Characterizations of stability of first order linear Hahn difference equations

Abstract

Hahn introduced the difference operator Dq;!f(t) = 􀀀 f(qt + !) 􀀀 f(t)=􀀀 t(q 􀀀 1) + ! in 1949, where 0 0 are xed real numbers. This operator extends the classical difference operator M! f(t) = (f(t + !) 􀀀 f(t))=! as well as Jackson q􀀀dierence operator Dqf(t) = (f(qt) 􀀀 f(t))=(t(q 􀀀 1)). In this paper, our objective is to establish characterizations of many types of stability, like (uniform, uniform exponential, -) stability of linear Hahn difference equations of the form Dq;!x(t) = p(t)x(t) + f(t). At the end, we give two illustrative examples.