New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations

Abstract

In this paper we define new concepts of fractional quantum calculus by defining a new q-shifting operator. After giving the basic properties we define the q-derivative and q-integral. New definitions of Riemann-Liouville fractional q-integral and q-difference on an interval $[a,b]$ are given and their basic properties are discussed. As applications of the new concepts, we prove existence and uniqueness results for first and second order initial value problems for impulsive fractional q-difference equations.