Generalized Hukuhara differentiability of interval-valued functions and interval differential equations

Nonlinear Analysis, Theory, Methods and Applications

Volumn 71, Issue 3-4, 2009, Pages 1311-1328

  Stefanini, Luciano ^a,b   Bede, Barnabás ^a,b  

^a UNIVERSITY OF URBINO   (Italy)

^b UNIVERSITY OF TEXAS PAN AMERICAN   (United States)

Author keywords

Generalized Hukuhara differentiability; Interval differential equations; Interval valued functions

Indexed keywords

CONVEX SETS; GENERALIZED HUKUHARA DIFFERENTIABILITY; HUKUHARA DIFFERENCES; INTERVAL DIFFERENTIAL EQUATIONS; INTERVAL-VALUED FUNCTIONS; LOCAL EXISTENCES;

SET THEORY;

DIFFERENTIAL EQUATIONS;

EID: 67349163185     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2008.12.005     Document Type: Article

References (52)

1. Moore R.E. Interval Analysis (1966), Prentice-Hall, Englewood Cliffs, NJ
2. Neumaier A. Interval Methods for Systems of Equations (1990), Cambridge University Press
3. Nedialkov N.S., Jackson K.R., and Pryce J.D. An effective high order interval method for validating existence and uniqueness of the solution of an IVP for an ODE. Reliab. Comput. 7 6 (2001) 449-465
4. Makino K., and Berz M. Efficient control of the dependency problem based on Taylor model methods. Reliab. Comput. 5 1 (1999) 3-12 D.
5. Neumaier A. Rigorous sensitivity analysis for parameter dependent systems of equations. J. Math. Anal. Appl. 114 (1989) 16-25
6. Alefeld G., and Mayer G. Interval analysis: Theory and applications. J. Comput. Appl. Math. 121 (2000) 421-464
7. Moens D., and Vandepitte D. A survey of non-probabilistic uncertainty treatment in finite element analysis. Comput. Methods Appl. Mech. Engrg. 194 14-16 (2005) 1527-1555
8. Skalna I., Rama Rao M.V., and Pownuk A. Systems of fuzzy equations in structural mechanics. J. Comput. Appl. Math. 218 (2008) 149-156
9. Johnson T., and Tucker W. Rigorous parameter reconstruction for differential equations with noisy data. Automatica 44 (2008) 2422-2426
10. In: Baker Kearfott R., and Kreinovich V. (Eds). Applications of Interval Computations (1996), Kluwer Academic Publishers
11. Aubin J.P., and Cellina A. Differential Inclusions (1984), Springer-Verlag, Berlin, Heidelberg, New York, Tokyo
12. Deimling K. Multivalued Differential Equations (1992), W. De Gruyter, Berlin
13. Feckan M. Chaos in ordinary differential equations with perturbations: Applications to dry friction problems. Nonlinear Anal. TMA 30 (1997) 1355-1364
14. Kunze M. Non-Smooth Dynamical Systems (2000), Springer
15. Sextro W. Dynamical contact problems with friction. Models, Methods, Experiments and Applications (2002), Springer
16. Abbasbandy S., Nieto J.J., and Alavi M. Tuning of reachable set in one-dimensional fuzzy differential inclusions. Chaos Solitons Fractals 26 (2005) 1337-1341
17. Hukuhara M. Intégration des applications mesurables dont la valeur est un compact convex. Funkcial. Ekvac. 10 (1967) 205-229
18. Gnana Bhaskar T., and Lakshmikantham V. Set differential equations and flow invariance. J. Appl. Anal. 82 2 (2003) 357-368
19. Lakshmikantham V., and Tolstonogov A.A. Existence and interrelation between set and fuzzy differential equations. Nonlinear Anal. 55 3 (2003) 255-268
20. De Blasi F.S., Lakshmikantham V., and Bhaskar T.G. An existence theorem for set differential inclusions in a semilinear metric space. Control Cybernet. 36 3 (2007) 571-582
21. Lakshmikantham V., and Mohapatra R. Theory of Fuzzy Differential Equations and Inclusions (2003), Taylor & Francis, London
22. Agarwal R.P., and O'Regan D. Existence for Set Differential Equations via Multivalued Operator Equations. Differential Equations and Applications Vol. 5 (2007), Nova Sci. Publ, New York 1-5
23. De Blasi F.S. Semifixed sets of maps in hyperspaces with application to set differential equations. Set-Valued Anal. 14 (2006) 263-272
24. Kloeden P.E., Sadovsky B.N., and Vasilyeva I.E. Quasi-flows and equations with nonlinear differentials. Nonlinear Anal. (2002) 1143-1158
25. Laksmikantham V., Leela S., and Vatsala A.S. Interconnection between set and fuzzy differential equations. Nonlinear Anal. 54 (2003) 351-360
26. Zadeh L.A. Fuzzy Sets. Inform. and Control 8 (1965) 338-353
27. Puri M., and Ralescu D. Differentials of fuzzy functions. J. Math. Anal. Appl. 91 (1983) 552-558
28. Kaleva O. Fuzzy differential equations. Fuzzy Sets Syst. 24 (1987) 301-317
29. Seikkala S. On the fuzzy initial value problem. Fuzzy Sets Syst. 24 (1987) 319-330
30. Diamond P. Stability and periodicity in fuzzy differential equations. IEEE Trans. Fuzzy Systems 8 (2000) 583-590
31. Hüllermeier E. An approach to modelling and simulation of uncertain dynamical systems. Int. J. Uncertainty, Fuzziness and Knowledge-Based Systems 5 (1997) 117-137
32. Kloeden P.E. Remarks on Peano theorem for fuzzy differential equations. Fuzzy Sets Syst. 44 (1991) 161-163
33. Stefanini L., Sorini L., and Guerra M.L. Parametric representation of fuzzy numbers and application to fuzzy calculus. Fuzzy Sets Syst. 157 (2006) 2423-2455
34. Stefanini L., Sorini L., and Guerra M.L. Fuzzy numbers and Fuzzy Arithmetic. In: Pedrycz W., Skowron A., and Kreinovich V. (Eds). Handbook of Granular Computing (2008), John Wiley & Sons, Ltd. Chapter 12
35. Nieto J.J., and Rodríguez-López R. Euler polygonal method for metric dynamical systems. Inform. Sci. 177 (2007) 4256-4270
36. Rodríguez-López R. Comparison results for fuzzy differential equations. Inform. Sci. 178 (2008) 1756-1779
37. Mizukoshi M.T., Barros L.C., Chalco-Cano Y., Román-Flores H., and Bassanezi R.C. Fuzzy differential equations and the extension principle. Inform. Sci. 177 (2007) 3627-3635
38. Chalco-Cano Y., and Román-Flores H. On new solutions of fuzzy differential equations. Chaos Solitons Fractals 38 (2008) 112-119
39. Lakshmikantham V., and Nieto J.J. Differential equations in metric spaces: An introduction and an application to fuzzy differential equations. Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 10 (2003) 991-1000
40. Nieto J.J., and Rodriguez-Lopez R. Hybrid metric dynamical systems with impulses. Nonlinear Anal. 64 (2006) 368-380
41. Nieto J.J., Rodríguez-López R., and Georgiou D.N. Fuzzy differential systems under generalized metric spaces approach. Dynam. Systems Appl. 17 (2008) 1-24
42. Stefanini L. A generalization of Hukuhara difference for interval and fuzzy arithmetic. In: Dubois D., Lubiano M.A., Prade H., Gil M.A., Grzegorzewski P., and Hryniewicz O. (Eds). Soft Methods for Handling Variability and Imprecision. Series on Advances in Soft Computing vol. 48 (2008), Springer. http://econpapers.repec.org/RAS/pst233.htm An extended version is available online at the RePEc service: http://econpapers.repec.org/RAS/pst233.htm
43. L. Stefanini, On the generalized LU-fuzzy derivative and fuzzy differential equations, in: Proc. of the 2007 IEEE International Conference on Fuzzy Systems, London, July 2007, pp. 710-715
44. Bede B., and Gal S.G. Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets Syst. 151 (2005) 581-599
45. Bede B., Rudas I.J., and Bencsik A. First order linear differential equations under generalized differentiability. Inform. Sci. 177 (2007) 1648-1662
46. Bede B. Note on "Numerical solutions of fuzzy differential equations by predictor-corrector method". Inform. Sci. 178 (2008) 1917-1922
47. Diamond P., and Kloeden P. Metric Spaces of Fuzzy Sets (1994), World Scientific, New Jersey
48. Nieto J.J., Rodríguez-López R., and Franco D. Linear first order fuzzy differential equations. Internat. J. Uncertain. Fuzziness Knowledge-Based Systems 14 (2006) 687-709
49. Bede B., Gnana Bhaskar T., and Lakshmikantham V. Perspectives of fuzzy initial value problems. Commun. Appl. Anal. 11 (2007) 339-358
50. Wu C., Song S., and Stanley Lee E. Approximate solutions, existence and uniqueness of the Cauchy problem of fuzzy differential equations. J. Math. Anal. Appl. 202 (1996) 629-644
51. Brenan K.E., Campbell S.L., and Petzold L.R. Numerical Solution of Initial-Value Problems in Differential Algebraic Equations (1989), North Holland, New York republished by SIAM, 1996
52. Campbell S.L., and Gear C.W. The index of general nonlinear DAEs. Numer. Math. 72 (1995) 173-196