The Diamond Integral on Time Scales

Abstract

We define a more general type of integral on time scales. The new diamond integral is a refined version of the diamond-alpha integral introduced in 2006 by Sheng et al. A mean value theorem for the diamond integral is proved, as well as versions of Holder's, Cauchy-Schwarz's and Minkowski's inequalities.Comment: This is a preprint of a paper whose final and definite form will appear in the Bulletin of the Malaysian Mathematical Sciences Society. Paper submitted 12-Feb-2013; revised 07-May-2013; accepted for publication 05-Jun-201

Some Dynamic Inequalities via Diamond Integrals for Function of Several Variables

“…(Second proof of (10)). Assume L.H.S of (11) and applying Minkowski's inequality ( [22], Theorem 5) and Fubini's Theorem, we get:…”

“…In [22], the authors provided the more refined form of diamond-α integrals, which are called diamond integrals, and are of tremendous interest including in the classical case T = R. Diamond Alpha Integral ( [22]) Consider l : T → R to be a continuous mapping and c, d ∈ T(c < d). The diamond alpha integral of l from c to d is defined by:…”

Nonsymmetric and symmetric fractional calculi on arbitrary nonempty closed sets

“…The symmetric calculus on time scales was recently proposed and investigated in [13,14,18]. We refer the reader to these references for the motivation to study the symmetric calculus and for a deep understanding of the theory.…”

Helmholtz theorem for Hamiltonian systems on time scales

“…The further extension of the Hamiltonian Helmholtz problem is to consider the derivative as combinations of ∆ and ∇ such that ⋄ = ∆+∇ 2 which is the diamond integral for which motivations and definitions can be found in [19], [11] and references therein.…”