by It can be shown (see [18]) that if , then the Cauchy integral exists, , and satisfies , . An infinite integral is defined as and the integration on discrete time scales is defined by Now, we define the Taylor monomials or generalized polynomials as defined originally by Agarwal and Bohner [21]. These types of monomials are important because they are intimately related to Cauchy functions for certain dynamic equations which are important in variations of constants formulas. The Taylor monomials , are defined recursively as follows. The function is defined by and given for , the function is defined by If we let denote for each fixed , the derivative of with respect to , then for each fixed . The above definition obviously implies In the following, we give some formulas of as determined in [18]. In the case when , then , , , and In the case when , we see that , , , and where is the so-called falling function (cf. [22]). When , we have , , , and If ,we see that , , , and In general for , we have that , and We also consider the Taylor monomials , , which are defined recursively as follows. The function is defined by and given for , the function is defined by If we let denote for each fixed , the derivative of with respect to , then for each fixed . By Theorem  1.112 in [18], we see that We denote by the space of all functions such that for for . For the function , we consider the second derivative provided is delta differentiable on with derivative . Similarly, we define the order derivative . Now, we are ready to state the Taylor formula that we will need to prove the main results in this paper. This formula as proved in [23] states the following. Assuming that and , then As a special case if , then

Now, we are ready to state and prove our main results in this paper. Throughout the rest of the paper, we will assume that all the integrals that will appear in the inequalities exist and are finite.

Theorem 2.1. Letting be a time scale with and . If , for , then

Proof. From the Taylor formula (2.22), since , for , we have This implies that Applying the Hölder inequality (1.6) with , we have Then Define . This implies that and . From this and (2.28), we have Applying the Hölder inequality again , we obtain From (1.7), we have (note that and that Then where . Substituting (2.32) into (2.30), we have which is the desired inequality (2.24). The proof is complete.

Remark 2.2. Let , but fixed, and let be such that , . Then from (2.24) it follows that Thus for , where , , ; then we have the following result.

Corollary 2.3. Let be a time scale with and . If , , then

Note that Theorem 2.1 can be extended to a general inequality with two different constants and that satisfy , to obtain the following result.

Theorem 2.4. Letting be a time scale with and . If , for , then

Theorem 2.5. Let be a time scale with and let , be positive real numbers such that , and . If , for ; then where

Proof. From the Taylor formula (2.22) and since , for , we have Applying the Hölder inequality with and , we have This implies that Then Define . This implies that , and From this and (2.42), we have Applying the Hölder inequality again and , we have From (1.7), we have (note that and ; see also, page 116 [17]) that Then where . Substituting (2.47) into (2.45), we have which is the desired inequality (2.37). The proof is complete.

Following Remark 2.2, we can obtain the following result.

Corollary 2.6. Let be a time scale with and let , be positive real numbers such that and . If , ; then where

Note that Theorem 2.5 cannot be applied when . In the following theorem we prove a new inequality which can be applied in this case.

Theorem 2.7. Let be a time scale with and let , be positive real numbers such that and . If , for , then

Proof. Using the fact that is increasing with respect to its first component for , we have from the Taylor formula (2.22) and , for , that This implies that Now applying the Hölder inequality (1.6) with and , we obtain Let . Then and so that and hence As in the proof of Theorem 2.5, we have that Substituting into (2.56), we have which is the desired inequality (2.51). The proof is complete.

Following Remark 2.2, we can obtain the following result.

Corollary 2.8. Let be a time scale with and let , be positive real numbers such that , and . If , , then

In the following, we will prove some inequalities with two different weighted functions.

Theorem 2.9. Let be a time scale with and let , , be positive real numbers such that and . Further, let and be positive rd-continuous functions defined on and . If , for , then where

Proof. From the Taylor formula (2.22), we see that Applying the Hölder inequality on the right hand side with and , we have This implies that Integrating from to , we have Let Then , and . This implies that Applying the Hölder inequality (1.6) with indices and , we obtain Substituting into (2.67), we have As in the proof of Theorem 2.5, we have This implies that which is the desired inequality (2.60) where is defined as in (2.61). The proof is complete.

Following Remark 2.2, we can obtain the following result.

Theorem 2.10. Let be a time scale with and let , be positive real numbers such that and . Further, let and be positive rd-continuous functions defined on and . If , , then where

Note that Theorem 2.10 cannot be applied when and . In the following theorem we prove an inequality which can be applied in this case.

Theorem 2.11. Let be a time scale with and let , be positive real numbers such that , and let , be nonnegative rd-continuous functions on and . If , for , then where

Proof. From the Taylor formula, we see that Now, since is nonnegative on , it follows from the Hölder inequality (1.6) with that Then, we get that Setting , we see that , and This gives us Since is nonnegative on , we have from (2.79) and (2.81) that This implies that Applying the Hölder inequality (1.6) with indices and , we have From (2.80), the chain rule (1.7) and the fact that , we obtain Substituting (2.85) into (2.84) and using the fact that , we have Using , we have from the last inequality that which is the desired inequality (2.74). The proof is complete.

Following Remark 2.2, we can obtain the following result.

Theorem 2.12. Let be a time scale with and let , be positive real numbers such that , and let , be nonnegative rd-continuous functions on and . If , , then where

Instead of (2.25), we can use the relation between and and define Proceeding as pervious by using the same arguments and using (2.90) one can obtain some results when , for . For example one can get the following results.

Theorem 2.13. Letting be a time scale with and . If , for , then

Theorem 2.14. Let be a time scale with and let , be positive real numbers such that . Let . If , for , then

Remark 2.15. Similar results as in Theorems 2.13 and 2.14 can be obtained from the results in the rest of the paper, but in this case one will use , for , instead of If , for , and instead of