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.
Iterated function systems on multifunctions / D. La Torre, F. Mendivil, E. Vrscay - In: Math Everywhere: Deterministic and Stochastic Modelling in Biomedicine, Economics and Industry : Dedicated to the 60th Birthday of Vincenzo Capasso / [a cura di] G. Aletti, M. Burger, A. Micheletti, D. Morale. - New York : Springer, 2007. - ISBN 978-3-540-44445-9. - pp. 125-138
Iterated function systems on multifunctions
D. La TorrePrimo
;
2007
Abstract
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.File | Dimensione | Formato | |
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