On q-Di®erence Equations

On q-Di®erence Equations Moemen H. Abu-Risha Department of Mathematics Cairo University, Eygpt E-mail: [email protected] We give an investigation of some properties of solutions of linear q¡di®erence equations de ̄ned on an interval [a;1): The cases a = 0 and a > 0 are essentially di®erent. The study of the latter case is necessary for the derivation of asymptotic formulae and oscillation criteria for solutions near 1: In the case a > 0; some q¡di®erence equations share with the delay di®erential equations the property that a set of initial conditions at the point a is insu±cient to guarantee the uniqueness of the solution. A set of initial functions de ̄ned on an interval left to a is needed. q-Sturm-Liouville Problems and Applications Mahmoud Annaby Department of Mathematics Cairo University, Eygpt E-mail: [email protected] In this talk we discuss some problems concerning q¡di®erence equations. An existence and uniqueness theorem based on successive approximations will be introduced. The n th order linear equation is also considered and the construction of a fundamental system of solutions is developed. The general q¡type Wronskian as well as a Liouville type formula will be given. The self adjoint Sturm-Liouville problem is de ̄ned and its properties are given. A detailed example will be presented with an application in sampling theory. The work presented in this talk is taken from some projects with di®erent colleagues. Richard Askey Department of Mathematics University of Wisconsin-Madison, USA E-mail: [email protected]