الفهرس 
The ??difference operator and its associated equationsOnly 14 pages are availabe for public view
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Abstract
Let D_β be the general quantum difference operator defined by
D_β y(t)=( y (β(t))–y (t))/(β(t)–t),t≠β(t),t∈I
and D_β y(t)=y ́(t) at t = β(t), where β(t) is a strictly increasing continuous function defined on I R.
In this thesis, we introduce the β-Laplace transform L_β and some of its main properties. Furthermore, we compute the β-Laplace transform of the β-exponential and the β-trigonometric functions. We give some examples to solve some β-difference equations. In addition, we deduce the inverse β-Laplace transform, L_β^(-1). Moreover, we investigate a β-convolution of two functions and study their shift, associativity and differentiability and also prove the β-convolution theorem. We study some properties of the β-exponential functions in a Banach algebra E with a unit e. Finally, we present the stability of the linear β-difference equations in a Banach algebra.
Keywords: A general quantum difference operator; β-difference equations; β-Laplace transform; β-convolution theorem; Banach algebra; β-exponential functions; stability.