Thesis (M.Sc.) - Cairo University - Faculty of Science - Department of Mathematics

This thesis introduces the theory of linear power quantum difference equations associated with the power quantum difference operator. We define the power quantum exponential and trigonometric (hyperbolic) functions and give some of their properties. We prove that they are solutions of difference equations of first and second order respectively. Next, we apply the method of successive approximations to obtain the existence and uniqueness of solutions of linear difference equations in Banach spaces in both local and global cases. Then, we study the theory of linear power quantum difference equations. We introduce the power quantum Wronskian and prove its properties. We show that it is an effective tool to determine whether set of solutions is a fundamental set or not. Hence, we obtain Liouville{u2019}s formula for power quantum difference equations. Also, we derive the solution of the first order linear power quantum difference equation with non-constant coefficients. We derive solutions of Euler-Cauchy difference equations as special cases of second order linear difference equations. Thereafter, we are concerned with constructing a fundamental set of solutions of homogeneous power quantum linear difference equations when the coefficients are constant. Finally, we establish many Pachpatte{u2019}s inequalities based on the power quantum difference operator and as special cases we obtain Gronwall{u2019}s and Bernoulli{u2019}s inequalities associated with this operator

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