A general quantum difference calculus

Abstract

In this paper, we consider a strictly increasing continuous function β, and we present a general quantum difference operator D β which is defined to be. This operator yields the Hahn difference operator when β(t) = qt + ω, the Jackson q-difference operator when β(t) = qt, q ∈ (0, 1), ω > 0 are fixed real numbers and the forward difference operator when β(t) = t + ω, ω > 0.A calculus based on the operator D β and its inverse is established. MSC: 39A10; 39A13; 39A70; 47B39Keywords: quantum difference operator; quantum calculus; Hahn difference operator; Jackson q-difference operator

General quantum variational calculus

“…We note here that there exist discontinuous functions that are β-differentiable (for an example see [14]), something that cannot happen in the classical theory and can be seen as an advantage of the general quantum calculus.…”

“…We remark that the general quantum difference operator D β can also be defined for strictly increasing and continuous functions β that have a unique fixed point s 0 and satisfy (t − s 0 ) (β (t) − t) 0, for all t ∈ I (see [14]). …”

q-Non uniform difference calculus and classical integral inequalities

“…We particularly aim to establish q-non uniform difference versions of the well-known in differential calculus integral inequalities of Hölder, Cauchy-Schwarz, Minkowski, Grönwall, Bernoulli, and Lyapunov. Another captivating work on the raised inequalities can be found in [8] for a calculus based on a derivative also generalizing (6) and (7), but one will remark that even if that work greatly inspired us, there is not any hierarchic relationship between the calculus considered there (see [9] for a general theory) and the one considered here (see also [10][11][12]) based on (8). We will note also that (8) is at our best knowledge the most general known divided difference derivative having the property of sending a polynomial of degree n in a polynomial of degree n -1.…”

Theory of nth-order linear general quantum difference equations

“…The function f is said to be β-differentiable on I if the ordinary derivative f exists at s 0 . The general quantum difference calculus was introduced in [10]. The exponential, trigonometric, and hyperbolic functions associated with D β were presented in [9].…”

“…[10]) The following statements are true:(i) The sequence of functions {β k (t)} ∞ k=0 converges uniformly to the constant function β(t) := s 0 on every compact interval V ⊆ I containing s 0 . (ii) The series ∞ k=0 |β k (t)β k+1 (t)| is uniformly convergent to |ts 0 | on every compact interval V ⊆ I containing s 0 .…”