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In the present article, we proved some new dynamic inequalities of Hardy‐type on an arbitrary time scale. We have used dynamic Minkowski inequality, dynamic Hölder's inequality, dynamic Jensen's inequality and Fubini theorem on time scales to prove our main results. As special case of our results, some Hardy‐type integral inequalities known in the literature have been recaptured, and some original discrete inequalities of Hardy‐type have been established. [ABSTRACT FROM AUTHOR]

We established some new α -conformable dynamic inequalities of Hardy–Knopp type. Some new generalizations of dynamic inequalities of α -conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy's inequality for several functions of α -conformable calculus. Our results are proved by using two-dimensional dynamic Jensen's inequality and Fubini's theorem on time scales. When α = 1 , then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we derived Hardy's inequality for T = R , T = Z and T = h Z . Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

The main objective of the present article is to prove some new delay nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. We introduce very important generalized results with the help of Leibniz integral rule on time scales. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Our results can be used as handy tools for the study of qualitative and quantitative properties of solutions of dynamic equations on time scales. Some examples are provided to demonstrate the applications of the results. [ABSTRACT FROM AUTHOR]

Throughout this article, generalizations of some Grónwall–Bellman integral inequalities for two real-valued unknown functions in n independent variables are introduced. We are looking at some novel explicit bounds of a particular class of Young and Pachpatte integral inequalities. The results in this paper can be utilized as a useful way to investigate the uniqueness, boundedness, continuousness, dependence and stability of nonlinear hyperbolic partial integro-differential equations. To highlight our research advantages, several implementations of these findings will be presented. Young's method, which depends on a Riemann method, will follow to prove the key results. Symmetry plays an essential role in determining the correct methods for solving dynamic inequalities. [ABSTRACT FROM AUTHOR]

Several inverse integral inequalities were proved in 2004 by Yong. It is our aim in this paper to extend these inequalities to time scales. Furthermore, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Our results are proved using some algebraic inequalities, inverse Hölder's inequality and inverse Jensen's inequality on time scales. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

In the present article, we prove some new generalizations of dynamic inequalities of Hardy-type by utilizing diamond- α dynamic integrals on time scales. Furthermore, new generalizations of dynamic inequalities of Hardy-type in two variables on time scales are proved. Moreover, we present Hardy inequalities for several functions by using the diamond- α dynamic integrals on time scales. The results are proved by using the dynamic Jensen inequality and the Fubini theorem on time scales. Our main results extend existing results of the integral and discrete Hardy-type inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [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]

We prove new Hardy–Copson-type (γ , a) -nabla fractional dynamic inequalities on time scales. Our results are proven by using Keller's chain rule, the integration by parts formula, and the dynamic Hölder inequality on time scales. When γ = 1 , then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we obtain new continuous and discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [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]

Our work is based on the multiple inequalities illustrated by Boudeliou and Khalaf in 2015. With the help of the Leibniz integral rule on time scales, we generalize a number of those inequalities to a general time scale. Besides that, in order to obtain some new inequalities as special cases, we also extend our inequalities to discrete, quantum, and continuous calculus. These inequalities may be of use in the analysis of some kinds of partial dynamic equations on time scales and their applications in environmental phenomena, physical and engineering sciences described by partial differential equations. [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 the present manuscript, we prove some new extensions of the dynamic Ostrowski inequality and its companion inequalities on an arbitrary time scale by using two parameters for functions whose second delta derivatives are bounded. In addition to these generalizations, some integral and discrete inequalities are obtained as special cases of main results. [ABSTRACT FROM AUTHOR]

In the present paper, we prove some new reverse type dynamic inequalities on T. Our main inequalities are proved by using the chain rule and Fubini's theorem on time scales T. Our results extend some existing results in the literature. As special cases, we obtain some new discrete inequalities, quantum inequalities and integral inequalities. [ABSTRACT FROM AUTHOR]

In this article, we will prove some new diamond alpha Hilbert-type dynamic inequalities on time scales which are defined as a linear combination of the nabla and delta integrals. These inequalities extend some known dynamic inequalities on time scales, and unify and extend some continuous inequalities and their corresponding discrete analogues. Our results will be proven by using some algebraic inequalities, diamond alpha Hölder inequality, and diamond alpha Jensen's inequality on time scales. [ABSTRACT FROM AUTHOR]

The main objective of the present article is to prove some new delay nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. We introduce very important generalized results with the help of Leibniz integral rule on time scales. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Our results can be used as handy tools for the study of qualitative and quantitative properties of solutions of dynamic equations on time scales. Some examples are provided to demonstrate the applications of the results. [ABSTRACT FROM AUTHOR]

The main aim of the present article is to introduce some new ∇-conformable dynamic inequalities of Hardy type on time scales. We present and prove several results using chain rule and Fubini's theorem on time scales. Our results generalize, complement, and extend existing results in the literature. Many special cases of the proposed results, such as new conformable fractional h-sum inequalities, new conformable fractional q-sum inequalities, and new classical conformable fractional integral inequalities, are obtained and analyzed. [ABSTRACT FROM AUTHOR]

This paper is devoted to the investigation of Ulam stability of first-order nonlinear impulsive dynamic equations on finite-time scale intervals. Our main objective is to formulate sufficient conditions under which the class of first-order nonlinear impulsive dynamic equations on time scales we consider exhibits Ulam stability. Our methods rely on the extended integral inequality on time scales for piecewise-continuous functions. We provide an example to support the validity of the results obtained. [ABSTRACT FROM AUTHOR]

Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and continuous inequalities as special cases. At the end of this paper, we apply our results to study the behaviour of the solution of an initial value problem. In selecting the best ways to solve dynamic inequalities, symmetry is crucial. [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 main objective of the present article is to prove some new ∇ dynamic inequalities of Hardy–Hilbert-type on time scales. We present and prove very important generalized results with the help of the Fenchel–Legendre transform, submultiplicative functions, and Hölder's and Jensen's inequality on time scales. We obtain some well-known time scale inequalities due to Hardy–Hilbert inequalities. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Symmetry plays an essential role in determining the correct methods for solutions to dynamic 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]

Building on the work of Josip Pečarić in 2013 and 1982 and on the work of Srivastava in 2017. We prove some new α -conformable dynamic inequalities of Steffensen-type on time scales. In the case when α = 1 , we obtain some well-known time scale inequalities due to Steffensen inequalities. For some specific time scales, we further show some relevant inequalities as special cases: α -conformable integral inequalities and α -conformable discrete inequalities. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities. [ABSTRACT FROM AUTHOR]

The motivation of this paper is to explore some new inequalities of Steffensen-type which were demonstrated by Pečarić and Kalamir in 2014. The main idea is to investigate a class of certain inequalities by employing diamond- α dynamic integral on time scales. In addition, to obtain some new inequalities as special cases, we also extend our results to continuous and discrete calculations. [ABSTRACT FROM AUTHOR]

In this paper, we will prove some new dynamic inequalities of Hardy-type on time scales. Some of the integral and difference inequalities that will be derived from our results in the continuous and discrete cases are original. The main results will be proved by using the dynamic Hölder inequality, integration by parts formula on time scales, and Keller's chain rule on time scales. We will apply the main results to the continuous calculus, discrete calculus, and q-calculus as special cases. [ABSTRACT FROM AUTHOR]

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results. [ABSTRACT FROM AUTHOR]

In this paper, some fractional Leindler and Hardy-type inequalities and their reversed will be proved by using integration by parts and Hölder inequality on conformable fractional calculus. As a special case, some classical integral inequalities will be obtained. Symmetrical properties play an essential role in determining the correct methods to solve inequalities. The new fractional inequalities in special cases yield some recent relevance, which also provide new estimates on inequalities of these type. [ABSTRACT FROM AUTHOR]