New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

Saker, S. H.; Sayed, A. G.; Sikorska-Nowak, A.; Abohela, I.

doi:https://doi.org/10.1155/2019/6757080

Discrete Dynamics in Nature and Society

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Volume 2019 | Article ID 6757080 | https://doi.org/10.1155/2019/6757080

New Characterizations of Weights in Hardy and Opial Type Inequalities via Solvability of Dynamic Equations

S. H. Saker,¹A. G. Sayed,²A. Sikorska-Nowak,³and I. Abohela⁴

Academic Editor: Pavel Rehak

Received26 Feb 2019

Accepted04 Jun 2019

Published01 Aug 2019

Abstract

In this paper, we prove that the solvability of dynamic equations of second order is sufficient for the validity of some Hardy and Opial type inequalities with two different weights on time scales. In particular, the results give new characterizations of two different weights in inequalities containing Hardy and Opial operators. The main contribution in this paper is the characterizations of weights in discrete inequalities that will be formulated from our results as special cases.

1. Introduction

In 1925, Hardy [1] proved thatwhere is a positive integrable function over any finite interval , is an integrable function over , and The constant in (1) is the best possible. Hardy proved his inequality by employing the calculus of variations which reduces the validation of the inequality to the existence of a positive solution of an Euler-Lagrange differential equation. In 1961 Beesack [2] followed the same technique and proved a generalized inequality of the formwhere the weighted functions and satisfy the Euler-Lagrange differential equationFor more related results for Hardy’s type inequalities that improved or extended the results of Beesack, we refer the reader to the papers by Bennett [3], Beesack [4], Beesack and Heing [5], Bliss [6], Bloom and Kerman [7], Bradly [8], Florkiewicz [9], Leindler [10, 11], Muckenhoupt [12], Shum [13], Sinnamon [14], Tomaselli [15], and Talenti [16, 17] and the references they cited.

In 1960, Opial [18] proved a new type of inequalities involving the relation between the function and its derivatives. In particular, he proved that if is an absolutely continuous function on with , thenThe constant in (4) is the best possible. In further the proof of the Opial inequality had already been simplified by Olech [19], Beesack [20], Levinson [21], Mallows [22], and Pederson [23]. In particular, it is proved that if is real absolutely continuous on and with , thenSince then much work has been done, and many papers which deal with new proofs, various generalizations, extensions, and their discrete analogues have appeared in the literature. The discrete analogy of (4) has been proved in [24] and the discrete analogy of (5) has been proved in [25, Theorem 5.2.2].

In 1976, Florkiewicz and Rybarski [26] used a new method to derive inequalities involving a function and its first derivative of Opial’s type with weights of the formwhere and satisfy a second order differential equation of the form This method makes it possible to obtain, in a natural way, the equality conditions important in differential equations and also makes it is possible, given a function and an auxiliary function , to define a function and an additional function and next using , , and to define a new class of functions for which (2) or (6) holds. Moreover it allows avoiding some assumptions on weights that have to be given in other methods.

In the last decades, some authors interested to find certain discrete inequalities analogues to integral inequalities in the analysis of different fields and as a result this subject becomes a topic of ongoing research. One reason for this upsurge of interest in discrete case is also due to the fact that discrete operators may even behave differently from their continuous counterparts. But the main challenge is that there are no general methods to study these questions on discrete spaces. Instead, these methods have to be developed starting from the basic definitions. In some cases it is possible to translate or adapt almost straightforward the objects and results from the continuous setting to the discrete setting or vice versa, however, in some other cases that is far from be trivial. In recent years, some authors have been interested in finding some characterizations of discrete Hardy’s type inequalities analogues for characterizations of Hardy’s type inequalities via differential equations; see, for example, the results by Chen [27, 28] and Liao [29] and the references they cited. To the best of the authors’ knowledge there are no characterizations obtained for discrete Opial type inequalities via difference equations.

In recent years, the study of dynamic inequalities on time scales has received a lot of attention and becomes a major field in pure and applied mathematics. The general idea is to prove a result for an inequality where the domain of the unknown function is a so-called time scale , which is an arbitrary nonempty closed subset of the real numbers . The study of dynamic inequalities on time scales helps avoid proving results twice, for differential inequalities as well as for difference inequalities. For more details of Hardy’s type inequalities on time scales, we refer the reader to the papers [30–39] and to the book [40] and to more results about Opial’s type inequalities on time scales, we refer to papers [41–44] and to the book [45].

The natural question which is expected now is Is it possible to prove new characterizations of the weighted functions in dynamic Hardy’s and Opial’s type inequalities by reducing the problem to the solvability of dynamic equations on time scales?

Our aim is to give an affirmative answer to this question in our present paper, which is organized as follows: In Section 2, we present some basic definition concerning the delta calculus on time scales. In Section 3, we prove a new characterization of weights on dynamic Hardy’s type inequality and prove that the characterization reduces to the solvability of dynamic equation of second order, where is the forward jump operator which is defined by , for , and In Section 4, we prove a new characterization of weighs on dynamic Opial’s type inequality and prove that the characterization reduces to the solvability of dynamic equation In Section 5, we prove that the characterization of weights in an inequality involving Hardy and Opial operator will be obtained from the solvability of a generalized dynamic equation of the form We will formulate the classical integral and discrete inequalities in this paper as special cases. Throughout this paper, we will assume that , , and are nonnegative rd-continuous functions and the integrals considered are assumed to exist.

2. Preliminaries

In this section, we present some basic definition concerning the delta calculus on time scales. For more details on time scale analysis, we refer the reader to the two books by Bohner and Peterson [46, 47] which summarize and organize much of time scales calculus. A time scale is an arbitrary nonempty closed subset of the real numbers . The forward jump operator and the backward jump operator are defined by , and , where . A point , which is said to be left-dense if and , is right-dense if , is left-scattered if , and is right-scattered if A function is said to be right-dense continuous (rd-continuous) provided is continuous at right-dense points and at left-dense points in , and left hand limits exist and are finite. The set of all such rd-continuous functions is denoted by Also, the set of functions that are differentiable and whose derivative is rd-continuous is denoted by The graininess function for a time scale is defined by , and for any function the notation denotes Without loss of generality, we assume that and define the time scale interval by Recall of the following product and quotient rules for the derivative of the product and the quotient (where , here ) of two differentiable functions and The first chain rule that we will use in this paper iswhich is a simple consequence of Keller’s chain rule [46, Theorem 1.90]. The second chain rule that we will use in this paper is given in the following. Let be continuously differentiable and suppose is delta differentiable, then is delta differentiable andIn 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 [46]) that if , then the Cauchy integral exists, , and it satisfies , An infinite integral is defined as Integration on discrete time scales is defined by The integration by parts formula on time scales readsHölder’s inequality [46, Theorem 6.13] states that, for , we havewhere , , and ,

3. Characterizations of Weights in Hardy’s Type Inequalities

In this section, we will establish some characterizations of two different weights in Hardy’s type inequalities on time scales and via the solvability of a dynamic equation of the formwhere and are continuous functions defined on In this section, we will definewhere and is a positive solution of (17).

Definition 1. We say that belongs to the class ifand

Lemma 2. Assume that is a time scale and . If is a positive solution of the dynamic equation (17), thenwhere

Proof. For convenience, we skip the argument sometimes in the computations. From the definition of and by using the product and the quotient rules of derivative on time scaleswe have for thatandFrom (17) and (23), we get thatSince , we get thatand thenSubstituting (25) and (27) into (24), we obtain thatSince , thenFrom (28) and (29), we have which is the desired equation (21). The proof is complete.

Theorem 3. Let be a time scale and . If is a positive solution of (17), thenwhere .

Proof. For convenience, we skip the argument sometimes in the computations. Since and by using (26) we getBy applying Lemma 2, we get thatSubstituting (33) into (32), we obtainSince , thenThen, from (34) and (35), we have that which is the desired identity (31). The proof is complete.

Theorem 4. Let be a time scale and and satisfy the dynamic equation (17). Then for every , we have thatFurthermore (37) becomes an equality if and only if where is a constant.

Proof. From Theorem 3, we have that Then Since , are nonnegative, we see that , which implies thatwhich is the desired inequality (37). Furthermore inequality (37) becomes equality if and only if , where This in fact gives us that , and then with The proof is complete.

Definition 5. We say that belongs to the class ifand

Remark 6. Obviously

Theorem 7. Let be a time scale and and satisfy the dynamic equation (17). Then for every function , we haveIf , then (43) becomes an equality if and only if , and

Proof. From (42), we have By applying Theorem 4, then we obtain which is the desired inequality (43). Furthermore inequality (43) becomes equality if and only if where This implies that , and with andSince , and , this implies that and then and The proof is complete.

Remark 8. As a special case of Theorem 7 when , we obtain the inequality where and satisfy the differential equation

Remark 9. As a special case of Theorem 7 when , we obtain the inequality where and satisfy the difference equation

Remark 10. For the differential form we get the Wirtinger type inequality where and satisfy the differential equation (51) with

Example 11. Let with , and . Put , and , then , , and By applying Theorem 7, we see that which is Hardy’s inequality with a sharp constant .

Example 12. Let with , and . Put , and , we see that . By applying Theorem 7, we get the inequality which is the Hardy-type inequality.

4. Characterizations of Weights in Opial’s Type Inequalities

In this section, we will establish some characterizations of two different weights in Opial’s type inequalities on time scales via the solvability of a dynamic equation of the formIn this section, we will definewhere is a positive solution of (58).

Definition 13. We say that belongs to the class , ifand

Lemma 14. Assume that is a time scale and If is a solution of the dynamic equation (58), then

Proof. For convenience we skip the argument sometimes in the computations. From the definition of and by using the product and the quotient rules (22) for , we haveSince , thenSubstituting (63) and (64) intowe obtain that which is (62). The proof is complete.

Lemma 15. Let be a time scale and If is a solution of the dynamic equation (58), then

Proof. For convenience we skip the argument sometimes in the computations. By using the product and the quotient rules (22) for , we have that and this implies thatMultiplying (69) by , we have that which is (67). The proof is complete.

Theorem 16. Assume that is a time scale and and satisfy the dynamic equation (58). Then for every and , we have thatFurthermore inequality (71) becomes equality if and only if with a constant

Proof. From Lemmas 14 and 15, we get that Since , then we obtain This implies thatwhere By using the dynamic equations (58) and (74) and , we see that This implies thatwhich is the desired inequality (71). Furthermore inequality (71) becomes equality if and only if , where This in fact gives us that and then with a constant The proof is complete.

Definition 17. We say that belongs to the class ifand

Remark 18. Obviously

Theorem 19. Assume that is a time scale and and satisfy the dynamic equation (58). Then for every function the inequalityholds. If , inequality (80) becomes equality if and only , , and

Proof. From (79), we have By applying Theorem 16, then we obtain which is the desired inequality (80). Furthermore inequality (80) becomes equality if and only if , where This in fact gives us that and then with a constant , and Since , and with a constant , this implies that and then , and The proof is complete.

Remark 20. As a special case of Theorem 19, we see that

Remark 21. As a special case of Theorem 19 when , we get the Opial’s type inequality where the weighted functions and satisfy the differential equation

5. A Combination between Hardy’s and Opial’s Inequalities

In this section, we will prove some characterizations of weights on a general inequality which contains Hardy’s type inequality and Opial’s type inequalities as special cases on time scales and reduces the characterizations of these weights to the solvability of a generalized Euler dynamic equation of the formIn this section, we definewhere is a positive solution of (90).

Definition 22. We say that the function belongs to the class ifand

Lemma 23. Assume that is a time scale and If is a positive solution of (90), then

Proof. For convenience, we skip the argument sometimes in the computations. By using the product rule in(22) for , we see thatandFrom the definition of and by using the quotient rule in (22) for , we have thatSubstituting (97) and (95), we obtain that which is (94). The proof is complete.

Lemma 24. Assume that be a time scale and let If is a positive solution of (90), then

Proof. For convenience, we skip the argument sometimes in the computations. Since , and by using the product and the quotient rules (22) for , we have that This implies thatMultiplying (101) by , we have that which is (99). The proof is complete.

Theorem 25. Let be a time scale and let If , , and satisfy (90), then

Proof. From Lemmas 23 and 24 we get that Since , then we obtain This implies thatwhere , and by using dynamic equation (90), we obtain From (106), we have thatwhich is the desired inequality (103). The proof is complete.

Definition 26. We say that belongs to the class ifand

Remark 27. Obviously

Theorem 28. Assume that is a time scale and , , and satisfy the dynamic equation (90). Then for every function the inequalityholds. Furthermore inequality (111) becomes equality if and only if with , and

Proof. From (110), we have By applying Theorem 25, we obtain which is the desired inequality (111). Furthermore inequality (111) becomes equality if and only if where This in fact gives us that , and then with And Since , and with , this implies that and then and The proof is complete.

Remark 29. As a special case of Theorem 28 when , we obtain inequality where , and satisfy the differential equation

Remark 30. As a special case of Theorem 28 when , let and on , then we get the Sturm-Liouville type inequality where and satisfy the differential equation

Remark 31. As a special case of Theorem 28 when , let , on , and or let , on , and Then we get the Opial type inequality where and satisfy the differential equation

Example 32. Let with , and . Put , and , where arbitrary constant, and let , and , where is an arbitrary constant. This in fact satisfies the equation By applying Theorem 28, if , we getIf , then inequality (128) becomes If , and , we get that which is Hardy’s inequality.

Data Availability

No data have been used.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2019 S. H. Saker et al. 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.

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