Set-valued analysis

Summary

Set-valued analysis is the study of maps $F \colon X \rightrightarrows Y$ that assign to each point $x$ a subset $F(x)$ of $Y$, possibly empty or with many points. Such maps are the multivalued functions of analysis, studied with tools of topology, convexity and differentiation. Its main notions are the graph and inverse of a set-valued map, its hemicontinuity, limits of sets, selections and derivatives of set-valued maps. Set-valued maps arise as solution sets of parametrized problems, as subdifferentials of nonsmooth functions, and as right-hand sides of differential inclusions in control theory. Typical results are the Kakutani fixed-point theorem and the Michael selection theorem.