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Global existence and uniqueness of solutions for fuzzy differential equations under dissipative-type conditions. (English) Zbl 1165.34303

Summary: Using the properties of a differential and integral calculus for fuzzy set valued mappings and completeness of metric space of fuzzy numbers, the global existence, uniqueness and the continuous dependence of a solution on a fuzzy differential equation are derived under the dissipative-type conditions. We also present the global existence and uniqueness of solutions for a fuzzy differential equation on a closed convex subset of fuzzy number space.

MSC:

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
26E50 Fuzzy real analysis
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[1] A. Kandel, W.J. Byatt, Fuzzy differential equations, in: Proc. Int. Conf. Cybern. and Society, Tokyo, Nov. 1978, pp. 1213-1216; A. Kandel, W.J. Byatt, Fuzzy differential equations, in: Proc. Int. Conf. Cybern. and Society, Tokyo, Nov. 1978, pp. 1213-1216
[2] Kaleva, O., Fuzzy diffrential equations, Fuzzy Sets and Systems, 24, 301-319 (1987)
[3] Nieto, J. J., The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 102, 259-262 (1999) · Zbl 0929.34005
[4] Kaleva, O., The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems, 35, 389-396 (1990) · Zbl 0696.34005
[5] Wu, C. X.; Song, S. J., Existence theorem to the Cauchy problem of fuzzy diffrential equations under compactness-type conditions, Inform. Sci., 108, 123-134 (1998)
[6] Wu, C. X.; Song, S. J.; Qi, Z. Y., Existence and uniqueness for a solution on the closed subset to the Cauchy problem of fuzzy diffrential equations, J. Harbin Inst. Tech., 2, 1-7 (1997)
[7] Wu, C. X.; Song, S. J.; Lee, E. S., Approximate solutions and Existence and uniqueness theorem to the Cauchy problem of fuzzy diffrential equations, J. Math. Anal. Appl., 202, 629-644 (1996) · Zbl 0861.34040
[8] Song, S. J.; Wu, C.; Xue, X. P., Existence and uniqueness theorem to the Cauchy problem of fuzzy diffrential equations under dissipative conditions, Comput. Math. Appl., 51, 1483-1492 (2006) · Zbl 1157.34002
[9] Park, J. Y.; Han, H. K., Fuzzy differential equations, Fuzzy Sets and Systems, 110, 69-77 (2000) · Zbl 0946.34055
[10] Song, S. J.; Guo, L.; Feng, C. B., Global existence of solution to fuzzy diffrential equations, Fuzzy Sets and Systems, 115, 371-376 (2000)
[11] Ding, Z. H.; Ma, M.; Kandel, A., Existence of the solutions of fuzzy differential equations with parameters, Inform. Sci., 99, 205-217 (1997) · Zbl 0914.34057
[12] Seikkala, S., On the initial value problem, Fuzzy Sets and Systems, 24, 319-330 (1987) · Zbl 0643.34005
[13] Aumann, R. J., Integrals of set-valued fuctions, J. Math. Anal. Appl., 12, 1-12 (1965) · Zbl 0163.06301
[14] Dubois, D.; Prade, H., Towards fuzzy differential calculus: Part 1, Integration of fuzzy mappings, Fuzzy Sets and Systems, 8, 1-17 (1982) · Zbl 0493.28002
[15] Radstrom, H., An embedding theorem for spaces on convex set, Proc. Amer. Math. Soc., 3, 165-169 (1952) · Zbl 0046.33304
[16] Diamond, P.; Kloeden, P., Characterization of compact subsets of fuzzy sets, Fuzzy Sets and Systems, 29, 341-348 (1989) · Zbl 0661.54011
[17] Puri, M. L.; Ralescu, D. A., Fuzzy random variables, J. Math. Anal. Appl., 114, 409-422 (1986) · Zbl 0592.60004
[18] Puri, M. L.; Ralescu, D. A., Differentials of fuzzy functions, J. Math. Anal. Appl., 91, 552-558 (1983) · Zbl 0528.54009
[19] Ma, M., On embedding problems of fuzzy number space: Part 5, Fuzzy Sets and Systems, 55, 313-318 (1993) · Zbl 0798.46058
[20] Dugundji, J., An extension of Tietzes theorem, Pacific J. Math., 1, 353-367 (1951) · Zbl 0043.38105
[21] Lakshmikantham; Leela, S., Nonlinear Diffrential Equations in Abstract Spaces (1981), Pergamon Press: Pergamon Press New York · Zbl 0456.34002
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