Abstract:
zz ifferen 6.1 Introduction Recall that the theory of fuzzy differential equations (FDEs) considered so far utilizes the Hukuhara derivative (H-derivative) for the formulation. We have investigated, in the previous chapters, several basic results of fuzzy differential equations via the comparison principle in the metric space (E n , d) with no complete linear structure. This approach for fuzzy differential equations which employs the H-derivative suffers from a disadvantage because the solution z(t) of an FDE has the property that diarn[z(t)]D is nonde-creasing in time, that is, the solution is irreversible in probabilistic terms. Consequently, it has been recently realized that this formulation of FDEs cannot really reflect any rich behavior of solutions of ODES such as stability, periodicity and bifurcation, and therefore is not well-suited for modeling. Alternative approaches have recently been introduced by Buckley and Feuring [8], Vorobiev and Seikkala [112], and Hullermeier [40]. A different and interesting framework suggested by Hullerrneier is more general than the others. It is based on a family of differential inclusions at each p-level, 0 < p < 1, namely where [G(., .)I0: R x Rn-t ICF, the space of nonempty compact convex subsets of Rn. The idea is that the set of all such solutions Sp(xo, T) would be the @-level of a fuzzy set S(zo,T), in the sense that all attainable sets A(xo, t) , 0 < t 5 T, are levels of the fuzzy set Ap(xo, t) on Rn. This framework captures both vagueness (uncertainty) and the rich properties of differential inclusions in one and the same technique. For example, with this
