Iterated function systems and attractors in the KM fuzzy metric spaces

Fractal attractors are considered as invariant sets for contraction mappings in hyperspaces of the KM fuzzy metric spaces. Using the properties of Hausdorff fuzzy metrics, several existence results of attractors are obtained for a finite family and a countable family of nonlinear contractions. In particular, unbounded invariant sets are also examined for the iterated function systems in KM fuzzy metric spaces. Our results are generalizations in some aspects and analogous to their counterparts in usual metric spaces.

[1]  Xiao-Yan Liu,et al.  An alternative characterization of probabilistic Menger spaces with H-type triangular norms , 2013, Fuzzy Sets Syst..

[2]  Shenghua Wang,et al.  Answers to some open questions on fuzzy ψψ-contractions in fuzzy metric spaces , 2013, Fuzzy Sets Syst..

[3]  B. Schweizer,et al.  The metrization of statistical metric spaces. , 1960 .

[4]  Mariusz Grabiec,et al.  Fixed points in fuzzy metric spaces , 1988 .

[5]  Radko Mesiar,et al.  A Fixed Point Theorem in Probabilistic Metric Spaces and an Application , 1996 .

[6]  Salvador Romaguera,et al.  An identification theorem for the completion of the Hausdorff fuzzy metric , 2013, Fuzzy Sets Syst..

[7]  Samuel Morillas,et al.  Examples of fuzzy metrics and applications , 2011, Fuzzy Sets Syst..

[8]  Valentín Gregori,et al.  Some properties of fuzzy metric spaces , 2000, Fuzzy Sets Syst..

[9]  Dorel Mihet,et al.  A Banach contraction theorem in fuzzy metric spaces , 2004, Fuzzy Sets Syst..

[10]  Salvador Romaguera,et al.  The Hausdorff fuzzy quasi-metric , 2010, Fuzzy Sets Syst..

[11]  Michael F. Barnsley,et al.  Fractals everywhere , 1988 .

[12]  Salvador Romaguera,et al.  The Hausdor$ fuzzy metric on compact sets , 2004 .

[13]  Valentín Gregory,et al.  Some properites of fuzzy metric spaces , 2000 .

[14]  Samuel Morillas,et al.  Some questions in fuzzy metric spaces , 2012, Fuzzy Sets Syst..

[15]  A. George,et al.  On some results in fuzzy metric spaces , 1994 .

[16]  B. Schweizer,et al.  Statistical metric spaces. , 1960 .

[17]  Russell J. Egbert,et al.  Products and quotients of probabilistic metric spaces , 1968 .

[18]  Ramakant Bhardwaj,et al.  On Fixed Point theorems in Fuzzy Metric Spaces , 2013 .

[19]  K. Menger.  Statistical Metrics. , 1942, Proceedings of the National Academy of Sciences of the United States of America.

[20]  Nicolae Adrian Secelean,et al.  The Existence of the Attractor of Countable Iterated Function Systems , 2012 .

[21]  Euclidean self-similar sets generated by geometrically independent sets , 2007 .

[22]  Xing-Hua Zhu,et al.  Note on Coupled fixed point theorems for contractions in fuzzy metric spaces , 2011 .

[23]  Endre Pap,et al.  Fixed Point Theory in Probabilistic Metric Spaces , 2001 .

[24]  Salvador Romaguera,et al.  The Banach fixed point theorem in fuzzy quasi-metric spaces with application to the domain of words , 2007 .

[25]  Dorel Miheţ.  On Fuzzy -Contractive Mappings in Fuzzy Metric Spaces , 2007 .

[26]  Ivan Kramosil,et al.  Fuzzy metrics and statistical metric spaces , 1975, Kybernetika.