Skip to main content
SpringerLink
Log in

Reich-type iterated function systems

  • Published:
Journal of Fixed Point Theory and Applications Aims and scope Submit manuscript

Abstract

In this paper, we introduce the concept of Reich-type iterated function system and prove the existence and uniqueness of the attractor of such a system. Moreover, we study the properties of the canonical projection from the code space onto the attractor of such a system. We also present an iterated function system consisting of continuous Reich contractions having more than one attractor.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Andres and M. Rypka, Multivalued fractals and hyperfractals. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 22 (2012), doi:10.1142/S02181127412500095.

  2. Boriceanu M., Bota M., Petruşel A.: Multivalued fractals in b-metric spaces. Cent. Eur. J. Math. 8, 367–377 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chifu C., Petruşel A.: Multivalued fractals and generalized multivalued contractions. Chaos Solitons Fractals 36, 203–210 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dumitru D.: Generalized iterated function systems containing Meir-Keeler functions. An. Univ. Bucur. Mat. 58, 109–121 (2009)

    MathSciNet  MATH  Google Scholar 

  5. Gwóźdź-Łukowska G., Jachymski J.: IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem. J. Math. Anal. Appl. 356, 453–463 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Kannan R.: Some results on fixed points. Bull. Calcutta Math. Soc. 60, 71–76 (1968)

    MathSciNet  MATH  Google Scholar 

  7. Klimek M., Kosek M.: Generalized iterated function systems, multifunctions and Cantor sets. Ann. Polon. Math. 96, 25–41 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Leśniak K.: Infinite iterated function systems: A multivalued approach. Bull. Pol. Acad. Sci. Math. 52, 1–8 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Llorens-Fuster E., Petruşel A., Yao J.-C.: Iterated function systems and well posedness. Chaos Solitons Fractals 41, 1561–1568 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Máté L.: The Hutchinson-Barnsley theory for certain noncontraction mappings. Period. Math. Hungar. 27, 21–33 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Mihail and R. Miculescu, Applications of fixed point theorems in the theory of generalized IFS. Fixed Point Theory Appl. 2008 (2008), Art. ID 312876, 11 pages.

  12. Mihail A., Miculescu R.: The shift space for an infinite iterated function system. Math. Rep. (Bucur.) 61, 21–31 (2009)

    MathSciNet  MATH  Google Scholar 

  13. A. Mihail and R. Miculescu, Generalized IFSs on noncompact spaces. Fixed Point Theory Appl. 2010 (2010), Art. ID 584215, 15 pages.

  14. A. Petruşel, Fixed point theory and the mathematics of fractals. In: Topics in Mathematics, Computer Science and Philosophy, Presa Univ. Clujeană, Cluj- Napoca, 2008, 165–172.

  15. Petruşel A., Rus I. A., M. A Şerban, Fixed points, fixed sets and iterated multifunction systems for nonself multivalued operators. Set-Valued Var. Anal. 23 (2015), 223–237.

  16. Reich S.: Some remarks concerning contraction mappings. Canad. Math. Bull. 14, 121–124 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  17. Sahu D. R., Chakraborty A.: K-Iterated function system. Fractals 18, 139–144 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. N. A. Secelean, Iterated function systems consisting of F-contractions. Fixed Point Theory Appl. 2013 (2013), doi:10.1186/1687-1812-2013-277, 13 pages.

  19. N. A. Secelean, Countable iterated function systems. Lambert Academic Publishing, 2013.

  20. N. A. Secelean: Generalized iterated function systems on the space \({l^\infty(X)}\). J. Math. Anal. Appl. 410, 847–458 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Singh S. L.: Remarks on Sarkhel’s comparison of Banach and Kannan contractions. Nat. Acad. Sci. Lett. 28, 117–118 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Singh S. L., Prasad B., Kumar A.: Fractals via iterated functions and multifunctions. Chaos Solitons Fractals 39, 1224–1231 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Strobin F., Swaczyna J.: On a certain generalization of the iterated function system. Bull. Aust. Math. Soc. 87, 37–54 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  24. Strobin F.: Attractors of generalized IFSs that are not attractors of IFSs. J. Math. Anal. Appl. 422, 99–108 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Uthayakumar R., Prabakar G. A.: An iterated function system for Reich contraction in complete b metric space. International Journal of Mathematical, Computational, Physical and Quantum Engineering 7, 1118–1121 (2013)

    Google Scholar 

  26. S. Xu, S. Cheng and Z. Zhou, Reich’s iterated function systems and wellposedness via fixed point theory. Fixed Point Theory Appl. 2015 (2015), doi:10.1186/s13663-015-0320-7.

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014, Bucharest, Romania

    Radu Miculescu & Alexandru Mihail

  2. University Politehnica of Bucharest, 313 Splaiul Independenţei, 060042, Bucharest, Romania

    Alexandru Mihail

Authors
  1. Radu Miculescu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alexandru Mihail
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Radu Miculescu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Miculescu, R., Mihail, A. Reich-type iterated function systems. J. Fixed Point Theory Appl. 18, 285–296 (2016). https://doi.org/10.1007/s11784-015-0264-x

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11784-015-0264-x

Mathematics Subject Classification

Keywords

Search

Navigation