First order linear fuzzy differential equations under generalized differentiability.

*(English)*Zbl 1119.34003Summary: First order linear fuzzy differential equations are investigated. We interpret a fuzzy differential equation by using the strongly generalized differentiability concept, because under this interpretation, we may obtain solutions which have a decreasing length of their support (which means a decreasing uncertainty). In several applications the behaviour of these solutions better reflects the behaviour of some real-world systems. Derivatives of the \(H\)-difference and the product of two functions are obtained and we provide solutions of first order linear fuzzy differential equations, using different versions of the variation of constants formula. Some examples show the rich behaviour of the solutions obtained.

##### MSC:

34A12 | Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations |

34A30 | Linear ordinary differential equations and systems |

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\textit{B. Bede} et al., Inf. Sci. 177, No. 7, 1648--1662 (2007; Zbl 1119.34003)

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##### References:

[1] | Anastassiou, G.A.; Gal, S.G., On a fuzzy trigonometric approximation theorem of Weierstrass-type, Journal of fuzzy mathematics, 9, 701-708, (2001) · Zbl 1004.42005 |

[2] | Bede, B.; Gal, S.G., Almost periodic fuzzy-number-valued functions, Fuzzy sets and systems, 147, 385-403, (2004) · Zbl 1053.42015 |

[3] | Bede, B.; Gal, S.G., Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy sets and systems, 151, 581-599, (2005) · Zbl 1061.26024 |

[4] | Buckley, J.J.; Feuring, T., Fuzzy initial value problem for Nth order linear differential equations, Fuzzy sets and systems, 121, 247-255, (2001) · Zbl 1008.34054 |

[5] | Buckley, J.J.; Feuring, T., Introduction to fuzzy partial differential equations, Fuzzy sets and systems, 105, 241-248, (1999) · Zbl 0938.35014 |

[6] | Diamond, P., Brief note on the variation of constants formula for fuzzy differential equations, Fuzzy sets and systems, 129, 65-71, (2002) · Zbl 1021.34048 |

[7] | Diamond, P., Stability and periodicity in fuzzy differential equations, IEEE transactions on fuzzy systems, 8, 583-590, (2000) |

[8] | Ding, Z.; Ma, M.; Kandel, A., Existence of the solutions of fuzzy differential equations with parameters, Information sciences, 99, 205-217, (1997) · Zbl 0914.34057 |

[9] | Dubois, D.; Prade, H., Fuzzy numbers: an overview, (), 112-148 |

[10] | Feng, Y., The solutions of linear fuzzy stochastic differential systems, Fuzzy sets and systems, 140, 541-554, (2003) · Zbl 1043.60045 |

[11] | Gal, C.S.; Gal, S.G., Semigroups of operators on spaces of fuzzy-number-valued functions with applications to fuzzy differential equations, Journal of fuzzy mathematics, 13, 647-682, (2005) · Zbl 1095.47059 |

[12] | Gal, S.G., Approximation theory in fuzzy setting, (), 617-666 · Zbl 0968.41018 |

[13] | Gnana Bhaskar, T.; Lakshmikantham, V.; Devi, V., Revisiting fuzzy differential equations, Nonlinear analysis, 58, 351-358, (2004) · Zbl 1095.34511 |

[14] | Hüllermeier, E., An approach to modelling and simulation of uncertain dynamical systems, International journal of uncertainty, fuzziness and knowledge-based systems, 5, 117-137, (1997) · Zbl 1232.68131 |

[15] | Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019 |

[16] | Puri, M.; Ralescu, D., Differentials of fuzzy functions, Journal of mathematical analysis and applications, 91, 552-558, (1983) · Zbl 0528.54009 |

[17] | Rodríguez-Muñiz, L.J.; López-Díaz, M.; Ángeles Gil, M.; Ralescu, D.A., The s-differentiability of a fuzzy-valued mapping, Information sciences, 151, 283-299, (2003) · Zbl 1027.26028 |

[18] | Seikkala, S., On the fuzzy initial value problem, Fuzzy sets and systems, 24, 319-330, (1987) · Zbl 0643.34005 |

[19] | Song, S.; Wu, C., Existence and uniqueness of solutions to the Cauchy problem of fuzzy differential equations, Fuzzy sets and systems, 110, 55-67, (2000) · Zbl 0946.34054 |

[20] | Wu, C.; Song, S., Existence theorem to the Cauchy problem of fuzzy differential equations under compactness-type conditions, Information sciences, 108, 123-134, (1998) · Zbl 0931.34041 |

[21] | Wu, C.; Song, S.; Stanley Lee, E., Approximate solutions, existence and uniqueness of the Cauchy problem of fuzzy differential equations, Journal of mathematical analysis and applications, 202, 629-644, (1996) · Zbl 0861.34040 |

[22] | Wu, C.; Gong, Z., On Henstock integral of fuzzy-number-valued functions I, Fuzzy sets and systems, 120, 523-532, (2001) · Zbl 0984.28010 |

[23] | Zadeh, L., Toward a generalized theory of uncertainty (GTU) - an outline, Information sciences, 172, 1-40, (2005) · Zbl 1074.94021 |

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