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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 objective of this paper is to prove some new dynamic inequalities of Hardy type on time scales which generalize and improve some recent results given in the literature. Further, we derive some new weighted Hardy dynamic inequalities involving many functions on time scales. As special cases, we get continuous and discrete inequalities. [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]

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]

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]