Peano type theorem for random fuzzy initial value problem

Marek T. Malinowski

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2011)

  • Volume: 31, Issue: 1, page 5-22
  • ISSN: 1509-9407

Abstract

top
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.

How to cite

top

Marek T. Malinowski. "Peano type theorem for random fuzzy initial value problem." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 31.1 (2011): 5-22. <http://eudml.org/doc/271141>.

@article{MarekT2011,
abstract = {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.},
author = {Marek T. Malinowski},
journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
keywords = {Random fuzzy differential equations; random fuzzy initial value problem; fuzzy stochastic process; fuzzy random variable; random fuzzy differential equations},
language = {eng},
number = {1},
pages = {5-22},
title = {Peano type theorem for random fuzzy initial value problem},
url = {http://eudml.org/doc/271141},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Marek T. Malinowski
TI - Peano type theorem for random fuzzy initial value problem
JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization
PY - 2011
VL - 31
IS - 1
SP - 5
EP - 22
AB - 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.
LA - eng
KW - Random fuzzy differential equations; random fuzzy initial value problem; fuzzy stochastic process; fuzzy random variable; random fuzzy differential equations
UR - http://eudml.org/doc/271141
ER -

References

top
  1. [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 Zbl1042.34027
  2. [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. [3] J.P. Aubin and H. Frankowska, Set-valued Analysis, Birkhäuser, Boston, 1990. Zbl0713.49021
  4. [4] B. Bede, T.G. Bhaskar and V. Lakshmikantham, Perspectives of fuzzy initial value problems, Communications in Applied Analysis 11 (2007), 339-358. Zbl1152.34041
  5. [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 Zbl1061.26024
  6. [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 Zbl1119.34003
  7. [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 Zbl1095.34511
  8. [8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Springer Verlag, Berlin, 1977. 
  9. [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 Zbl1142.34309
  10. [10] P. Diamond and P. Kloeden, Metric Spaces of Fuzzy Sets: Theory and Applications, World Scientific, Singapore, 1994. Zbl0873.54019
  11. [11] P. Diamond, Stability and periodicity in fuzzy differential equations, IEEE Trans. Fuzzy Systems 8 (2000), 583-590. empty 
  12. [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 Zbl1129.60063
  13. [13] Y. Feng, Fuzzy stochastic differential systems, Fuzzy Sets and Systems 115 (2000), 351-363. doi: 10.1016/S0165-0114(98)00389-3 
  14. [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 Zbl1232.68131
  15. [15] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems 24 (1987), 301-317. doi: 10.1016/0165-0114(87)90029-7 
  16. [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. [17] O. Kaleva, A note on fuzzy differential equations, Nonlinear Analysis 64 (2006), 895-900. doi: 10.1016/j.na.2005.01.003 Zbl1100.34500
  18. [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 Zbl1071.60060
  19. [19] M. Kisielewicz, Differential Inclusions and Optimal Control, Kluwer Acad. Publ., Dordrecht, 1991. Zbl0731.49001
  20. [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. [21] V. Lakshmikantham and R.N. Mohapatra, Theory of Fuzzy Differential Equations and Inclusions, Taylor & Francis, London, 2003. doi: 10.1201/9780203011386 Zbl1072.34001
  22. [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 Zbl1035.34064
  23. [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 Zbl1119.60039
  24. [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 Zbl1184.34011
  25. [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 Zbl1147.34311
  26. [26] C.V. Negoita and D.A. Ralescu, Applications of Fuzzy Sets to System Analysis, Wiley, New York, 1975. Zbl0326.94002
  27. [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. [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 Zbl1330.34039
  29. [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 Zbl1048.34015
  30. [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 Zbl0528.54009
  31. [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. [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 Zbl1225.34008
  33. [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 Zbl1225.34023
  34. [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. [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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.