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_{beta } which is defined by D_{beta }{f(t)}= (f(beta (t))-f(t) )/ (beta (t)-t ), beta (t)neq t, where β is a strictly increasing continuous function defined on an interval Isubseteq mathbb{R} that has only one fixed point s_{0}in {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_{beta }, 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.
Highlights
Quantum difference operator allows us to deal with sets of non-differentiable functions
Its applications are used in many mathematical fields such as the calculus of variations, orthogonal polynomials, basic hypergeometric functions, quantum mechanics, and the theory of scale relativity; see, e.g., [3, 5, 7, 13, 14]
T = s0, t = s0, where f : I → X is a function defined on an interval I ⊆ R, X is a Banach space, and β : I → I is a strictly increasing continuous function defined on I that has only one fixed point s0 ∈ I and satisfies the inequality (t – s0)(β(t) – t) ≤ 0 for all t ∈ I
Summary
Quantum difference operator allows us to deal with sets of non-differentiable functions. In [6], the existence and uniqueness of solutions of the β-Cauchy problem of the second-order β-difference equations were proved. A fundamental set of solutions for the secondorder linear homogeneous β-difference equations when the coefficients are constants was constructed, and the different cases of the roots of their characteristic equations were studied. 3, we give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of the nth-order β-difference equations. We construct the fundamental set of solutions for the homogeneous linear β-difference equations when the coefficients aj (0 ≤ j ≤ n) are constants. We study the undetermined coefficients, the variation of parameters, and the annihilator methods for the non-homogeneous linear β-difference equations. We use the symbol T for the transpose of the vector or the matrix
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