A General Quantum Difference Operator and its Calculus

Abstract

The general quantum difference operator, D_β , is defined by D_β f(t)={█(( f (β(t))–f (t))/(β(t)–t), t≠s_0,@ f^́ (s_0 ), t=s_0,)┤ where f:I→X is a function defined on an interval I  R and β∶I→ I is a strictly increasing continuous function defined on I which has only one fixed point 〖 s〗_0 ∈ I and satisfies the inequality: (t – s_0)(β(t) – t) ≤0 for all t ∈ I. In this thesis, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations a_0 (t) D_β^2 y(t)+ a_1 (t) D_β y(t)+ a_2 (t)y(t)= b(t),t ∊ I, a_0 (t)  0, in a neighborhood of the unique fixed point s_0 of the function β. 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. Then we drive the Euler-Cauchy β-difference equation. Furthermore, we give the sufficient conditions for the existence and uniqueness of solutions of the β-Cauchy problem of β-difference equations. We also establish the fundamental set of solutions when the coefficients are constants, the β-Wronskian associated with D_β, and Liouville’s formula for β-difference equations. In addition, we deduce the undetermined coefficients, the variation of parameters and the annihilator methods for the non-homogeneous β-difference equations. Finally, we introduce the solutions of homogenous and non-homogenous systems of linear β-difference equations.