الفهرس 
Hahn difference calculusOnly 14 pages are availabe for public view
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Abstract
In this thesis, we introduce a general quantum difference operator D
which is defined by ),(,)()())(()(tttttftftfDwhere β is a strictly increasing continuous function, and we deduce the β-calculus associated with D . β may be linear or nonlinear then it has either no fixed points or at least one. Therefore, β has many types according to the number of its fixed points that belong to I. The fixed points of the function β play an essential role in the β-calculus. Every choice of the function β gives a new difference operator. Thus, we can obtain a wide class of quantum difference operators with the corresponding quantum calculi. The advantage of this study is that it provides a general quantum calculus and allows us to avoid repetition in proving results, once for the Jackson q-difference operator, once for the Hahn difference operator and once for any difference operator in the form of D of the same type. For the type of β that has a unique fixed point 0s, two classes can be considered. The first class is the family of all functions β that has a unique fixed point and satisfies the following inequality:
,0))()((0ttst the second class is the family of all functions β that has a unique fixed point 0 s and satisfies the following inequality: .0))()((0ttstWe presented the β-exponential and β-trigonometric (β-hyperbolic) functions and prove that they satisfy the first and second order β-difference equations, respectively. Also, we deduced some integral inequalities based on D . Finally, we established the existence and uniqueness of solutions of the initial value β-difference equations.
The thesis consists of six Chapters. We summarize the main aims of each Chapter as follows. In Chapter 1, we present the basic results about the Hahn quantum difference operator and the associated calculus.
Also, we give some recent results concerning with some quantum difference operators. In Chapter 2, the definition of the β-derivative is presented, and its main properties are established. For instance, we deduce the chain rule and Leibniz’ formula. Also, we introduce the β-integral and deduce the mean value theorem. Furthermore, we establish the fundamental theorem of β-calculus. Finally, we prove some results concerning with the β-differentiation under the integral sign.