Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations.

*(English)*Zbl 1061.26024Summary: The usual concept of differentiability of fuzzy-number-valued functions has the following shortcoming: if \(c\) is a fuzzy number and \(g:[a,b]\to \mathbb R\) is a usual real-valued function differentiable in \(x_{0}\in (a,b)\) with \(g'(x_{0})\leq 0\), then \(f(x)=c\odot g(x)\) is not differentiable in \(x_{0}\). In this paper we introduce and study generalized concepts of differentiability (of any order \(n \in \mathbb N)\), which solves this shortcoming. A Newton-Leibniz-type formula is obtained and existence of the solutions of fuzzy differential equations involving generalized differentiability is studied. Also, some concrete applications to partial and ordinary fuzzy differential equations with fuzzy input data of the form \(c\odot g(x)\) are given.

##### MSC:

26E50 | Fuzzy real analysis |

34A99 | General theory for ordinary differential equations |

35A99 | General topics in partial differential equations |

##### Keywords:

fuzzy-number-valued functions; generalized differentiability; ordinary fuzzy differential equations; fuzzy partial differential equations
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\textit{B. Bede} and \textit{S. G. Gal}, Fuzzy Sets Syst. 151, No. 3, 581--599 (2005; Zbl 1061.26024)

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##### References:

[1] | Anastassiou, G.A., On H-fuzzy differentiation, Math. balkanica, 16, 155-193, (2002) · Zbl 1070.26026 |

[2] | Anastassiou, G.A.; Gal, S.G., On a fuzzy trigonometric approximation theorem of Weierstrass-type, J. fuzzy math., 9, 3, 701-708, (2001) · Zbl 1004.42005 |

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

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

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

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

[7] | 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 |

[8] | D. Dubois, H. Prade, Fuzzy numbers: an overview, Analysis of Fuzzy Information, vol.1: Math. Logic, CRC Press, Boca Raton, 1987, 3-39. · Zbl 0663.94028 |

[9] | S.G. Gal, Approximation theory in fuzzy setting, in: G.A. Anastassiou (Ed.), Handbook of Analytic-Computational Methods in Applied Mathematics, Chapman & Hall/CRC, Boca Raton, London, New York, Washington DC, 2000, pp. 617-666. · Zbl 0968.41018 |

[10] | Hüllermeier, E., An approach to modelling and simulation of uncertain dynamical systems, Internat. J. uncertainty, fuzzyness and knowledge-based systems, 5, 117-137, (1997) · Zbl 1232.68131 |

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

[12] | Kloeden, P.E., Remarks on Peano theorem for fuzzy differential equations, Fuzzy sets and systems, 44, 161-163, (1991) · Zbl 0742.34058 |

[13] | Puri, M.; Ralescu, D., Differentials of fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009 |

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

[15] | 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 |

[16] | Vorobiev, D.; Seikkala, S., Towards the theory of fuzzy differential equations, Fuzzy sets and systems, 125, 231-237, (2002) · Zbl 1003.34046 |

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

[18] | Wu, C.; Song, S.; Stanley Lee, E., Approximate solutions, Existence and uniqueness of the Cauchy problem of fuzzy differential equations, J. math. anal. appl., 202, 629-644, (1996) · Zbl 0861.34040 |

[19] | Zadeh, L.A., Fuzzy sets, Inform. and control, 8, 338-353, (1965) · Zbl 0139.24606 |

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