Armand, A., Allahviranloo, T., Gouyandeh, Z. (2018). SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS. Iranian Journal of Fuzzy Systems, 15(3), 27-46. doi: 10.22111/ijfs.2018.3948

Atefeh Armand; Tofigh Allahviranloo; Zienab Gouyandeh. "SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS". Iranian Journal of Fuzzy Systems, 15, 3, 2018, 27-46. doi: 10.22111/ijfs.2018.3948

Armand, A., Allahviranloo, T., Gouyandeh, Z. (2018). 'SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS', Iranian Journal of Fuzzy Systems, 15(3), pp. 27-46. doi: 10.22111/ijfs.2018.3948

Armand, A., Allahviranloo, T., Gouyandeh, Z. SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS. Iranian Journal of Fuzzy Systems, 2018; 15(3): 27-46. doi: 10.22111/ijfs.2018.3948

^{1}Young Researchers and Elites Club, Yadegar-e-Imam Khomeini (RAH) Shahre Rey Branch, Islamic Azad University, Tehran, Iran

^{2}Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran

^{3}Department of Mathematics, Najafabad Branch, Islamic Azad University, Najafabad, Isfahan, Iran

Abstract

In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for fuzzy integrals. Furthermore, we prove three essential theorems such as a fuzzy intermediate value theorem, fuzzy mean value theorem for integral and mean value theorem for $gH$-derivative. We derive a Bolzano's theorem, Rolle's theorem and some properties for $gH$-differentiable functions. To illustrate and explain these rules and theorems, we have provided several examples in details.

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