定义和讨论了模糊数值函数的距离导数, 给出了模糊有界变差函数全变差的积分表示. 发现模糊绝对连续函数是几乎处处距离可导的, 距离导数的积分等于其原函数的总变差, 从而给出了模糊有界变差函数全变差的积分表示.
Abstract
The metric derivative of the fuzzy-number-valued functions and the representation of the total variation for the fuzzy-number-valued function which is of bounded variation are defined and discussed. It is proved that the fuzzy absolutely continuous functions are metrically differentiable almost everywhere, and the integration of its metric derivative equals to the total variation of the primitive. Finally, the representation of the total variation for the fuzzy-number-valued functions which is of bounded variation is given.
关键词
模糊数 /
模糊数值函数 /
距离导数
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Key words
fuzzy numbers /
fuzzy-number-valued functions /
metric derivative
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参考文献
[1] Puri M. L., Ralescu D. A., Fuzzy random variables, Journal of Mathematics Analysis and Applications, 1986, 114(2): 409-422.
[2] Kaleva O., Fuzzy differential equations, Fuzzy Sets and Systems, 1987, 24(3): 301-317.
[3] Wu C. X., Ma M., On embedding problem of fuzzy number space: Part 2, Fuzzy Sets and Systems, 1992, 45(2): 189-202.
[4] Wu C. X., Gong Z. T., On Henstock integrals of interval-valued functions and fuzzy-valued functions, Fuzzy Sets and Systems, 2000, 115(3): 377-391.
[5] Wu C. X., Gong Z. T., On Henstock integrals of fuzzy-valued functions (I), Fuzzy Sets and Systems, 2001, 120(3): 523-532.
[6] Gong Z. T., Wu C. X., On the problem of characterizing derivatives for the fuzzy-valued functions, Fuzzy Sets and Systems, 2002, 127(3): 315-322.
[7] Gong Z. T., Wu C. X., Bounded variation, absolute continuous and absolute integrability for fuzzy-numbervalued functions, Fuzzy Sets and Systems, 2002, 129(1): 83-94.
[8] Feng Y. H., A note on indefinite integrals and absolute continuity for fuzzy-valued mappings, Fuzzy Sets and Systems, 2004, 147(3): 405-415.
[9] Gong Z. T., On the problem of characterizing derivatives for the fuzzy-valued functions (II), Fuzzy Sets and Systems, 2004, 145(3): 381-393.
[10] Kirchheim B., Retifiable metric spaces:local structure and regularity of the Hausdorff measure, Proceedings of the American Mathematical Society, 1994, 121(1): 113-123.
[11] Federer H., Geometric Measure Theory, Grundlehren Math. Wiss. 153, New York: Springer, 1969.
[12] Lee P. Y., Lanzhou Lectures on Henstock Integration, Singapore, New Jersey, London, Hong Kong: World Scientific, 1989.
[13] Gong Z. T., Nonabsolute fuzzy integrals, absolute integrability and its absolute-value inequality, Journal of Mathematical Reseach and Exposition, 2008, 28(3): 479-488.
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脚注
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基金
国家自然科学基金(71061013, 10771171);西北师范大学知识创新工程(NWNU-KJCXGC-03-61)
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