Introducing set differential equations
by
V. Lakshmikantham
Florida Institute of Technology

Let K_c(Rⁿ) be the collection of all nonempty, compact, convex subsets of Rⁿ. Define the Hausdorff metric D[A, B]=max[sup_{x in B}d(x, A), sup_{y in A}d(y, B)]. Then (K_c(Rⁿ), D] is a complete metric space. Consider the set differential equation

DHU=F(t, U), U(t0)=U0 in Kc(Rn),

(1)

where D_hU is the Hukuhara derivative of the function J:R₊ --> K_c(Rⁿ). The investigation of the basic and qualitative results of solutions of 1 as an independent subject area has several advantages. If U(t) is a single-valued mapping, it is easy to see that 1 reduces to ordinary differential systems. Here we have only semilinear metric space compared to normed linear space one employs in the study of ODE. Moreover, 1, which can be generated by differential inclusions which do not possess convex values, can be utilized as a tool to study differential inclusions. Furthermore, one can profitably use 1, for investigating fuzzy differential equations, the original formulation of which suffers from great disadvantages. In this talk, we shall explore these ideas.

Date received: October 25, 2002