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Fuzzy differential equations. (English) Zbl 0646.34019
A differential and integral calculus for fuzzy-set-valued mappings was developed in recent papers of Dubois and Prade, and Puri and Ralescu. The purpose of this paper is to study differential equations for fuzzy-set- valued mappings of a real variable whose values are normal, convex, upper semi-continuous and compactly supported fuzzy sets in \(R^ n\). The differentiability and integrability properties of such functions are studied and an existence and uniqueness theorem for a solution to a fuzzy differential equation is given.

MSC:
34A99 General theory for ordinary differential equations
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[1] Aumann, R.J., Integrals of set-valued functions, J. math. anal. appl., 12, 1-12, (1965) · Zbl 0163.06301
[2] Berge, C., Escapes topologiques; fonctions multivoques, (1959), Dunod Paris · Zbl 0088.14703
[3] Bradley, M.; Datko, R., Some analytic and measure theoretic properties of set-valued mappings, SIAM J. control optim., 15, 625-635, (1977) · Zbl 0354.28001
[4] Castaing, C.; Valadier, M., Convex analysis and measurable multifunctions, (1977), Springer Berlin · Zbl 0346.46038
[5] Debreu, G., Integration of correspondences, (), 351-372, Part 1 · Zbl 0211.52803
[6] Dubois, D.; Prade, H., Towards fuzzy differential calculus — part 1, Fuzzy sets and systems, 8, 1-17, (1982) · Zbl 0493.28002
[7] Dubois, D.; Prade, H., Towards fuzzy differential calculus — part 2, Fuzzy sets and systems, 8, 105-116, (1982) · Zbl 0493.28003
[8] Dubois, D.; Prade, H., Towards fuzzy differential calculus — part 3, Fuzzy sets and systems, 8, 225-234, (1982)
[9] Hukuhara, M., Integration des applications measurables dont la valeur est un compact convexe, Funkcialaj. ekvacioj., 10, 205-223, (1967) · Zbl 0161.24701
[10] Kaleva, O., On the convergence of fuzzy sets, Fuzzy sets and systems, 17, 53-65, (1985) · Zbl 0584.54004
[11] McShane, E.J., Stochastic differential equations, J. multivariate anal., 5, 121-177, (1975) · Zbl 0323.60059
[12] Negoita, C.V.; Ralescu, D.A., Applications of fuzzy sets to systems analysis, (1975), Wiley New York · Zbl 0326.94002
[13] Nguyen, H.T., A note on the extension principle for fuzzy sets, J. math. anal. appl., 64, 369-380, (1978) · Zbl 0377.04004
[14] Puri, M.L.; Ralescu, D.A., Differentials for fuzzy functions, J. math. anal. appl., 91, 552-558, (1983) · Zbl 0528.54009
[15] Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004
[16] Royden, H.L., Real analysis, (1968), Macmillan London · Zbl 0197.03501
[17] Rådström, H., An embedding theorem for spaces of convex sets, (), 165-169 · Zbl 0046.33304
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