Using the properties of superquadratic and subquadratic functions, we establish some new refinement multidimensional dynamic inequalities of Hardy’s type on time scales. Our results contain some of the recent results related to classical multidimensional Hardy’s and Polya–Knopp’s inequalities on time scales. To show motivation of the paper, we apply our results to obtain some particular multidimensional cases and provide refinements of some Hardy-type inequalities known in the literature.

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Refinement multidimensional dynamic inequalities with general kernels and measures

We will prove some new Opial dynamic inequalities involving higher order derivatives on time scales. The results will be proved by making use of Holder'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.

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Some Opial Dynamic Inequalities Involving Higher Order Derivatives on Time Scales

For a fourth-order differential equation, we will establish some new Lyapunov-type inequalities, which give lower bounds of the distance between zeros of a nontrivial solution and also lower bounds of the distance between zeros of a solution and/or its derivatives. The main results will be proved by making use of Hardy’s inequality and some generalizations of Opial-Wirtinger-type inequalities involving higher-order derivatives. Some examples are considered to illustrate the main results.

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Lyapunov's Type Inequalities for Fourth-Order Differential Equations

In this paper, for a fourth-order differential equation, we will establish some lower bounds for the distance between zeros of a nontrivial solution and also lower bounds for the distance between zeros of a solution and/or its derivatives. We also give some new results related to some boundary value problems in the theory of bending of beams. The main results will be proved by making use of some generalizations of Opial and Wirtinger-type inequalities. Some examples are considered to illustrate the main results. MSC: 34K11; 34C10

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New gaps between zeros of fourth-order differential equations via Opial inequalities

In this paper, we study dynamic inequalities where the domain of the unknown function is a so-called time scale T . The cases when the time scale equals to the reals or to the integers represent the classical theories of differential and of difference inequalities. A cover story article in New Scientist [24] discusses several other possible applications. Continuous and discrete inequalities are important in the analysis of qualitative properties of solutions of differential and difference equations [3, 18, 19] and as a result we believe that dynamic inequalities on time scales will be important in the analysis of qualitative properties of solutions of dynamic equations [20, 21]. In this paper, we will prove some new dynamic inequalities involving higher order on time scales. The three most popular examples of calculus on time scales are differential calculus, difference calculus, and quantum calculus (see Kac and Cheung [9]), i.e, when T = R, T = N and T = q0 = {qt : t ∈ N0} where q > 1. For more details of time scale analysis we refer the reader to the two books by Bohner and Peterson [6], [7] which summarize and organize much of the time scale calculus. The study of dynamic inequalities of Opial type on time scales was initiated by Bohner and Kaymakcalan [5] in 2001; see also the recent papers [10], [23] and [25] and the references cited therein. In [5] the authors showed that if x : [0,b]∩T → R is delta differentiable with x(0) = 0, then ∫ b

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Higher order dynamic inequalities on time scales

In this paper, for a third order differential equation, we will establish some new inequalities of Lyapunov type. These inequalities give implicit lower bounds on the distance between zeros of a nontrivial solution and also lower bounds for the spacing between zeros of a solution and/or its derivatives. The main results will be proved by making use of the Holder inequality and some generalizations of Opial and Wirtinger type inequalities. Some examples are considered to illustrate the main results. AMS (MOS) Subject Classification. 34K11, 34C10

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Papers

Applications of opial and wirtinger inequalities on zeros of third order differential equations

Global Attractivity of a Multiplicative Delay Population Dynamics Model