Iterated Function Systems on Multifunctions

We introduce a method of iterated function systems (IFS) over the space of set-valued mappings (multifunctions). This is done by first considering a couple of useful metrics over the space of multifunctions F(X,Y). Some appropriate IFS-type fractal transform operators T:F(X,Y)->F(X,Y) are then defined which combine spatially-contracted and range-modified copies of a multifunction u to produce a new multifunction v = Tu. Under suitable conditions, the fractal transform T is contractive, implying the existence of a fixed-point set-valued mapping u. Some simple examples are then presented. We then consider the inverse problem of approximation of set-valued mappings by fixed points of fractal transform operators T and present some preliminary results.