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
In this paper, some integral inequalities based on the general quantum difference operatorwhere β is a strictly increasing continuous function, defined on an interval I ⊆ R, that has one fixed point s 0 ∈ I. The β-Hölder and β-Minkowski inequalities are proved. Also, the β-Gronwall, β-Bernoulli, and some related inequalities are shown. Finally, the β-Lyapunov inequality is established.
In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations
, in a neighborhood of the unique fixed point of the strictly increasing continuous function β, defined on an interval . These equations are based on the general quantum difference operator , which is defined by , . We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.
In this paper, we introduce a general quantum Laplace transform $\mathcal{L}_{\beta }$
L
β
and some of its properties associated with the general quantum difference operator ${D}_{\beta }f(t)= ({f(\beta (t))-f(t)} )/ ({ \beta (t)-t} )$
D
β
f
(
t
)
=
(
f
(
β
(
t
)
)
−
f
(
t
)
)
/
(
β
(
t
)
−
t
)
, β is a strictly increasing continuous function. In addition, we compute the β-Laplace transform of some fundamental functions. As application we solve some β-difference equations using the β-Laplace transform. Finally, we present the inverse β-Laplace transform $\mathcal{L}_{\beta }^{-1}$
L
β
−
1
.
In this paper, we derive the solutions of homogeneous and non-homogeneous nth-order linear general quantum difference equations based on the general quantum difference operator D β which is defined by D β f (t) = (f (β(t))-f (t))/(β(t)-t), β(t) = t, where β is a strictly increasing continuous function defined on an interval I ⊆ R that has only one fixed point s 0 ∈ I. We also give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of these equations. Furthermore, we present the fundamental set of solutions when the coefficients are constants, the β-Wronskian associated with D β , and Liouville's formula for the β-difference equations. Finally, we introduce the undetermined coefficients, the variation of parameters, and the annihilator methods for the non-homogeneous β-difference equations.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.