On the fractional differential equations with uncertainty.(English)Zbl 1219.34004

Summary: This paper is based on the concept of fuzzy differential equations of fractional order introduced by R. P. Agarwal, V. Lakshmikantham and J. J. Nieto [Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 6, 2859–2862 (2010; Zbl 1188.34005)]. Using this concept, we prove some results on the existence and uniqueness of solutions to fuzzy fractional differential equations.

MSC:

 34A07 Fuzzy ordinary differential equations 34A08 Fractional ordinary differential equations

Zbl 1188.34005
Full Text:

References:

 [1] Oldham, K.B.; Spanier, J., The fractional calculus, (1974), Academic Press New York · Zbl 0428.26004 [2] Samko, S.G.; Kilbas, A.A.; Marichev, O.I., Fractional integrals and derivatives, theory and applications, (1993), Gordon and Breach Sci. Publishers London, New York · Zbl 0818.26003 [3] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), A Wiley-Interscience Publication, John Wiley & Sons, Inc. New York · Zbl 0789.26002 [4] Podlubny, I., Fractional differential equations, (1999), Academic Press San Diego · Zbl 0918.34010 [5] Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J., Theory and applications of fractional differential equations, (2006), Elsevier Science B.V. Amsterdam · Zbl 1092.45003 [6] Lakshmikantham, V.; Leela, S.; Vasundhara Devi, J., Theory of fractional dynamic systems, (2009), Cambridge Academic Publishers Cambridge · Zbl 1188.37002 [7] Ahmad, B.; Nieto, J.J., Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions, Comput. math. appl., 58, 1838-1843, (2009) · Zbl 1205.34003 [8] Araya, D.; Lizama, C., Almost automorphic mild solutions to fractional differential equations, Nonlinear anal., 69, 3692-3705, (2008) · Zbl 1166.34033 [9] Belmekki, M.; Nieto, J.J.; Rodríguez-López, R., Existence of periodic solutions for a nonlinear fractional differential equation, Bound. value probl., 2009, (2009), Art. ID. 324561 · Zbl 1181.34006 [10] Bonilla, B.; Rivero, M.; Rodríguez-Germá, L.; Trujillo, J.J., Fractional differential equations as alternative models to nonlinear differential equations, Appl. math. comput., 187, 79-88, (2007) · Zbl 1120.34323 [11] Chen, M.; Li, D.; Xue, X., Periodic problems of first order uncertain dynamical systems, Fuzzy sets and systems, 162, 67-78, (2011) · Zbl 1214.34003 [12] A.D.R. Choudary, T. Donchev, On Peano theorem for fuzzy differential equations, Fuzzy Sets and Systems, doi:10.1016/j.fss.2011.01.005 (in press). · Zbl 1246.34003 [13] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005 [14] Diethelm, K.; Ford, N.J., Analysis of fractional differential equations, J. math. anal. appl., 265, 229-248, (2002) · Zbl 1014.34003 [15] Dubois, D.; Prade, H., Towards fuzzy differential calculus — part 1, Fuzzy sets and systems, 8, 1-17, (1982) · Zbl 0493.28002 [16] Dubois, D.; Prade, H., Towards fuzzy differential calculus — part 2, Fuzzy sets and systems, 8, 105-116, (1982) · Zbl 0493.28003 [17] Dubois, D.; Prade, H., Towards fuzzy differential calculus — part 3, Fuzzy sets and systems, 8, 225-234, (1982) [18] Chang, Y.-K.; Nieto, J.J., Some new existence results for fractional differential inclusions with boundary conditions, Math. comput. modelling, 49, 605-609, (2009) · Zbl 1165.34313 [19] Lakshmikantham, V., Theory of fractional functional differential equations, Nonlinear anal., 69, 3337-3343, (2008) · Zbl 1162.34344 [20] Lakshmikantham, V.; Leela, S., Nagumo-type uniqueness result for fractional differential equations, J. nonlinear anal., 71, 7-8, 2886-2889, (2009) · Zbl 1177.34003 [21] Lakshmikanthama, V.; Vatsala, A.S., Basic theory of fractional differential equations, Nonlinear anal., 69, 2677-2682, (2008) · Zbl 1161.34001 [22] Lakshmikantham, V.; Vatsala, A.S., General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. lett., 21, 8, 828-834, (2008) · Zbl 1161.34031 [23] Nieto, J.J.; Rodríguez-López, R.; Georgiou, D.N., Fuzzy differential systems under generalized metric spaces approach, Dyn. syst. appl., 17, 1-24, (2008) · Zbl 1168.34005 [24] Nieto, J.J., Maximum principles for fractional differential equations derived from Mittag-Leffler functions, Appl. math. lett., 23, 1248-1251, (2010) · Zbl 1202.34019 [25] Shuqin, Z., Monotone iterative method for initial value problem involving Riemann Liouville fractional derivatives, Nonlinear anal., 71, 2087-2093, (2009) · Zbl 1172.26307 [26] Puri, M.L.; Ralescu, D.A., Differentials for fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009 [27] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004 [28] Kaleva, O., A note on fuzzy differential equations, Nonlinear anal., 64, 895-900, (2006) · Zbl 1100.34500 [29] Lakshmikantham, V.; Mohapatra, R.N., Theory of fuzzy differential equations and inclusions, (2003), Taylor & Francis London · Zbl 1072.34001 [30] Khastan, A.; Nieto, J.J., A boundary value problem for second order fuzzy differential equations, Nonlinear anal., 72, 3583-3593, (2010) · Zbl 1193.34004 [31] Lupulescu, V., On a class of fuzzy functional differential equations, Fuzzy sets and systems, 160, 1547-1562, (2009) · Zbl 1183.34124 [32] Xu, J.; Liao, Z.; Nieto, J.J., A class of linear differential dynamical systems with fuzzy matrices, J. math. anal. appl., 368, 54-68, (2010) · Zbl 1193.37025 [33] Yu, J.; Chena, B.; Yua, H.; Gao, J., Adaptive fuzzy tracking control for the chaotic permanent magnet synchronous motor drive system via backstepping, Nonlinear anal.: real world appl., 12, 671-681, (2011) · Zbl 1203.93112 [34] Agarwal, R.P.; Lakshmikantham, V.; Nieto, J.J., On the concept of solution for fractional differential equations with uncertainty, Nonlinear anal., 72, 2859-2862, (2010) · Zbl 1188.34005 [35] Negoita, C.V.; Ralescu, D.A., Applications of fuzzy sets to system analysis, (1975), Birkhauser Basel · Zbl 0326.94002 [36] Seikkala, S., On the fuzzy initial value problem, Fuzzy sets and systems, 24, 319-330, (1987) · Zbl 0643.34005
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