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 ${cal F}(X,Y)$. Some appropriate IFS-type fractal transform operators $T : {cal F}(X,Y) o {cal F}(X,Y)$ 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 $ar u$. 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.