Bulletin of the Australian Mathematical Society Bull. Austral. Math. Soc. 0004-9727 ALNBAB 2005 72 3 10.wxyz/CV72P3 http://www.austms.org.au/Publ/Bulletin/V72P3/ Gertruda Gwóźdź-Łukawska Jacek Jachymski 13 February 2006 2006 2 13 441 454 723-5209-GwJa-2005 10.wxyz/C2005V72P3p441 http://www.austms.org.au/Publ/Bulletin/V72P3/723-5209-GwJa/ We show that some results of the Hutchinson--Barnsley theory for finite iterated function systems can be carried over to the infinite case. Namely, if $\{F_i:i\in \N \}$ is a family of Matkowski's contractions on a complete metric space $(X,d)$ such that $(F_ix_0)_{i\in \N }$ is bounded for some $x_0\in X$, then there exists a non-empty bounded and separable set $K$ which is invariant with respect to this family, that is, $K=\bcu \limits _{i\in \N }F_i(K)$. Moreover, given $\si \in \N ^{\N }$ and $x\in X$, the limit $\lim \limits _{n\ra \iy }F_{\si _1}\co F_{\si _n}(x)$ exists and does not depend on $x$. 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