Search

In this paper, we prove some new dynamic inequalities similar to Hardy's inequality on time scales T. The results as special cases when T = R contain continuous inequalities similar to Hardy's inequality, and when T = Z, the results contain discrete inequalities that are essentially new to the best of the authors' knowledge. R and Z denote the set of all real numbers and the set of all integers, respectively. Our main results are proved by using the Holder € inequality, Chain rule, and some properties of multiple integrals on time scales. Furthermore, some applications and examples are given to illustrate the investigated results. [ABSTRACT FROM AUTHOR]

We investigated several novel conformable fractional gamma-nabla dynamic Hardy–Hilbert inequalities on time scales in this study. Several continuous inequalities and their corresponding discrete analogues in the literature are combined and expanded by these inequalities. Hölder's inequality on time scales and a few algebraic inequalities are used to demonstrate our findings. [ABSTRACT FROM AUTHOR]

In this article, we will prove some new conformable fractional Hilbert-type dynamic inequalities on time scales. These inequalities generalize some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using some algebraic inequalities, conformable fractional H¨older inequalities, and conformable fractional Jensen's inequalities on time scales. [ABSTRACT FROM AUTHOR]

In the present paper, some new generalizations of dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales are established. Some integral and discrete Gronwall–Bellman–Pachpatte-type inequalities that are given as special cases of main results are original. The main results are proved by using the dynamic Leibniz integral rule on time scales. To highlight our research advantages, several implementations of these findings are presented. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

We prove some new dynamic inequalities of the Gronwall–Bellman–Pachpatte type on time scales. Our results can be used in analyses as useful tools for some types of partial dynamic equations on time scales and in their applications in environmental phenomena and physical and engineering sciences that are described by partial differential equations. [ABSTRACT FROM AUTHOR]

As a new usage of Leibniz integral rule on time scales, we proved some new extensions of dynamic Gronwall–Pachpatte-type inequalities on time scales. Our results extend some existing results in the literature. Some integral and discrete inequalities are obtained as special cases of the main results. The inequalities proved here can be used in the analysis as handy tools to study the stability, boundedness, existence, uniqueness and oscillation behavior for some kinds of partial dynamic equations on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

We established several novel inequalities of Gronwall–Pachpatte type on time scales. Our results can be used as handy tools to study the qualitative and quantitative properties of the solutions of the initial boundary value problem for a partial delay dynamic equation. The Leibniz integral rule on time scales has been used in the technique of our proof. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

In this article, with the help of Leibniz integral rule on time scales, we prove some new dynamic inequalities of Gronwall–Bellman–Pachpatte-type on time scales. These inequalities can be used as handy tools to study the qualitative and quantitative properties of solutions of the initial boundary value problem for partial delay dynamic equation. [ABSTRACT FROM AUTHOR]

In this article, using a (γ ,a)-nabla conformable integral on time scales, we study several novel Hilbert-type dynamic inequalities via nabla time scales calculus. Our results generalize various inequalities on time scales, unifying and extending several discrete inequalities and their corresponding continuous analogues. We say that symmetry plays an essential role in determining the correct methods with which to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

In this work, we prove several new (γ, a)-nabla Bennett and Leindler dynamic inequalities on time scales. The results proved here generalize some known dynamic inequalities on time scales, unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proved by using integration by parts, chain rule and Hölder inequality for the (γ, a)-nablafractional derivative on time scales. [ABSTRACT FROM AUTHOR]

In this article, we prove new retarded dynamic inequalities on time scales that contain some integral and discrete inequalities reported in the literature. These inequalities can be used as handy tools for the study of qualitative properties of solutions of dynamic equations on time scales. Some examples are included to demonstrate the applications of our results. [ABSTRACT FROM AUTHOR]

The primary goal of this research is to prove some new Hardy-type ∇-conformable dynamic inequalities by employing product rule, integration by parts, chain rule and (γ , a) -nabla Hölder inequality on time scales. The inequalities proved here extend and generalize existing results in the literature. Further, in the case when γ = 1 , we obtain some well-known time scale inequalities due to Hardy inequalities. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities and new classical conformable fractional integral inequalities. [ABSTRACT FROM AUTHOR]

The primary goal of this our research is to prove several new ∇-conformable dynamic Steffensen inequalities that were demonstrated in recent works. Our results generalize and extend existing results in the literature. Many special cases of the proposed results are obtained and analyzed such as new conformable fractional sum inequalities and new classical conformable fractional integral inequalities. [ABSTRACT FROM AUTHOR]