# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $x'(t)=f(t,x(t)),$ $x(t\sb 0)=x\sb 0$, 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 ${\bbfR}\sp n$ and assume that $f: [t\sb 0,t\sb 0+a]\times U\to U$ is continuous. We show that the Cauchy problem has a solution if and only if U is locally compact.

##### MSC:
 34A12 Initial value problems for ODE, existence, uniqueness, etc. of solutions
##### Keywords:
Cauchy problem; Banach space; fuzzy differential equations
Full Text:
##### References:
 [1] Aubin, J. P.; Cellina, A.: Differential inclusions. (1984) · Zbl 0538.34007 [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. Proc. fifth Berkeley symp. Math. stat. Probab. 2, 351-372 (1967) · 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) · 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. Proc. roy. Soc. lond. A 407, 171-182 (1986) · Zbl 0605.60038 [14] Lang, S.: Analysis II. (1969) · Zbl 0176.00504 [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