Abstract

The paper derives some new time-scale (TS) dynamic inequalities for multiple integrals. The obtained inequalities are special cases of Copson integral using Steklov operator in (TS) version with high dimension. We prove the inequalities with several formulas for the operator and in different cases and for every , using time-scales (TSs) setting for integral properties, chain rules, Fubini’s theorem, and Hölder’s inequality.

1. Introduction

Equations and inequalities are the core of scientific study and have a great influence on a huge number of applications. A large number of physical phenomena and engineering studies have been analyzed and explained through equations and inequalities. For this reason, the study in this field developed rapidly and many types of inequalities and equations appeared. Dynamic inequalities on (TS) are some of the important inequalities that were extended by a lot of researchers and have interesting applications. Furthermore, dynamic inequalities are used to study the behaviour of dynamic equations.

Mathematical analysis has been the most important study in mathematics for the past three decades. Integral inequalities are one of the main studies and the core of mathematical analysis. In the 20th century, a significant part of science was numerical inequalities as the first composition to be released in 1934, through the published study by P’olya et al. [1]. This framework of inequalities played a vital role in the improvement processes and various applications of mathematics.

A large number of essential studies of integral inequalities appeared in the twentieth century, including pure and applied mathematics study. In 1920, Hardy produced the discrete Hardy inequality [2]. This inequality was also proved by himself in [3] (see also [4]), using the variations calculus to obtain the following inequality that is very valuable across both technological sciences and mathematics. If and in andthenwhere is the best possible constant (BPC). Several important assessments and their implementation are done by inequality (1). Furthermore, the inequality is true in case ,where . The classical inequality of Hardy declares that if and is nonnegative and measurable on , then (2) is valid except a.e. in , considering the (BPC).

Integral inequalities (3) and (4) are established in 1928 by Hardy [5].

Let be a nonnegative measurable function on :Then,Later, in 1976, Copson studied the integral inequalities ([6], Theorem 1, Theorem 3) as follows.

Let and be functions such that they are nonnegative measurable on ;Then,

Many papers included new extensions and generalizations for the inequalities above in more general settings. For instance, in 1979, some generalizations of Hardy-type inequality were proved by Chan [7]. Then, in 1992, Pachpatte [8] generalized the inequalities that were produced by Chan [7]. In 2005, P. Rehak used (TS) setting to extend Hardy’s inequalities [9]. In 2015, Pachpatte’s inequalities [8] were extended by Saker and O’Regan [10], with setting of (TSs). Later, some extensions of (TSs) Hardy inequalities were done for functions with high dimensions (see, for example, [1114]).

In 2021, Albalawi and Khan generalized the main integral of Hardy and Copson inequalities, using the Steklov operator. The operator is defined in the following formulas with considering conditions in two cases (for more details, see [15]).

The aim of this paper is extending the study in [16] that was used for some new Hardy-type inequalities to obtain new special Copson inequalities with the Steklov operator (see [15]) in (TS) versions with high dimension. The results below are proved in two cases and by considering some general conditions that can be applied for any variable in the integral. To achieve this paper, we use (TSs) settings in integrals properties, chain rules, Hölder’s inequality, and Fubini’s theorem.

The paper takes the following structure: After introduction, the main concepts of (TSs) are presented in Section 2. Then, in Section 3, we generalized a class of Copson inequalities pertaining the Steklov operator with (TS) in high dimension. Lastly, conclusion of our results is presented.

2. Preliminaries and Lemmas on Time Scales

We state the main concepts of (TSs) that are used in this paper (for more details about (TS) calculus, see [17, 18]).

(TS) calculus in continuous case and discrete analysis was introduced by Hilger [19] in 1988. We denote to a subset (TS) of the real numbers by . Hence, the sets of numbers , , and can be considered as (TSs).

Let be a forward jump operator, such that , while is the backward jump operator, given by for all .

If , then t is right-scattered, and if , t is left-scatted. In the case if points are right-scattered and left-scattered at the same time, then they will be isolated. The point t is right-dense if and , while t is left-dense if and .

Let be a continuous function and if it satisfied the continuity at all right-dense points in and the limits of the left-sided exist (finite) at all left-dense points in , then is known rd-continuous. We use to denote the space of all rd-continuous.

A function is -differentiable at , if there is a real number and for all , there exists a neighbor. of satisfies

The -derivative of a function in high order is given by

If the -derivative of exists, the following examples show that the delta derivative for every number set of (TSs).

If , then

If , then

Let ; if is continuous at right-scattered , then it is delta-derivative of the function , given by

In the case of is not right-scattered, then the derivative of is given by

Here, the limit exists. Note that if , we have

If , we have

Lemma 1. Let be delta-differentiable. Then,

The Cauchy integral of a delta-differential function of is defined by

The time-scale integration by parts formula is given by

The infinite integrals are defined by

If , we have

If , we get

Lemma 2 (chain rule [16]). Assume a continuous function, , a delta-differentiable, and , on and a continuous differentiable . Then, there exists with

Lemma 3 (dynamic Hölder inequality). Let and . If with , then

Theorem 4 (Fubini’s theorem [20]). Let and be (TS) measure spaces with finite dimension. Consider as the measure space, where is the algebra product that is generated by andThen, Fubini’s theorem satisfied.
To be more accurate, if is —integrable,andThen,

3. Main Results

A new (TS) version of Copson-type inequality with Steklov operator for multiple integrals is obtained in this section. We consider the nonnegative rd-continuous functions , , , and are -integrable and defined integrals. Throughout this paper, we set as the Copson–Steklov-type operator considering the existence of the integral and also finite.

Theorem 5. Let be a (TS) and , for with . In addition, let , , , and be nonnegative and rd-continuous functions on . Furthermore, assume there exist such thatandwhere for every ,andDefine the operatorThenwhere and .

Proof. We write the left side of (10) as follows:where is the -termUsing formula (6) for integration by parts to compute , we havewhere and then and , implying that , and hence,Assume such thatwhere , and since , we haveSubstituting the previous quantities in (12) and since and , then we haveAssume such thatThen, we obtainSince , then we haveThen, Hölder’s inequality (8) with indices and can be applied:Substituting in (11) and applying Fubini’s Theorem 4, then we obtain the inequality

Corollary 6. If in Theorem 5, inequality (10) becomes

Corollary 7. If in Corollary 6, we obtain

Remark 8. Assume and in Corollary 7; then, we have Corollary 3 in [15]

Theorem 9. Let be (TS) and , for with . In addition, let , , , and be nonnegative and rd-continuous functions on . Furthermore, assume there exist such thatandwhere ; for every ,andDefine the operatorThen,where and .

Proof. We write the left side of (17) as follows:Use formula (6) to calculate the following term:where and .
Using (7) and the product rule (5), there exists such thatAssume such thatSince , , and , thenBy integration, we haveNow, we calculate and we obtainAssume such thatwhereand since and , then we haveThen,Hence, we haveUsing Hölder’s inequality, where and , we obtainSubstituting (22) in (17), we have

Corollary 10. If and in Theorem 9, we getwhere

Remark 11. Assume and in Corollary 10; we have Corollary 5 in [15].

Theorem 12. Let be (TS) and , for with . In addition, let , , , and be nonnegative rd-continuous functions on . Furthermore, assume there exist such thatandwhere for every ,andDefine the operatorThen,where and .

Proof. We write the left side of (24) as follows:Apply (6) to calculate the following term:where . Using the chain rule on (TS) (7) and product rule (5), there exist such thatAssume such thatSince , , and , thenimplyingWe calculate , and we obtainAssume such thatwhereand since , then we haveSince , thenHence, we haveThen, Hölder’s inequality (8) can be applied with indices and :Substituting (28) in (26), we have

Corollary 13. If and in Theorem 12, we obtainwhere

Example 14. Choose , , and in Theorem 12. Hence, we get

4. Conclusions

(TSs) calculus is used in this paper to prove special cases of (TS) Copson–Steklov-type inequalities with several variables. The obtained inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Furthermore, the inequalities can be discussed in calculus, discrete calculus, and quantum calculus. As a perspective, we propose to study these results for other kinds of operators and solve the singularity that appeared in Theorem 12 with case .

Data Availability

All data that support the findings of this study are included within the article.

Conflicts of Interest

The author declares no conflicts of interest.

Acknowledgments

This work was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R157), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.