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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 for ODE, existence, uniqueness, etc. of solutions 26E50 Fuzzy real analysis
Full Text:
##### References:
 [1] 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) · Zbl 0646.34019 · doi:10.1016/0165-0114(87)90029-7 [3] Nieto, J. J.: The Cauchy problem for fuzzy differential equations, Fuzzy sets and systems 102, 259-262 (1999) · Zbl 0929.34005 · doi:10.1016/S0165-0114(97)00094-8 [4] Kaleva, O.: The Cauchy problem for fuzzy differential equations, Fuzzy sets and systems 35, 389-396 (1990) · Zbl 0696.34005 · doi:10.1016/0165-0114(90)90010-4 [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) · Zbl 0931.34041 · doi:10.1016/S0020-0255(97)10064-0 [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 · doi:10.1006/jmaa.1996.0338 [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 · doi:10.1016/j.camwa.2005.12.001 [9] Park, J. Y.; Han, H. K.: Fuzzy differential equations, Fuzzy sets and systems 110, 69-77 (2000) · Zbl 0946.34055 · doi:10.1016/S0165-0114(98)00150-X [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) · Zbl 0963.34056 · doi:10.1016/S0165-0114(99)00046-9 [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 · doi:10.1016/S0020-0255(96)00279-4 [12] Seikkala, S.: On the initial value problem, Fuzzy sets and systems 24, 319-330 (1987) · Zbl 0643.34005 · doi:10.1016/0165-0114(87)90030-3 [13] Aumann, R. J.: Integrals of set-valued fuctions, J. math. Anal. appl. 12, 1-12 (1965) · Zbl 0163.06301 · doi:10.1016/0022-247X(65)90049-1 [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 · doi:10.1016/0165-0114(82)90025-2 [15] Radstrom, H.: An embedding theorem for spaces on convex set, Proc. amer. Math. soc. 3, 165-169 (1952) · Zbl 0046.33304 · doi:10.2307/2032477 [16] Diamond, P.; Kloeden, P.: Characterization of compact subsets of fuzzy sets, Fuzzy sets and systems 29, 341-348 (1989) · Zbl 0661.54011 · doi:10.1016/0165-0114(89)90045-6 [17] Puri, M. L.; Ralescu, D. A.: Fuzzy random variables, J. math. Anal. appl. 114, 409-422 (1986) · Zbl 0592.60004 · doi:10.1016/0022-247X(86)90093-4 [18] Puri, M. L.; Ralescu, D. A.: Differentials of fuzzy functions, J. math. Anal. appl. 91, 552-558 (1983) · Zbl 0528.54009 · doi:10.1016/0022-247X(83)90169-5 [19] Ma, M.: On embedding problems of fuzzy number space: part 5, Fuzzy sets and systems 55, 313-318 (1993) · Zbl 0798.46058 · doi:10.1016/0165-0114(93)90258-J [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) · Zbl 0456.34002