Abstract
Hahn introduced the difference operator D q , ω f(t)=(f(qt+ω)−f(t))/(t(q−1)+ω) in 1949, where 0<q<1 and ω>0 are fixed real numbers. This operator extends the classical difference operator Δ ω f(t)=(f(t+ω)−f(t))/ω as the Jackson q-difference operator D q f(t)=(f(qt)−f(t))/(t(q−1)).In this paper, we present new results of the calculus based on the Hahn difference operator. Also, we establish an existence and uniqueness result of solutions of Hahn difference equations by using the method of successive approximations.
Highlights
1 Introduction and preliminaries Hahn introduced his difference operator which is defined by f – f (t) Dq,ωf (t) = t(q – ) + ω at t = ω/( – q) and the usual derivative at t = ω/( – q), where < q < and ω > are fixed real numbers [, ]
In [ ], Annaby et al gave a rigorous analysis of the calculus associated with Dq,ω. They stated and proved some basic properties of such a calculus. They defined a right inverse of Dq,ω in terms of both the Jackson q-integral; see [ ], which contains the right inverse of Dq and Nörlund sum; cf. [ ], which involves the right inverse of ω
Each φk(t) is continuous at t = θ . (ii) we show by induction that φm+ (t) – φm(t) ≤ KAmhm+, t ∈ [θ, θ + h]
Summary
R, and f is a real valued function defined on I. Dq,ω y(t)E–r(t) = Dq,ωy(t)E–r h(t) + y(t)Dq,ωE–r(t) = Dq,ωy(t) E–r h(t) + y(t) –rE–r h(t) = E–r h(t) Dq,ωy(t) – ry(t) ≤ E–r h(t) f (t) Integrating both sides from θ to t in the inequality above, we obtain t y(t)E–r(t) – y(θ )E–r(θ ) ≤ E–r h(τ ) f (τ ) dq,ωτ. This implies that t y(t) ≤ y(θ )er(t) + er(t) E–r h(τ ) f (τ ) dq,ωτ. T y(t) ≤ f (t) + ry(τ ) dq,ωτ θ for all t ∈ I implies that t y(t) ≤ f (t) + rer(t) E–r h(τ ) f (τ ) dq,ωτ.
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