q-Difference Equations

Abstract

In this thesis, we study the Cauchy problem of q--difference equations. We distinguish between two cases. The first case is when the initial conditions are defined at a = 0. The other is when the initial conditions are defined at a > 0. Unlike the case of differential equations, the two cases are different in the case of q--difference equations. We derive existence and uniqueness theorems for both cases based on q-analogues of Picard-Lindel~f method of successive approximations. When a = 0, we study the linear q--difference equation of order n. A fundamental set of solutions is derived when the coefficients are all constants. The q-type Wronskian is defined and a q-type Liouville's formula is given. Several illustrative examples are given including q-type Legendre polynomials. The asymptotic formulae for solutions as well as a Sturm-type separation theorem will be given at the end of the thesis.