Alternative views on fuzzy numbers and their application to fuzzy differential equations

Document Type : Research Article

Authors

1 Department of Mathematics, University of Science and Technology of Mazandaran, Behshahr, Iran

2 Young Researchers and Elite Club, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran

3 Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Abstract

 We consider fuzzy valued functions from two parametric representations of $\alpha$-level sets. New concepts are introduced and compared with available notions. Following the two proposed approaches, we study fuzzy differential equations. Their relation with Zadeh's extension principle and the generalized Hukuhara derivative is discussed. Moreover,
we prove existence and uniqueness theorems for fuzzy differential equations. Illustrative examples are given.

Keywords

Main Subjects


[1] L.C. Barros, L.T. Gomes, P.A. Tonelli, Fuzzy differential equations: an approach via fuzzification
of the derivative operator, Fuzzy Sets and Systems 230 (2013) 39–52.
[2] B. Bede, Mathematics of Fuzzy Sets and Fuzzy Logic, Studies in Fuzziness and Soft Computing,
295, Springer, Heidelberg, 2013.
[3] B. Bede, S.G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with
applications to fuzzy differential equations, Fuzzy Sets and Systems 151 (2005) 581–599.
[4] B. Bede, S.G. Gal, Solutions of fuzzy differential equations based on generalized differentiability,
Commun. Math. Anal. 9 (2010) 22–41.
[5] B. Bede, L. Stefanini, Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Sys-
tems 230 (2013) 119–141.
[6] A. Bencsik, B. Bede, J. Tar, J. Fodor, Fuzzy differential equations in modeling hydraulic differ-
ential servo cylinders, Third Romanian-Hungarian Joint Symposium on Applied Computational
Intelligence, SACI, Timisoara, Romania, 2006.
[7] J.J. Buckley, T. Feuring, Fuzzy differential equations, Fuzzy Sets and Systems 110 (2000), 43–54.
[8] Y. Chalco-Cano, W.A. Lodwick, B. Bede, Single level constraint interval arithmetic, Fuzzy Sets
and Systems 257 (2014), 146–168.
[9] Y. Chalco-Cano, H. Rom´an-Flores, On new solutions of fuzzy differential equations, Chaos Solitons
Fractals 38 (2008) 112–119.
[10] Y. Chalco-Cano, H. Rom´an-Flores, Comparison between some approaches to solve fuzzy differen-
tial equations, Fuzzy Sets and Systems 160 (2009) 1517–1527.
[11] Y. Chalco-Cano, H. Rom´an-Flores, Some remarks on fuzzy differential equations via differential
inclusions, Fuzzy Sets and Systems 230 (2013) 3–20.
[12] Y. Chalco-Cano, H. Rom´an-Flores, M. Rojas-Medar, O.R. Saavedra, M.D. Jim´enez-Gamero, The
extension principle and a decomposition of fuzzy sets, Inform. Sci. 177 (2007) 5394–5403.
[13] Y. Chalco-Cano, A. Rufi´an-Lizana, H. Rom´an-Flores, M.D. Jim´enez-Gamero, Calculus for
interval-valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and
Systems 219 (2013) 49–67.
[14] P. Diamond, Time-dependent differential inclusions, cocycle attractors and fuzzy differential equa-
tions, IEEE Trans. Fuzzy Syst. 7 (1999) 734–740.
[15] P. Diamond, Stability and periodicity in fuzzy differential equations, IEEE Trans. Fuzzy Syst. 8
(2000), 583–590.
[16] D. Dubois, H. Prade, Operations on fuzzy numbers, Internat. J. Systems Sci. 9 (1978), 613–626.
[17] R. E. Giachetti, R. E. Young, A parametric representation of fuzzy numbers and their arithmetic
operators, Fuzzy Sets and Systems 91 (1997), 185–202.
[18] R. Goetschel Jr., W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18 (1986) 31–43.
[19] L.T. Gomes, L.C. Barros, A note on the generalized difference and the generalized differentiability,
Fuzzy Sets and Systems 280 (2015) 142–145.
[20] M. Heidari, M. Ramezanzadeh, A.H. Borzabadi, O.S. Fard, Solutions to fuzzy variational problems:
necessary and sufficient conditions, Int. J. Model. Identif. Control. 28 (2017) 187–198.
[21] M. Heidari, M.R. Zadeh, O.S. Fard, A.H. Borzabadi, On unconstrained fuzzy-valued optimization
problems, Int. J. Fuzzy Syst. 18 (2016) 270–283.
[22] H. Huang, C. Wu, Approximation of fuzzy functions by regular fuzzy neural networks, Fuzzy Sets
and Systems 177 (2011) 60–79.
[23] E. H¨ullermeier, An approach to modelling and simulation of uncertain dynamical systems, Internat.
J. Uncertain. Fuzziness Knowledge-Based Systems 5 (1997), 117–137.
[24] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987) 301–317.
[25] G. J. Klir, Fuzzy arithmetic with requisite constraints, Fuzzy Sets and Systems 91 (1997) 165–175.
[26] G.J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic, Prentice Hall PTR, Upper Saddle River, NJ, 1995.
[27] W.A. Lodwick, D. Dubois, Interval linear systems as a necessary step in fuzzy linear systems,
Fuzzy Sets and Systems 281 (2015) 227–251.
[28] V. Lupulescu, On a class of fuzzy functional differential equations, Fuzzy Sets and Systems 160
(2009) 547–1562.
[29] S. Melliani, Semi-linear equation with fuzzy parameters, Notes IFS 5 (1999) 42–47.
[30] M.T. Mizukoshi, L.C. Barros, Y. Chalco-Cano, H. Rom´an-Flores, R.C. Bassanezi, Fuzzy differential
equations and the extension principle, Inform. Sci. 177 (2007) 3627–3635.
[31] C.V. Negoit¸˘a D.A. Ralescu, Applications of Fuzzy Sets to Systems Analysis, John Wiley & Sons,
New York, 1975.
[32] M. Oberguggenberger, S. Pittschmann, Differential equations with fuzzy parameters, Math. Mod-
elling Syst. 5 (1999) 181–202.
[33] M. L. Puri, D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983) 552–558.
[34] M. Ramezanadeh, M. Heidari, O.S. Fard, A.H. Borzabadi, On the interval differential equation:
novel solution methodology, Adv. Difference Equ. 2015 (2015) 23.
[35] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic,
Fuzzy Sets and Systems 161 (2010) 1564–1584.
[36] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and in-
terval differential equations, Nonlinear Anal. 71 (2009) 1311–1328.
[37] L. Stefanini, L. Sorini, M.L. Guerra, Parametric representation of fuzzy numbers and application
to fuzzy calculus, Fuzzy Sets and Systems 157 (2006) 2423–2455.
[38] L.A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning. I,
Information Sci. 8 (1975) 199–249.
[39] L.A. Zadeh, The role of fuzzy logic in modeling, identification and control, MIC—Model. Identif.
Control 15 (1994) 191–203.
[40] L.A. Zadeh, Toward a generalized theory of uncertainty (GTU)—an outline, Inform. Sci. 172 (2005)
1–40.