This is fractionalising of the n-iterated nabla
Probably more work has been developed for the backward or nabla
difference and we refer the readers to the papers , .
He describes the right delta and right nabla
fractional calculus on time scales; right delta and right nabla
discrete fractional calculus in the Caputo sense; representations formulae of functions on time scales and Ostrowski type inequalities, Landau type inequalities, GrEss type, and comparison of means inequalities; integral operator inequalities and their multivariate vectorial versions using convexity of functions over time scales; general GrEss and Ostrowski inequalities using s-convexity; essential and convexity Ostrowski and GrEss inequalities using several functions; general fractional Hermite-Hadamard inequalities using m-convexity and convexity; and the reduction method in fractional calculus and fractional Ostrowski type inequalities.
In this article, we extend this study to linear fractional nabla
In 2001 the time scale nabla
calculus (or simply nabla
calculus) was introduced by Atici and Guseinov .
Ltd., The Mews Pickett's Lodge, Pickett's Lane, Salfords, Surrey RH1 5RG, England; www.nabla.co.uk.
Throughout this work a knowledge and understanding of time scales and time-scale notation is assumed; for an excellent introduction to the calculus on time scales, see Bohner and Peterson [5,6] for a general overview, the paper introducing nabla
derivatives by Atici and Guseinov , and the introduction of multiple integrals on time scales by Bohner and Guseinov .
Keywords: Time scale, delta and nabla
derivatives and integrals, Green's function, completely continuous operator, eigenfunction expansion.
but he does not use a name such as gradient, del, or nabla
for this operator in his Treatise, even though he corresponds with his lifelong friend P.