Applications of Fixed Point Theorems in the Theory of Generalized IFS

Abstract

We introduce the notion of a generalized iterated function system GIFS , which is a finite family of functions f k : X m → X, where X, d is a metric space and m ∈ N. In case that X, d is a compact metric space and the functions f k are contractions, using some fixed point theorems for contractions from X m to X, we prove the existence of the attractor of such a GIFS and its continuous dependence in the f k 's.

Reich-type iterated function systems

“…For example, Máté [10] studied the case of weaker contractions, Leśniak [8] the case of multivalued contractions, Petruşel [14] and Dumitru [4] the case of Meir-Keeler contractions, Llorens-Fuster, Petruşel and Yao [9] the case of mixed-type contractions, Boriceanu, Bota and Petruşel [2] the case of set-valued contractions on a b-metric space, Gwóźdź-� Lukowska and Jachymski [5] the case of contractions on a metric space endowed with a directed graph and Secelean [18] the case of F -contractions. For other related results see [1,3,7,11,13,15,20,23,24].…”

Topological version of generalized (infinite) iterated function systems

“…In papers [15,[17][18][19][20]22,29,27,[30][31][32][33] the theory of generalized IFSs was developed. We will recall it here.…”

“…Our construction base on idea behind the one given by Mihail and Miculescu in [19] (cf. also [31]), who proved that the Hilbert cube ½0; 1 Â ½0; 1 2 Â ½0; 1 4 Â Á Á Á is the fractal generated by a certain GIFS of order 2.…”

Iterated Function Systems Consisting of Generalized Convex Contractions in the Framework of Complete Strong b-metric Spaces

“…-to work with more general contractive conditions on the constitutive functions of the iterated function systems (see, for example, [18], [34], [36], [37], [38], [39], [45], [48], [52], [53], [57] and [58]). …”