Analysis on Measure Chains — A Unified Approach to Continuous and Discrete Calculus

Abstract

In this paper, some various types of Opial's inequality involving several functions and their higher-order derivatives are presented on arbitrary time scales. The well-known Muirhead's inequality is employed to obtain very interesting results. While dealing with higher-order derivatives , the generalized Taylor's formula and the generalized polynomials are used to simplify our proofs too. Our new results generalize and extend the existing results in the literature, and some of the works done by Pachpatte. Moreover, our results are not only new for arbitrary time scales, but also new for the continuous and the discrete cases. Mathematics subject classification (2010): 34A40, 39A13, 26D15.

Oscillation of Second-Order Neutral Dynamic Equations with Mixed Arguments

“…In this section, using the Riccati transformation technique, we establish some oscillation criteria for (1). In the sequel, we let…”

“…The study of dynamic equations on time scales, which goes back to its founder Hilger [1], is an area of mathematics that has recently received a lot of attention. Several authors have expounded on various aspects of this new theory, see the survey paper by Agarwal et al [2] and the references cited therein.…”

“…The purpose of this paper is to derive new oscillation criteria for (1) in the cases where (2) holds or…”

Properties and Characterizations of Convex Functions on Time Scales

“…The theory of time scales was introduced by S. Hilger in his PhD thesis [4]. The calculus and applications of dynamic derivatives on time scales provide an unification and an extension of traditional differential and difference equations.…”

“…The incoming notation and definitions are from [2] and [4]. A time scale (or measure chain) is any nonempty closed subset T of R (together with the topology of subspace of R).…”