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Proyecciones (Antofagasta)

versión impresa ISSN 0716-0917

Proyecciones (Antofagasta) v.21 n.2 Antofagasta ago. 2002

http://dx.doi.org/10.4067/S0716-09172002000200003 

Proyecciones
Vol. 21, N o 2, pp. 141-153, August 2002.
Universidad Católica del Norte
Antofagasta - Chile

 

A NOTE ON ASYMPTOTIC
SMOOTHNESS OF THE
EXTENSIONS OF ZADEH

 

LAECIO C. BARROS
Universidade Estadual de Campinas - Brasil

SUZANA A. OLIVEIRA
Universidade de Sao Paulo - Brasil

and
PEDRO A. TONELLI
Universidade de Sao Paulo - Brasil

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 n® R n is asymptotically smooth in the complete metric space F (R n ) of normal fuzzy sets with the induced Hausdorff metric d 1 (see Kloeden and Diamond [8] ).
Keywords : Fuzzy Dynamical Systems, Global Attractors.

Subjclass: 37B25, 54A40, 54H20.

References

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

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[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).        [ Links ]

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

[8] P. Diamond and P. Kloeden - "Metric Spaces of Fuzzy Sets: The-ory and Applications" - World Scientific Pub. (1994).        [ Links ]

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

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

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

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

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

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

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

[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).        [ Links ]

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

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

 

Received : May, 2002.

 

Laécio C. Barros
Departamento de Matemática Aplicada
Universidade estadual de Campinas
13081-970 CP 6065, Campinas
Brasil
e-mail : laeciocb@ime.unicamp.br

Suzana A. Oliveira Souza
Departamento de Matemática Aplicada
Universidade de S˜ao Paulo
05508-900 CP 66281
Sao Paulo
Brasil
e-mail : suzanabreu@uol.com.br

and

Pedro A. Tonelli
Departamento de Matemática Aplicada
Universidade de Sao Paulo
CP 66.281
CEP 05311 - 970
Sao Paulo
Brasil
e-mail : tonelli@ime.usp.br

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