Multivalued Analysis

Summary

^*Multivalued analysis deals with the study of maps whose values are sets. Multivalued analysis is closely related to nonsmooth analysis and a symbiotic relationship exists between them which feeds both with new ideas and directions to grow. This chapter presents in detail the main aspects of nonsmooth analysis. We study the continuity and measurability properties of multifunctions (set-valued functions) and we present the main selection theorems, for both continuous and measurable selectors (Michael’s theorem and the Kuratowski–Ryll Nardzewski and Yanlov–von Neumann–Aumann theorems). Then we deal with the set of integrable selectors of a multifunction and develop the main properties of the set-valued integral. We also prove fixed point theorems and study Carathéodory multifunctions. Finally we examine the different modes of convergence of sets and of multifunctions.