zbMATH — the first resource for mathematics

The Cauchy problem for fuzzy differential equations. (English) Zbl 0696.34005
Summary: The classical Peano theorem states that in finite dimensional spaces the Cauchy problem \(x'(t)=f(t,x(t)),\) \(x(t_ 0)=x_ 0\), has a solution provided f is continuous. In addition, Godunov has shown that each Banach space in which the Peano theorem holds true is finite dimensional. For differential inclusions, the existence of a solution to the Cauchy problem is also guaranteed under various assumptions on the right-hand side. In this paper, we study the Cauchy problem for fuzzy differential equations. To be more specific, let U be a subspace of normal, convex, upper semicontinuous, compactly supported fuzzy sets defined in \({\mathbb{R}}^ n\) and assume that \(f: [t_ 0,t_ 0+a]\times U\to U\) is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.

34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
Full Text: DOI
[1] Aubin, J.P.; Cellina, A., Differential inclusions, (1984), Springer-Verlag Berlin
[2] De Blasi, F.R.; Lasota, A., Daniell’s method in the theory of the Aumann-Hukuhara integral of set-valued functions, Atti acad. naz. lincei rendiconti ser. 8, 45, 252-256, (1968) · Zbl 0174.09202
[3] De Blasi, F.R.; Pianigiani, G., Differential inclusions in Banach spaces, J. differential equations, 66, 208-229, (1987) · Zbl 0609.34013
[4] Debreu, G., Integration of correspondences, (), 351-372, Part 1 · Zbl 0211.52803
[5] Diamond, P.; Kloeden, P., Characterization of compact subsets of fuzzy sets, Fuzzy sets and systems, 29, 341-348, (1989) · Zbl 0661.54011
[6] Dubois, D.; Prade, H., Fuzzy sets and systems: theory and applications, (1980), Academic Press New York · Zbl 0444.94049
[7] Feron, R., Ensembles aleatoires dont la fonction d’appartenance prend ses valeurs dans un treillis distributif ferme, Publ. econometriques, 12, 81-118, (1979) · Zbl 0406.46010
[8] Godunov, A.N., Peano’s theorem in Banach spaces, Funct. anal. appl., 9, 53-55, (1975) · Zbl 0314.34059
[9] Goetschel, R.; Voxman, W., Topological properties of fuzzy numbers, Fuzzy sets and systems, 10, 87-99, (1983) · Zbl 0521.54001
[10] Himmelberg, C.J.; Van Vleck, F.S., Existence of solutions for generalized differential equations with unbounded right-hand side, J. differential equations, 61, 295-320, (1986) · Zbl 0582.34002
[11] Hukuhara, M., Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj. ekvacioj., 10, 205-223, (1967) · Zbl 0161.24701
[12] Kaleva, O., Fuzzy differential equations, Fuzzy sets and systems, 24, 301-317, (1987) · Zbl 0646.34019
[13] Klement, E.P.; Puri, M.L.; Ralescu, D.A., Limit theorems for fuzzy random variables, (), 171-182 · Zbl 0605.60038
[14] Lang, S., Analysis II, (1969), Addison-Wesley Reading, MA
[15] Nguyen, H.T., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004
[16] Puri, M.L.; Ralescu, D.A., Differentials for fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009
[17] Puri, M.L.; Ralescu, D.A., The concept of normality for fuzzy random variables, Ann. probab., 13, 1373-1379, (1985) · Zbl 0583.60011
[18] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[19] Tolstonogov, A.A., Differential inclusions in Banach space with nonconvex right-hand side, Siberian J. math., 22, 625-637, (1981) · Zbl 0529.34058
[20] Vitale, R.A., Lp metrics for compact, convex sets, J. approx. theory, 45, 280-287, (1985) · Zbl 0595.52005
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.