×

A boundary value problem for second order fuzzy differential equations. (English) Zbl 1193.34004

The authors consider the two-point boundary value problem for fuzzy differential equations in the space of fuzzy real numbers
\[ y'' (t)=f(t,y(t),y' (t)), \,\, y(0)=y_0,\,\, y(1)=y_1, \]
where \(y_0,y_1\in E^1\) and \(f:[0,1]\times E\times E\to E\) is a continuous fuzzy function, and show that the set of solutions for the case when the derivative is understood in the sense of Hukuhara [O. Kaleva, Fuzzy Sets Syst. 24, 301–317 (1987; Zbl 0646.34019)] is included in the set of solutions when the derivative is understood in the generalized sense [B. Bede and S. G. Gal, Fuzzy Sets Syst. 151, No. 3, 581–599 (2005; Zbl 1061.26024)]. They present some examples to illustrate their results.

MSC:

34A07 Fuzzy ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] R.P. Agarwal, V. Lakshmikantham, J.J. Nieto, On the concept of solution for fractional differential equations with uncertainty, Nonlinear Anal., in press (doi:10.1016/j.na.2009.11.029)
[2] Buckley, J.J.; Feuring, T., Fuzzy differential equations, Fuzzy sets and systems, 110, 43-54, (2000) · Zbl 0947.34049
[3] Chalco-Cano, Y.; Román-Flores, H., Comparation between some approaches to solve fuzzy differential equations, Fuzzy sets and systems, 160, 1517-1527, (2009) · Zbl 1198.34005
[4] Hüllermeier, E., An approach to modelling and simulation of uncertain systems, Int. J. uncertain. fuzz., knowledege-based system, 5, 117-137, (1997) · Zbl 1232.68131
[5] Lakshmikantham, V.; Nieto, J.J., Differential equations in metric spaces: an introduction and application to fuzzy differential equations, Dyn. contin., discrete impuls. syst., 10, 991-1000, (2003) · Zbl 1057.34061
[6] Nieto, J.J.; Rodríguez-López, R., Euler polygonal method for metric dynamical systems, Inform. sci., 177, 4256-4270, (2007) · Zbl 1142.65068
[7] Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019
[8] Bede, B.; Gal, S.G., Generalizations of the differentibility of fuzzy number value functions with applications to fuzzy differential equations, Fuzzy sets and systems, 151, 581-599, (2005) · Zbl 1061.26024
[9] Diamond, P.; Kloeden, P., Metric spaces of fuzzy sets, (1994), World Scientific Singapore · Zbl 0843.54041
[10] Bede, B.; Gal, S.G., Almost periodic fuzzy-number-valued functions, Fuzzy sets and systems, 147, 385-403, (2004) · Zbl 1053.42015
[11] Bede, B.; Rudas, I.J.; Bencsik, A.L., First order linear fuzzy differential equations under generalized differentiability, Inform. sci., 177, 1648-1662, (2007) · Zbl 1119.34003
[12] Chalco-Cano, Y.; Román-Flores, H., On new solutions of fuzzy differential equations, Chaos, solitons & fractals, 38, 112-119, (2008) · Zbl 1142.34309
[13] A. Khastan, F. Bahrami, K. Ivaz, New Results on Multiple Solutions for \(N\)th-order Fuzzy Differential Equations under Generalized Differentiability, preprint · Zbl 1198.34006
[14] Nieto, J.J.; Khastan, A.; Ivaz, K., Numerical solution of fuzzy differential equations under generalized differentiability, Nonlinear anal. hybrid syst., 3, 700-707, (2009) · Zbl 1181.34005
[15] Bede, B., A note on two-point boundary value problems associated with non-linear fuzzy differential equations, Fuzzy sets and systems, 157, 986-989, (2006) · Zbl 1100.34511
[16] Lakshmikantham, V.; Murty, K.N.; Turner, J., Two-point boundary value problems associated with non-linear fuzzy differential equations, Math. inequal. appl., 4, 527-533, (2001) · Zbl 1022.34051
[17] O’Regan, D.; Lakshmikantham, V.; Nieto, J., Initial and boundary value problems for fuzzy differential equations, Nonlinear anal., 54, 405-415, (2003) · Zbl 1048.34015
[18] Negoita, C.V; Ralescu, D.A., Aplications of fuzzy sets to systems analysis, (1975), Wiley New York · Zbl 0326.94002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.