In this paper we consider a particular case of a contractive self-mapping on a complete metric space, namely the F-contraction introduced by Wardowski (Fixed Point Theory Appl. 87, 2012, doi:10.1186/1687-1812, and provide some new properties of it. As an application, we investigate the iterated function systems (IFS) composed of F-contractions extending some fixed point results from the classical Hutchinson-Barnsley theory of IFS consisting of Banach contractions. Some illustrative examples are given. MSC: Primary 28A80; secondary 47H10; 54E50

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The theory of iterated function systems (IFS) and of infinite iterated function systems consisting of contraction mappings has been studied in the last decades. Some extensions of the spaces and the contractions concern many authors in fractal theory. In this paper there are described some results in that topic concerning the existence and uniqueness of nonempty compact set which is a set "fixed point" of a countable iterated function system (CIFS). Moreover, some approximations of the attractor of a CIFS by the attractors of the partial IFSs are given.

The aim of this paper is to extend the result of Wardowski (Fixed Point Theory Appl 2012:94, 2012) by introducing a new class of Picard operators which strictly includes the family of F -contractions. For such operators some fixed point theorems are proved. It is showed that there exist Picard self-mappings on a complete metric space that are neither nonexpansive nor expansive. Mathematics Subject Classification. Primary 47H09; Secondary 47H10, 45D05.

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