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Multivalued fractals

Author

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  • Andres, Jan
  • Fišer, Jiří
  • Gabor, Grzegorz
  • Leśniak, Krzysztof

Abstract

Multivalued fractals are considered as fixed-points of certain induced union operators, called the Hutchinson–Barnsley operators, in hyperspaces of compact subsets of the original spaces endowed with the Hausdorff metric. Various approaches are presented for obtaining the existence results, jointly with the information concerning the topological structure of the set of multivalued fractals. According to the applied fixed-point principles, we distinguish among metric, topological and Tarski’s multivalued fractals. Finite families of condensing and (locally) compact maps as well as of different sorts of contractions are examined with this respect. In particular, continuation principle for multivalued fractals is established for (locally) compact maps. Multivalued fractals are also generated implicitly by means of differential inclusions. A randomization of the deterministic results is indicated. Numerical aspects of computer generated multivalued fractals are discussed in detail.

Suggested Citation

  • Andres, Jan & Fišer, Jiří & Gabor, Grzegorz & Leśniak, Krzysztof, 2005. "Multivalued fractals," Chaos, Solitons & Fractals, Elsevier, vol. 24(3), pages 665-700.
    • Handle: RePEc:eee:chsofr:v:24:y:2005:i:3:p:665-700
    DOI: 10.1016/j.chaos.2004.09.029
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    Cited by:

    1. Zhang, Yongping & Sun, Weihua & Liu, Shutang, 2009. "Control of generalized Julia sets," Chaos, Solitons & Fractals, Elsevier, vol. 42(3), pages 1738-1744.
    2. Jaroszewska, Joanna, 2013. "A note on iterated function systems with discontinuous probabilities," Chaos, Solitons & Fractals, Elsevier, vol. 49(C), pages 28-31.
    3. Chifu, Cristian & Petruşel, Adrian, 2008. "Multivalued fractals and generalized multivalued contractions," Chaos, Solitons & Fractals, Elsevier, vol. 36(2), pages 203-210.
    4. Ullah, Kifayat & Katiyar, S.K., 2023. "Generalized G-Hausdorff space and applications in fractals," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    5. Andres, Jan & Rypka, Miroslav, 2013. "Dimension of hyperfractals," Chaos, Solitons & Fractals, Elsevier, vol. 57(C), pages 146-154.
    6. Singh, S.L. & Prasad, Bhagwati & Kumar, Ashish, 2009. "Fractals via iterated functions and multifunctions," Chaos, Solitons & Fractals, Elsevier, vol. 39(3), pages 1224-1231.
    7. Prithvi, B.V. & Katiyar, S.K., 2023. "Revisiting fractal through nonconventional iterated function systems," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    8. Prithvi, B.V. & Katiyar, S.K., 2022. "Interpolative operators: Fractal to multivalued fractal," Chaos, Solitons & Fractals, Elsevier, vol. 164(C).
    9. Llorens-Fuster, Enrique & Petruşel, Adrian & Yao, Jen-Chih, 2009. "Iterated function systems and well-posedness," Chaos, Solitons & Fractals, Elsevier, vol. 41(4), pages 1561-1568.

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