New concepts of Hahn calculus and impulsive Hahn difference equations

Jessada Tariboon, Sotiris K Ntouyas, Weerawat Sudsutad
2016 Advances in Difference Equations  
In this paper, we introduce new concepts of Hahn difference operator, the q k , ω k -Hahn difference operator. We aim to establish a calculus of differences based on the q k , ω k -Hahn difference operator. We construct a right inverse of the q k , ω k -Hahn operator and study some of its properties. As applications, we establish existence and uniqueness results for first-and second-order impulsive q k , ω k -Hahn difference equations. MSC: 34A08; 34A12; 34A60; 39A10; 39A13 Keywords: Hahn
more » ... ence operator; Jackson q-difference operator; Jackson q-integral; Nörlund sums; impulsive difference equations Preliminaries Let q ∈ (, ) and ω > . Define and let I be a real interval containing ω  . The function f is called q, ω-differentiable on I, if D q,ω f (t) exists for all t ∈ I. Note that when q →  we obtain the forward ω-difference operator and when ω =  we obtain the Jackson q-difference operator provided that f () exists. Here f is supposed to be defined on a q-geometric set A ⊂ R, for which qt ∈ A whenever t ∈ A. Hence, we can state that the D q,ω operator generalizes (in the limit) the forward ωdifference and the Jackson q-difference operators [, ]. Notice also that, under appropriate conditions, lim q→,ω→ D q,ω f (t) = f (t). The Hahn difference operator has the following properties.
doi:10.1186/s13662-016-0982-4 fatcat:5yqxk34lurgbvfgyqrresfxixe