JID:FSS AID:6958 /FLA [m3SC+; v1.221; Prn:18/12/2015; 13:47] P.1(1-15) Available online at www.sciencedirect.com ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 18 18 19 19 20 20 21 21 22 22 23 23 24 24 25 25 26 26 27 27 28 28 29 29 30 30 31 31 32 32 33 33 34 34 35 35 36 36 37 37 38 38 39 39 40 40 41 41 42 42 43 43 44 44 45 45 46 46 47 47 48 48 49 49 50 50 51 51 52 52 Relationship between Bede–Gal differentiable set-valued functions and their associated support functions ¸ Sahin Emrah Amrahov a,∗ , Alireza Khastan b , Nizami Gasilov c , Afet Golayoglu Fatullayev c a Computer Engineering Department, Ankara University, 06830 Turkey b Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan 45137-66731, Iran c Baskent University, Ankara, 06810 Turkey Received 24 October 2014; received in revised form 26 November 2015; accepted 4 December 2015 Abstract In this study, we adapt the concept of the Bede–Gal derivative, which was initially suggested for fuzzy number-valued functions, to set-valued functions. We use an example to demonstrate that this concept overcomes some of the shortcomings of the Hukuhara derivative. We prove some properties of Bede–Gal differentiable set-valued functions. We also study the relationship between a Bede–Gal differentiable set-valued function and its value’s support function, which we call the associated support function. We provide examples of set-valued functions that are not Bede–Gal differentiable whereas their associated support functions are differentiable. We also present some applications of the Bede–Gal derivative to solving set-valued differential equations.  2015 Published by Elsevier B.V. Keywords: Bede–Gal differentiability; Set-valued differential equation; Set-valued function; Support function 1. Introduction Recently, many studies have examined the properties of set-valued functions [15,34,42]. In fact, this is the second time in mathematical history. Previously, at the beginning of the 1960s, set-valued functions were explored to study the properties of controllable dynamic systems. Reformulating differential equations that describe the behavior of controllable systems in the form of differential inclusion is a very important idea. Indeed, the notion of differential inclusion was actually expressed in earlier studies during the 1930s, e.g., Marchaud [33] and Zaremba [45]. However, these studies yielded no applications and thus they did not attract the attention of other researchers. In the 1960s, prov- ing the Pontryagin maximum principle for optimal control [37] inspired further studies of the properties of set-valued functions and many important results were reported [5,6,8,11,13,14,17,43]. The proposal of fuzzy logic and fuzzy sets * Corresponding author. Tel.: +90 312 2033300x1771. E-mail address: emrah@eng.ankara.edu.tr (¸ S.E. Amrahov). http://dx.doi.org/10.1016/j.fss.2015.12.002 0165-0114/ 2015 Published by Elsevier B.V.