Document Type : Research Paper


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


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.


[1] S. Abbasbandy and B. Asady, Ranking of fuzzy numbers by sign distance, Information Sciences,
176 ( 2006), 2405–2416.
[2] S. Abbasbandy and S. Hajighasemi, A Fuzzy Distance between Two Fuzzy Numbers, Communications
in Computer and Information Science, 81 (2010), 376–382.
[3] T. Allahviranloo, S. Abbasbandy and S. Hajighasemi, A new similarity measure for generalized
fuzzy numbers, Neural Computing and Applications, 21 (2012), 289–294.
[4] T. Allahviranloo, A. Armand and Z. Gouyandeh, Fuzzy fractional differential equations under
generalized fuzzy Caputo derivative, Journal of Intelligent and Fuzzy Systems, 26 (2014),
[5] T. Allahviranloo, A. Armand, Z. Gouyandeh and H. Ghadiri, Existence and uniqueness of
solutions for fuzzy fractional Volterra–Fredholm integro–differential equations, Journal of
Fuzzy Set Valued Analysis, 2013 (2013), 1–9.
[6] A. Armand and Z. Gouyandeh, Solving two–point fuzzy boundary value problem using variational
iteration method, Communications on Advanced Computational Science with Applications,
2013 (2013), 1–10.
[7] G. A. Anastassiou, Fuzzy Mathematics:Approximation Theory, Studies in Fuzziness and Soft
Computing, Springer–Verlag Berlin Heidelberg,2010.
[8] R. J. Aumann, Integrals of set–valued functions, Journal of Mathematical Analysis and Applications,
12 (1965), 1–12.
[9] B. Bede, I. J. Rudas and A. L. Bencsik, First order linear fuzzy differential equations under
generalized differentiability, Information Sciences, 177 (2007), 1648–1662.
[10] B. Bede and S. Gal, Generalizations of the differentiability of fuzzy number valued functions
with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 ( 2005), 581–
[11] B. Bede and L. Stefanini, Generalized differentiability of fuzzy–valued functions, Fuzzy Sets
and Systems, 230 (2013), 119–141.
[12] Y. Chalco-Canoa, A. Rufi´an-Lizanab, H. Rom´an–Floresa and M. D. Jim´enez-Gameroa, Calculus
for interval–valued functions using generalized Hukuhara derivative and applications,
Fuzzy Sets and Systems, 219 (2013), 49–67.
[13] L. H. Chen and H. W. Lu, The preference order of fuzzy numbers, Computers and Mathematics
with Applications, 44 (2002), 1455–1465.
[14] A. Chwastyk and W. Kosiski, Fuzzy calculus with aplications, Mathematica applicanda, 41
(2013), 47–96.
[15] W. Congxin and M. Ming, On embedding problem of fuzzy number space: part 3, Fuzzy Sets
and Systems, 16 (1992), 281–286.
[16] D. Dubois and H. Prade, Towards fuzzy differential calculus–part 1, Fuzzy Sets and Systems,
8 (1982), 1–17.
[17] D. Dubois and H. Prade, Towards fuzzy differential calculus–part 2, Fuzzy Sets and Systems,
8 (1982), 105–116.
[18] D. Dubois and H. Prade, Towards fuzzy differential calculus–part 3, Fuzzy Sets and Systems,
8 (1982), 225–234.
[19] D. Dubois and H. Prade, Operations on fuzzy numbers, International Journal of Systems
Science, 9 (1978), 613–626.
[20] Z. Gouyandeh, T. Allahviranloo, S. Abbasbandy and A. Armand, A fuzzy solution of heat
equation under generalized Hukuhara differentiability by fuzzy Fourier transform, Fuzzy Sets
and Systems, 309 (2017), 81–97.
[21] M. Hukuhara, Int´egration des applications mesurables dont la valeur est un compact convexe,
Funkcialaj Ekvacioj, 10 (1967), 205–229.
[22] R. Jain, Decision making in the presence of fuzzy variables, IEEE Trans. Systems Man
Cybernetics, 6 (1976), 698–703.
[23] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301–317.
[24] O. Kaleva and S. Seikkala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984),
[25] A. Khastan and J. J. Nieto, A boundary value problem for second order fuzzy differential
equations, Nonlinear Analysis, 72 (2010), 3583–3593.
[26] V. Lakshmikantham and R. N. Mohapatra, Theory of fuzzy differential equations and inclusions,
Taylor and Francis, 2003.
[27] V. Lakshmikantham, T. GnanaBhaskar and J. VasundharaDevi, Theory of Set Differential
Equations in Metric Spaces, Cambridge Scientific Publishers, 2006.
[28] X. W. Liu and S. L. Han, Ranking fuzzy numbers with preference weighting function expectations,
Computers and Mathematics with Applicatmns, 49 (2005), 1731–1753.
[29] M. Ma, M. Friedman and A. Kandel, Numerical solutions of fuzzy differential equations,
Fuzzy Sets and Systems, 105 (1999), 133–138.
[30] M. Mizumoto and K. Tanaka, The four operations of arithmetic on fuzzy numbers, Systems
Computers Controls, 7 (1976), 73–81.
[31] D. Mon, C. Cheng and J. Lin, Evaluating weapon system using fuzzy analytic hierarchy
process based on entropy weight, Fuzzy Sets and Systems, 62 (1994), 127–134.
[32] C. V. Negoita and D. Ralescu, Applications of Fuzzy Sets to Systems Analysis, Wiley, New
York, 1975.
[33] M. L. Puri and D. A. Ralescu, Differentials of fuzzy functions, Journal of Mathematical
Analysis and Applications, 114 (1986), 409–422.
[34] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy
arithmetic, Fuzzy Sets and Systems, 161 (2010), 1564–1584.
[35] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval–valued functions
and interval differential equations, Nonlinear Analysis, 71 (2009), 1311–1328.
[36] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338–353.