Differential calculus for set-valued maps

11 ABSTRACT Master of Science Thesis DIFFERENTIAL CALCULUS FOR SET-VALUED MAPS SERKAN ALI DUZCE Anadolu University Graduate School of Natural And Applied Sciences Mathematics Program Supervisor: Prof. Orhan OZER 2000, Pages 95 In this thesis, extensions of some properties of the single valued functions to set-valued maps are considered. Firstly, some basic theorems of convex analysis are presented, which are connected with notions as subdifferential of convex function, support function of sets and marginal function. Upper and lower semi-continuity of the set- valued maps are studied. Some equivalent forms of these notions are considered. At the last chapter continuity properties of the set-valued maps are investigated. It is proved that, if the lower derivative sets of the set-valued maps are non-empty then the set-valued map is continuous. Finally, the convex continuation of the set- valued map is considered. It is showed that, if V», V* C Rn are compact, convex sets, mi(V*) ^ 0, int(V*) ^ 0 then the set-valued map t - > V(t), defined as ^vw = (1-?^)-v-+(^)-v*',ei'»ri has a convex continuation. Keywords: Subdifferential, Lower Semi-Continuity, Upper Semi-Continuity, Contingent Cone, Differential of Set- Valued Map