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Recent Advances in Function Spaces and its Applications in Fractional Differential Equations 2020

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Volume 2020 |Article ID 5417084 | https://doi.org/10.1155/2020/5417084

H. A. Abd El-Hamid, H. M. Rezk, A. M. Ahmed, Ghada AlNemer, M. Zakarya, H. A. El Saify, "Dynamic Inequalities in Quotients with General Kernels and Measures", Journal of Function Spaces, vol. 2020, Article ID 5417084, 12 pages, 2020. https://doi.org/10.1155/2020/5417084

Dynamic Inequalities in Quotients with General Kernels and Measures

Guest Editor: Lishan Liu
Received05 Jun 2020
Accepted14 Jul 2020
Published03 Aug 2020

Abstract

By utilizing the precepts related to the punctuation of time scales , we present some nouveau forms in quotients for Hardy’s and related inequalities on time scales. In particular, some recent results for the Pólya-Knopp, Hardy-Hilbert, and Hardy-Littlewood-Pólya-type inequalities are presented.

1. Introduction

In [1], Hardy stated and proved the following integral inequality: where is a positive function and the constant is the best ever. We rewrite (1) with the function rather than , and by assuming limit we acquired the limiting instance of the inequality of Hardy known as the inequality of Pólya-Knopp (see [2]), that is,

Lately, Kaijser et al. in [3] pointed out that both (1) and (2) are just special states of the much more general inequality of Hardy-Knopp for positive function : where is a convex function. A popularization of inequality (3) with two weight functions is proved in [4]. Particularly, it was proved that if , is a nonnegative function and is a convex on , then the inequality is available for all integrable functions , such that and is defined by

Using the inequality of Jensen for convex functions and the theorem of Fubini, Kaijser et al. [5] established an inviting popularization of (1). Particularly, they proved that if and are nonnegative functions such that , for and then where is a convex function and

For further popularization of (7), Krulić et al. [6] proved that if and are two measure spaces with positive -finite measures and which are nonnegative measurable functions such that , , , and is defined by then the inequality is available for all measurable functions such that , where is a convex function and is defined by

In [7], Iqbal et al. checked some new weighted Hardy-type inequalities on and measure spaces with positive -finite measures by replacing by and by , where are measurable functions in (10) as follows: where , is a convex function, is a nonnegative measurable function, and are defined by

In [8], the author founded the time scale version of (1). Particularly, he proved that if , is a nonnegative function and exists, then

In [9], the authors proved the time scale version of (4), which is given by where is a convex function, is a nonnegative function, and is defined by

In [10], the authors outstretched a number of Hardy-type inequalities with certain kernels on time scale. Namely, they proved that if and are two time scale measure spaces, and , which are nonnegative measurable functions such that and is defined by then the inequality is available for all -integrable such that and is a convex function. For development of dynamic inequalities on time scale calculus, we refer the reader to articles [1118].

The article is regimented as follows. In Section 2, we recall the precepts related to the punctuation of time scales. In Section 3, we prove our results and give some remarks. Particularly, we prove a general dynamic weighted Hardy-type inequality with a nonnegative kernel. In Section 4, we critique a few particular states of the obtained inequalities, related to power and exponential functions and to the most simplest shapes of kernels.

2. Preliminaries

In this section, we will premise some fundamental precepts and effects on time scales which will be beneficial for deducing our major results. The following definitions and theorems are referred from [19, 20].

A nonempty arbitrary locked subplot of the real numbers is called a time scale which is denoted by For , if and , then the forward jump operator and the backward jump operator are defined as respectively. The -derivative of at is the number that enjoys the property that for all , there exists a neighborhood of such that

Furthermore, is called a delta differentiable on if it is delta differentiable at every . Similarly, for , we define the -derivative of at as the number that enjoys the property that for all , there exists a neighborhood of such that

Moreover, is called a nabla differentiable on if it is nabla differentiable at every . For , the delta integral of is defined as

Similarly, for the nabla integral of is defined as

Now, let be differentiable on in the and senses. For , where , the diamond- dynamic derivative is defined by

The diamond- derivative debases to the standard -derivative for or the standard -derivative for

Next, we recall the inequality of Minkowski and the inequality of Jensen on time scales which are utilized in the proof of the major results.

Theorem 1. Suppose and are two finite-dimensional time scale measure spaces, and let , , and be positive functions on , , and respectively. If then the inequality is available for all integrals in (26). If and are available, then (26) is reversed. For in addition with (27), if is available, then again (26) is reversed.

Theorem 2. Let and . Suppose that and are nonnegative with If is convex, then

3. Inequalities with General Kernels

In this section, we state and prove our major results. Before presenting the results, we labeled the following hypotheses. (H1) and are two time scale measure spaces(H2) is a positive measurable kernel and(H3) is a -integrable andwhere

In what follows, we will prove the foundation theorem that will be the decisive step in establishing our major result.

Theorem 3. Assume (H1)–(H3).
If is a positive convex, then the following inequality is available for all nonnegative -integrable function such that , where is defined by

Proof. We begin with an evident identity Utilizing the inequality of Jensen (29) and the theorem of Minkowski (2) on (34), we find that Taking into computation definition (31) of , it follows that Finally, elevating (36) to the th power, we acquired (33).

Remark 4. For the Lebesgue scale measures , , and , inequality (32) reduces to inequality (19) premised in Section 1. Choosing in Theorem 3, then inequality (32) reduces to inequality (10).

In what follows, we labeled few particular convex functions starting with power functions.

Corollary 5. Suppose the supposition of Theorem 3 be gratified only with for . Then, inequality (32) yields the following result:

Remark 6. For the Lebesgue scale measures , , and , Corollary 5 coincides with Corollary 3.3 in [10].

Now, considering Theorem 3 with and , we get the following inviting result.

Corollary 7. Suppose the supposition of Theorem 3 is gratified only with . If , then inequality (32) yields the following result:

Remark 8. For the Lebesgue scale measures , , and , Corollary 7 coincides with Corollary 3.4 in [10]. Also, in the special case , Remark 8 coincides with Corollary 3.5 in [10].

Now, by using a special substitution, we obtain our central result; that is, if we replace by and by , where are measurable functions, we obtain these results.

Theorem 9. Assume (H1) and (H2) and be defined on by where If is a positive convex, then the following inequality is available for all measurable functions and

Remark 10. Choosing in Theorem 9, then inequality (40) reduces to inequality (12) established by Iqbal et al. (Theorem 2.1, [7]).

As a special case of Theorem 9 for , we get the next corollary. Also, we note that the function need not to be positive.

Corollary 11. Assume (H1) and (H2) and be defined on by If is a positive convex, then the following inequality is available for all measurable functions and is defined by (41).

Corollary 12. Suppose the supposition of Theorem 9 is gratified only with for . Then, inequality (40) yields the following result:

Corollary 13. Suppose the supposition of Theorem 9 is gratified only with . If , then inequality (40) yields the following result:

Remark 14. Choosing in Corollary 11, then (43) takes the form where which is the same result due to Iqbal et al. (Corollary 2.2, [7]).

Corollary 15. For the Lebesgue diamond- scale measures , , , and , inequality (40) takes the form where , is a positive convex, and

Remark 16. If we set , then inequality (48) takes the form which coincided with inequality (7) in [21].

Remark 17. If we set , then inequality (50) takes the form which coincided with Corollary 4 in [21].

Remark 18. In Remark 16, if we set , , , and be a convex function, then We assume that and define Then, inequality (50) takes the form which coincided with inequality (2.7) in [22] (Theorem 2.2).

Remark 19. Apply Remark 8 for , which is convex with domain . Then, inequality (54) takes the form which coincided with inequality (2.10) in [22] (Corollary 2.3).

Remark 20. Apply Remark 18 for . Then, inequality (54) takes the form which coincided with inequality (2.11) in [22] (Corollary 2.4).

Remark 21. If we set , then the delta version form of inequality (50) takes the form which coincided with inequality (8) in [21].

Remark 22. If we set , then the nabla version form of inequality (50) takes the form which coincided with inequality (9) in [21].

Remark 23. In particular, if and , then inequality (50) takes the form where This result is given in [23, 24].

4. Inequalities with Special Kernels

The next theorem states the general result for Hardy’s inequality in quotient.

Theorem 24. Suppose and be a weight function. Define on by If is a positive convex, then the following inequality is available for all measurable functions .

Proof. Rewrite inequality (40) with , , and . Define the kernel by Then, in (41) takes the form Substituting in (40), we acquired (47).

Corollary 25. If we apply Theorem 24 for , , then (62) takes the form where

Remark 26. If we put in Corollary 25, , and , then we acquired Hardy’s inequality on time scale where

Remark 27. If we put in Theorem 24, , and , we see that and then (62) takes the form where which is the same result due to Iqbal et al. (Theorem 2.4, [7]).

Example 1. If we put , , and a particular weight function in (62), we obtain and inequality (62) becomes For , inequality (73) reduces to

Remark 28. When , , , and , we see that , and then (74) takes the form

Now, for the convex function defined by , we can give the general form of Pólya-Knopp’s inequality in quotient.

Theorem 29. Suppose and be a weight function defined on . Define on by Then, the following inequality: is available for all positive measurable functions .

Proof. We rewrite inequality (40) with , , , and defined by ; we obtain Define such as Theorem 24. Substituting defined by (64) in (78), we acquired Replacing by in (79), we acquired (77).

Remark 30. If we put in Theorem 29, , and , we see that