ON THE THEORY OF SET VALUED FUNCTIONS WITH APPLICATIONS IN APPROXIMATION, OPTIMIZATION AND CONTROL THEORY

The aim of this work is to show that the theory of set valued functions can be used in order to produce results in various areas of mathematics like approximation, optimization and control theory.; We use the theory of nonsmooth and convex analysis to obtain approximation and stochastic approximation results. We deal with f-best approximations, f-farthest points, f-proximinal sets and with the multifunctions involved in this theory. Also the stability of the notions associated with approximation theory and nonsmooth analysis is examined.; After establishing results on the integrable selectors and the Aumann integral of measurable multifunctions we prove several Liapunov type theorems originating in control theory. We also prove existence theorems for differential inclusions defined in a separable infinite dimensional Banach space and then we use them in order to obtain existence results for random differential inclusions.; Finally we present stability results and several versions of the bang-bang theorem for linear and nonlinear infinite dimensional control systems which are proved using techniques from the general theory of multifunctions developed earlier in this thesis.