Abstract:
We investigate some new nonlinear dynamic inequalities on time scales. Our results unify and extend some integral inequalities and their corresponding discrete analogues. The inequalities given here can be used to investigate the properties of certain dynamic equations on time scales.

Abstract:
We study some nonlinear dynamic integral inequalities on time scales by introducing two adjusting parameters, which provide improved bounds on unknown functions. Our results include many existing ones in the literature as special cases and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales.

Abstract:
We establish some delay integral inequalities on time scales, which on one hand provide a handy tool in the study of qualitative as well as quantitative properties of solutions of certain delay dynamic equations on time scales and on the other hand unify some known continuous and discrete results in the literature.

Abstract:
We present here some general fractional Schl?milch’s
type and Rogers-H?lder’s
type dynamic inequalities for convex functions harmonized on time scales. First
we present general fractional Schl?milch’s type dynamic inequalities and
generalize it for convex functions of several variables by using Bernoulli’s
inequality, generalized Jensen’s inequality and Fubini’s theorem on diamond-αcalculus. To conclude
our main results, we present general fractional Rogers-H?lder’s type dynamic
inequalities for convex functions by using general fractional Schl?milch’s type
dynamic inequality on diamond-αcalculus for p_{i}>1 with .

Abstract:
Our aim in this paper is to establish some explicit bounds of the unknown function in a certain class of nonlinear dynamic inequalities in two independent variables on time scales which are unbounded above. These on the one hand generalize and on the other hand furnish a handy tool for the study of qualitative as well as quantitative properties of solutions of partial dynamic equations on time scales. Some examples are considered to demonstrate the applications of the results.

Abstract:
By using the theory of calculus on time scales and some mathematical methods, several dynamic inequalities on time scales are established. Based on these results, we derive some sufficient conditions for permanence of predator-prey system incorporating a prey refuge on time scales. Finally, examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

Abstract:
In the present paper we establish new inequalities of Ostrowski and Gruss type for triple integrals involving three functions and their partial derivatives. The discrete Ostrowski and Gruss type inequalities for triple sums are also given.

Abstract:
We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of H？lder's inequality, a simple consequence of Keller's chain rule and Taylor monomials on time scales. Some continuous and discrete inequalities will be derived from our results as special cases. 1. Introduction In the past decade a number of Opial dynamic inequalities have been established by some authors which are motivated by some applications; we refer to the papers [1–3]. The general idea is to prove a result for a dynamic inequality where the domain of the unknown function is a so-called time scale , which may be an arbitrary closed subset of the real numbers , to avoid proving results twice, once on a continuous time scale which leads to a differential inequality and once again on a discrete time scale which leads to a difference inequality. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see [4]), that is, when , and where . A cover story article in New Scientist [5] discusses several possible applications of time scales. In this paper, we will assume that and define the time scale interval by . Since the continuous and discrete inequalities involving higher order derivatives are important in the analysis of qualitative properties of solutions of differential and difference equations [6–8], we also believe that the dynamic inequalities involving higher order derivatives on time scales will play the same effective act in the analysis of qualitative properties of solutions of dynamic equations [2, 3, 9]. To the best of the author’s knowledge there are few inequalities involving higher order derivatives established in the literature [10–13]. In the following, we recall some of these results that serve and motivate the contents of this paper. In [13] the authors proved that if is delta differentiable times with , for , and is a positive rd-continuous function on , then In [10] it is proved that if is delta differentiable times ( odd) with , for , then where and satisfy . Also in [10] it is proved that if is delta differentiable times with , for , and is increasing, then where and satisfy . As a generalization of (1.3) it is proved in [10] that if is delta differentiable times with , for , and is increasing, then where and satisfy . In [12] the authors proved that if and are positive rd-continuous functions on such that is nonincreasing, and is delta differentiable times with , for , then where and . For contributions of

Abstract:
In this paper we derive a new inequality of Ostrowski-Gruss type on time scales and thus unify corresponding continuous and discrete versions. We also apply our result to the quantum calculus case.