What is a Multi-Valued Mapping?

Dr. Claudio Morales

Department of Mathematical Sciences
University of Alabama in Huntsville

November 17, 2006
202 Madison Hall
3:00 (Coffee and Cookies at 2:30)

Abstract

By the 1940’s, the time of John von Neumann and the development of Game Theory, multi-valued operators were already studied. In Economics, these operators occur very naturally. Today, many mathematical problems may be formulated in terms of multi-valued mappings defined on various types of spaces. For example, it would be interesting to find out whether the classical Banach contraction principle has a multi-valued mapping formulation. To address this question, we need to figure out the topological structure in the new space where these specimens live. A very important question arises in the study of these mappings, that can be stated as follows: under what condition does a selection exist? The answer to this question would allow us to use the theory of single-valued mappings to derive existence of solutions for various problems.

Additionally, some open questions concerning contractions and monotone operators will be presented.