A note on asymptotic smoothness of the extensions of Zadeh

Authors

  • Laécio C. Barros Universidade Estadual de Campinas.
  • Suzana A. Oliveira Souza Universidade de Sao Paulo.
  • Pedro A. Tonelli Universidade de Sao Paulo.

DOI:

https://doi.org/10.4067/S0716-09172002000200003

Keywords:

Fuzzy dynamical systems, global attractors, sistemas dinámicos difusos, atractores globales.

Abstract

The concept of asymptotic smooth transformation was introduced by J. Hale [10]. It is a very important property for a transformation between complete metric spaces to have a global attractor. This property has also consequences on asymptotic stability of attractors. In our work we study the conditions under which the Zadeh’s extension of a continuous map f : R??R? is asymptotically smooth in the complete metric space F(R?) of normal fuzzy sets with the induced Hausdorff metric d?(see Kloeden and Diamond [8]).

Author Biographies

Laécio C. Barros, Universidade Estadual de Campinas.

Departamento de Matemática Aplicada.

Suzana A. Oliveira Souza, Universidade de Sao Paulo.

Departamento de Matemática Aplicada.

Pedro A. Tonelli, Universidade de Sao Paulo.

Departamento de Matemática Aplicada.

References

[1] L. C. Barros, R. C. Bassanezi and P. A. Tonelli - “On the continuity of Zadeh’s extension” - Proceedings Seventh IFSA World Congress, Prague, Vol. II, pp. 3-8, (1997).

[2] L. C. Barros, R. C. Bassanezi and P. A. Tonelli - “Fuzzy modeling in populations dynamics” - Ecological Modeling- 128, pp. 27-33 (2000).

[3] W. E. Brumley - “On the asymptotic behavior of solutions of differential difference equations of neutral type” - J. of Differential Equations 7, pp. 175-188 (1970).

[4] C. A. Cabrelli, B. Forte, U. Molter and E. Vrscay - “Iterated Fuzzy Sets Systems: A new approach to the inverse for fractals and other sets” - J. of Math. Anal. and Appl. 171, pp. 79-100 (1992).

[5] G. Cooperman - “fi-Condensing maps and dissipative processes” - Ph. D. Thesis, Brown University, Providence, R. I. (1978).

[6] P. Diamond - “Chaos in iterated fuzzy systems” - J. of Mathematical Analysis and Applications- 184, pp. 472-484 (1994).

[7] P. Diamond - “Time Dependent Differential Inclusions, Cocycle Attractors and Fuzzy Differential Equations” , IEEE Trans. On Fuzzy Systems - Vol. 7, pp. 734-740. (1999).

[8] P. Diamond and P. Kloeden - “Metric Spaces of Fuzzy Sets: Theory and Applications” - World Scientific Pub. (1994).

[9] M. Friedmann, M. Ma and A. Kandel - “Numerical solutions of fuzzy differential and integral equations” - Fuzzy Sets and Systems 106, pp. 35-48 (1999).

[10] J. K. Hale - “Asymptotic Behavior of Dissipative Systems”- Math. Surveys and Monographs 25, American Mathematical Society, Providence (1988).

[11] E. Hüllermeier “An Approach to Modeling and Simulation of Uncertain Dynamical Systems”-Int. J. Uncertainty, Fuzziness, Knowledge-Bases Syst. Vol. 5, pp. 117-137 (1997).

[12] P. E. Kloeden - “Fuzzy dynamical systems” - Fuzzy Sets and Systems- 7, pp. 275–296 (1982).

[13] P. E. Kloeden - “Chaotic iterations of fuzzy sets” - Fuzzy Sets and Systems- 42, pp. 37–42 (1991).

[14] H. T. Nguyen - “A note on the extension principle for fuzzy sets” - J. Math. Anal. Appl. 64, pp. 369-380 (1978).

[15] M. L. Puri and D. A. Ralescu - “Fuzzy Random Variables” - J. of Mathematical Analysis and Applications- 114, pp. 409-422 (1986).

[16] H. Roman-Flores, L. C. Barros and Bassanezzi, R. - “A note on Zadeh’s Extensions” - Fuzzy Sets and Systems-117, pp. 327-331 (2001).

[17] H. Roman-Flores - “On the Compactness of E(X)” - Appl. Math. Lett. 11, pp. 13–17. (1998).

[18] L. A. Zadeh - “Fuzzy sets” - Inform. Control- 8, pp. 338-353 (1965).

Published

2017-05-22

How to Cite

[1]
L. C. Barros, S. A. Oliveira Souza, and P. A. Tonelli, “A note on asymptotic smoothness of the extensions of Zadeh”, Proyecciones (Antofagasta, On line), vol. 21, no. 2, pp. 141-153, May 2017.

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