Inequalities on Time Scales: A Survey

Abstract

The study of dynamic equations on time scales, which goes back to its founder Stefan Hilger (1988), is an area of mathematics which is currently receiving considerable attention. Although the basic aim of this is to unify the study of differential and difference equations, it also extends these classical cases to cases “in between”. In this paper we present time scales versions of the inequalities: Holder, Cauchy-Schwarz, Minkowski, Jensen, Gronwall, Bernoulli, Bihari, Opial, Wirtinger, and Lyapunov. 1. Unifying Continuous and Discrete Analysis In 1988, Stefan Hilger [13] introduced the calculus on time scales which unifies continuous and discrete analysis. A time scale is a closed subset of the real numbers. We denote a time scale by the symbol T . For functions y defined on T , we introduce a so-called delta derivative y∆ . This delta derivative is equal to y (the usual derivative) if T = R is the set of all real numbers, and it is equal to ∆y (the usual forward difference) if T = Z is the set of all integers. Then we study dynamic equations f (t; y; y∆; y∆ 2 ; : : : ; y∆ n ) = 0; which may involve higher order derivatives as indicated. Along with such dynamic equations we consider initial values and boundary conditions. We remark that these dynamic equations are differential equations when T = R and difference equations when T = Z . Other kinds of equations are covered by them as well, such as q difference equations, where T = q := fqkj k 2 Zg[ f0g for some q > 1 and difference equations with constant step size, where T = hZ := fhkj k 2 Zg for some h > 0: Particularly useful for the discretization purpose are time scales of the form T = ftkj k 2 Zg where tk 2 R; tk < tk+1 for all k 2 Z: Mathematics subject classification (2000): 34A40, 39A13.