Peano type theorem for random fuzzy initial value problem

Malinowski, Marek

Discussiones Mathematicae. Differential Inclusions, Control and Optimization, Tome 31 (2011) no. 1, pp. 5-22.

Résumé

In this paper we consider the random fuzzy differential equations and show their application by an example. Under suitable conditions the Peano type theorem on existence of solutions is proved. For our purposes, a notion of ε-solution is exploited.

Détail

Keywords: Random fuzzy differential equations, random fuzzy initial value problem, fuzzy stochastic process, fuzzy random variable

Bibliographie

[1] R.P. Agarwal, D. O'Regan and V. Lakshmikantham, A stacking theorem approach for fuzzy differential equations, Nonlinear Analysis 55 (2003), 299-312. doi: 10.1016/S0362-546X(03)00241-4

[2] R.P. Agarwal, D. O'Regan and V. Lakshmikantham, Viability theory and fuzzy differential equations, Fuzzy Sets and Systems 151 (2005), 563-580. doi: 10.1016/j.fss.2004.08.008

[3] J.P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990.

[4] B. Bede, T.G. Bhaskar and V. Lakshmikantham, Perspectives of fuzzy initial value problems, Communications in Applied Analysis 11 (2007), 339-358.

[5] B. Bede and 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. doi: 10.1016/j.fss.2004.08.001

[6] B. Bede, I.J. Rudas and A.L. Bencsik, First order linear fuzzy differential equations under generalized differentiability, Information Sciences 177 (2007), 1648-1662. doi: 10.1016/j.ins.2006.08.021

[7] T.G. Bhaskar, V. Lakshmikantham and V. Devi, Revisiting fuzzy differential equations, Nonlinear Analysis 58 (2004), 351-358. doi: 10.1016/j.na.2004.05.007

[8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer Verlag, Berlin, 1977.

[9] Y. Chalco-Cano and H. Román-Flores, On new solutions of fuzzy differential equations, Chaos Solitons Fractals 38 (2008), 112-119. doi: 10.1016/j.chaos.2006.10.043

[10] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994.

[11] P. Diamond, Stability and periodicity in fuzzy differential equations, IEEE Trans. Fuzzy Systems 8 (2000), 583-590. empty

[12] W. Fei, Existence and uniqueness of solution for fuzzy random differential equations with non-Lipschitz coefficients, Information Sciences 177 (2007), 4329-4337. doi: 10.1016/j.ins.2007.03.004

[13] Y. Feng, Fuzzy stochastic differential systems, Fuzzy Sets and Systems 115 (2000), 351-363. doi: 10.1016/S0165-0114(98)00389-3

[14] E. Hüllermeier, An approach to modelling and simulation of uncertain dynamical systems, Int. J. Uncertainty Fuzziness Knowledge-Based Syst. 5 (1997), 117-137. doi: 10.1142/S0218488597000117

[15] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7

[16] O. Kaleva, The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Systems 35 (1990), 389-396. doi: 10.1016/0165-0114(90)90010-4

[17] O. Kaleva, A note on fuzzy differential equations, Nonlinear Analysis 64 (2006), 895-900. doi: 10.1016/j.na.2005.01.003

[18] J.H. Kim, On fuzzy stochastic differential equations, J. Korean Math. Soc. 42 (2005), 153-169. doi: 10.4134/JKMS.2005.42.1.153

[19] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad. Publ., Dordrecht, 1991.

[20] V. Lakshmikantham, S. Leela and A.S. Vatsala, Interconnection between set and fuzzy differential equations, Nonlinear Analysis 54 (2003), 351-360. doi: 10.1016/S0362-546X(03)00067-1

[21] V. Lakshmikantham and R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor Francis, London, 2003. doi: 10.1201/9780203011386

[22] V. Lakshmikantham and A.A. Tolstonogov, Existence and interrelation between set and fuzzy differential equations, Nonlinear Analysis 55 (2003), 255-268. doi: 10.1016/S0362-546X(03)00228-1

[23] Sh. Li and A. Ren, Representation theorems, set-valued and fuzzy set-valued Ito integral, Fuzzy Sets and Systems 158 (2007), 949-962. doi: 10.1016/j.fss.2006.12.004

[24] M.T. Malinowski, On random fuzzy differential equations, Fuzzy Sets and Systems 160 (2009), 3152-3165. doi: 10.1016/j.fss.2009.02.003

[25] M.T. Mizukoshi, L.C. Barros, Y. Chalco-Cano, H. Román-Flores and R.C. Bassanezi, Fuzzy differential equations and the extension principle, Information Sciences 177 (2007), 3627-3635. doi: 10.1016/j.ins.2007.02.039

[26] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to System Analysis, Wiley, New York, 1975.

[27] J.J. Nieto, The Cauchy problem for continuous fuzzy differential equations, Fuzzy Sets and Systems 102 (1999), 259-262. doi: 10.1016/S0165-0114(97)00094-8

[28] J.J. Nieto and R. Rodríguez-López, Bounded solutions for fuzzy differential and integral equations, Chaos Solitons Fractals 27 (2006), 1376-1386. doi: 10.1016/j.chaos.2005.05.012

[29] D. O'Regan, V. Lakshmikantham and J.J. Nieto, Initial and boundary value problems for fuzzy differential equations, Nonlinear Analysis 54 (2003), 405-415. doi: 10.1016/S0362-546X(03)00097-X

[30] M.L. Puri and D.A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl. 91 (1983), 552-558. doi: 10.1016/0022-247X(83)90169-5

[31] M.L. Puri and D.A. Ralescu, Fuzzy random variables, J. Math. Anal. Appl. 114 (1986), 409-422. doi: 10.1016/0022-247X(86)90093-4

[32] R. Rodríguez-López, Periodic boundary value problems for impulsive fuzzy differential equations, Fuzzy Sets and Systems 159 (2008), 1384-1409. doi: 10.1016/j.fss.2007.09.005

[33] R. Rodríguez-López, Monotone method for fuzzy differential equations, Fuzzy Sets and Systems 159 (2008), 2047-2076. doi: 10.1016/j.fss.2007.12.020

[34] S. Seikkala, On the fuzzy initial value problem, Fuzzy Sets and Systems 24 (1987), 319-330. doi: 10.1016/0165-0114(87)90030-3

[35] J. Zhang, Set-valued stochastic integrals with respect to a real valued martingale, in: Soft Methods for Handling Variability and Imprecision, Springer, Berlin 2008, 253-259.