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:10.1186/1687-1812-2012-94), 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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Iterated function systems consisting of F-contractions

In this paper, we initiate the notion of generalized multivalued ( α K * , Υ , Λ ) -contractions and provide some new common fixed point results in the class of α K -complete partial b-metric spaces. The obtained results are an improvement of several comparable results in the existing literature. We set up an example to elucidate our main result. Moreover, we present applications dealing with the existence of a solution for systems either of functional equations or of nonlinear matrix equations.

https://www.mdpi.com/2073-8994/11/1/86/pdf

Hybrid Multivalued Type Contraction Mappings in αK-Complete Partial b-Metric Spaces and Applications

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

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Applications of Fixed Point Theorems in the Theory of Generalized IFS

A Banach type fixed point theorem for multi-valued mapping

The aim of this paper is to continue the research work that we have done in a previous paper published in this journal (see Mihail and Miculescu, 2008). We introduce the notion of GIFS, which is a family of functions , where is a complete metric space (in the above mentioned paper the case when is a compact metric space was studied) and . In case that the functions are Lipschitz contractions, we prove the existence of the attractor of such a GIFS and explore its properties (among them we give an upper bound for the Hausdorff-Pompeiu distance between the attractors of two such GIFSs, an upper bound for the Hausdorff-Pompeiu distance between the attractor of such a GIFS, and an arbitrary compact set of and we prove its continuous dependence in the 's). Finally we present some examples of attractors of GIFSs. The last example shows that the notion of GIFS is a natural generalization of the notion of IFS.

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Generalized IFSs on Noncompact Spaces

In this paper, we indicate a way to generalize a series of fixed point results in the framework of b-metric spaces and we exemplify it by extending Nadler’s contraction principle for set-valued functions (see Nadler, Pac J Math 30:475–488, 1969) and a fixed point theorem for set-valued quasi-contraction functions due to Aydi et al. (see Fixed Point Theory Appl 2012:88, 2012).

New fixed point theorems for set-valued contractions in b-metric spaces

In this paper we indicate a way to generalize a series of fixed point results in the framework of b-metric spaces and we exemplify it by extending Nadler's contraction principle for set-valued functions (see Multi-valued contraction mappings, Pac. J. Math., 30 (1969), 475-488) and a fixed point theorem for set-valued quasi-contractions functions due to H. Aydi, M.F. Bota, E. Karapinar and S. Mitrovic (see A fixed point theorem for set-valued quasi-contractions in b-metric spaces, Fixed Point Theory Appl. 2012, 2012:88).

The aim of the paper is to define the shift space for an infinite iterated function systems (IIFS) and to describe the relation between this space and the attractor of the IIFS. We construct a canonical projection (which turns out to be continuous) from the shift space of an IIFS on its attractor and provide sucient conditions The shift (or code) space of an iterated function system (IFS for short) and the address of the points lying on the attractor of the IFS are very good tools to get a more precise description of the invariant dynamics of the IFS. The theory of fractal tops provides a useful mapping from an IFS attractor into the associated code space that may be applied to assign colors to the IFS attractor via a method introduced by M.F. Barnsley and J. Hutchinson (which they refer as colour-stealing) and to construct homeomorphisms between attractors (roughly speaking, if the symbolic dynamical systems associated with the tops of two IFSs are topologically conjugate, then the attractors of the IFSs are homeomorphic). Moreover, Barnsley (2) proved that if two hyperbolic IFS attractors are homeomorphic, then they have the same entropy. In this paper we present a generalization of the notion of shift space associated with an IFS. More precisely, we define the shift space of an infinite iterated function systems (IIFS) and describe the relation between this space and the attractor of the IIFS. We construct a canonical projection (which turns out to be continuous) from the shift space of an IIFS on its attractor and provide sucient conditions for this function to be onto.

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Alexandru Mihail

Papers

New fixed point theorems for set-valued contractions in b-metric spaces

New fixed point theorems for set-valued contractions in b-metric spaces

Applications of Fixed Point Theorems in the Theory of Generalized IFS

Generalized IFSs on Noncompact Spaces

The shift space for an infinite iterated function system