Open Access Open Access  Restricted Access Subscription or Fee Access

Multivalued Fractals in Fuzzy Metric Spaces

R. Uthayakumar, D. Easwaramoorthy

Abstract


The purpose of this paper is to initiate the notion of multivalued fractals in fuzzy metric spaces generated by the iterated multifunction system.  We extend the Hutchinson-Barnsley theory for an iterated multifunction system of fuzzy contractions in order to define and construct the multivalued fractals in the fuzzy metric spaces.  Also we discuss some results on multivalued fuzzy fractals in the standard fuzzy metric spaces. 


Keywords


Fractal Analysis, Fuzzy Contraction, Iterated Multifunction System, Fuzzy Metric Space, Hausdorff Fuzzy Metric

Full Text:

PDF

References


B.B. Mandelbrot, The Fractal Geometry of Nature, W.H. Freeman and Company, New York, 1983.

J.E. Hutchinson, “Fractals and self similarity”, Indiana University Mathematics Journal, vol. 30, pp. 713–747, 1981.

M. Barnsley, Fractals Everywhere, 2nd ed., Academic Press, USA, 1993.

M. Barnsley, SuperFractals, Cambridge University Press, New York, 2006.

G. Edgar, Measure, Topology, and Fractal Geometry, 2nd ed., Springer, New York, 2008.

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, 2nd ed., John Wiley & Sons Ltd., England, 2003.

J. Andres, and J. Fiser, “Metric and topological multivalued fractals”, International Journal of Bifurcation and Chaos, vol.14, no. 4, pp. 1277–1289, 2004.

J. Andres, J. Fiser, G. Gabor and K. Lesniak, “Multivalued fractals”, Chaos, Solitons and Fractals, vol. 24, pp. 665–700, 2005.

C. Chifu and A. Petrusel, “Multivalued fractals and generalized multivalued contractions”, Chaos, Solitons and Fractals, vol. 36, pp. 203–210, 2008.

J. Fiser, “Numerical aspects of multivalued fractals”, Fixed Point Theory, vol. 5, no. 2, pp. 249-264, 2004.

S.L. Singh, Bhagwati Prasad and Ashish Kumar, “Fractals via iterated functions and multifunctions”, Chaos, Solitons and Fractals, vol. 39, pp. 1224–1231, 2009.

J. Rodriguez-Lopez and S. Romaguera, “The Hausdorff fuzzy metric on compact sets”, Fuzzy Sets and Systems, vol. 147, pp. 273–283, 2004.

L.A. Zadeh, “Fuzzy Sets”, Information and Control, vol. 8, pp. 338–353, 1965.

I. Kramosil and J. Michalek, “Fuzzy metrics and statistical metric Spaces”, Kybernetika, vol. 11, pp. 336–344, 1975.

A. George and P. Veeramani, “On some results in fuzzy metric spaces”, Fuzzy Sets and Systems, vol. 64, pp. 395–399, 1994.

A. George and P. Veeramani, “On some results of analysis for fuzzy metric spaces”, Fuzzy Sets and Systems, vol. 90, pp. 365–368, 1997.

V. Gregori and A. Sapena, “On fixed-point theorems in fuzzy metric spaces”, Fuzzy Sets and Systems, vol. 125, pp. 245–252, 2002.

B. Schweizer and A. Sklar, “Statistical metric spaces”, Pacific Journal of Mathematics, vol. 10, pp. 314–334, 1960.


Refbacks

  • There are currently no refbacks.


Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.