Theory of nth-order linear general quantum difference equations

Abstract

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.

A β -Convolution Theorem Associated with the General Quantum Difference Operator

“…Let p, q : I ⟶ ℂ be continuous functions at s 0 and satisfy the condition 1 + ðβðtÞ − tÞpðtÞ ≠ 0 for all t ∈ I, then the following properties hold [24]: Journal of Function Spaces Recently, Cardoso [25] investigated the β-Lagrange's identity for the β-Sturm-Liouville eigenvalue problem and proved that it is self-adjoint in L 2 β ð½a, bÞ. For more results in β-calculus, we refer the readers to see [26][27][28][29].…”

The Directional Derivative in General Quantum Calculus

“…Moreover, the β-Laplace transform and the β-convolution theorem were given in [24,25]. For more details about the β-calculus associated with D β , see [26][27][28][29]. The benefit of using the β-calculus is that, from its results, we can deduce the other quantum calculi results and, this way, exhibit the properties of quantum calculus in a unified way.…”

A $β$-Sturm Liouville problem associated with the general quantum operator

“…Later, in [30], some new results on homogeneous second order linear general quantum difference equations were presented and, in [36], the exponential, trigonometric and hyperbolic functions were introduced. The theory of nth-order linear general quantum difference equations was developed in [31] while the general quantum variational calculus was build up in [24]. In [22], properties of the β-Lebesgue spaces were produced and, recently, in [43], a general quantum Laplace transform was displayed and studied.…”