We here study functional-differential equations which can be written in the form

~(z)= f (x',~),(1.1) where x (z) is, for each z, in a Banach space V,~(~) stands for dx (z)/dz, x', called the" history of x up to time z", is the function on R+--[0, oo) defined by x'(s)= x (zs), s~ IR+,(1.2) and f is a preassigned function mapping~ x [0, T] into V with T> 0 and~ a subset of a Banach function space 11, with norm [[. 1 [, formed from V-valued functions on IR+. We assume that f (., z) is locally Lipschitz continuous on with Lipschitz constants L that are independent of z for z in [0, T]. If b in~ and t in (0, T] are specified, by a solution up to t of (1.1) with initialhistory I1 we mean a function x on [0, t] which can be extended to (-oo, t] in such a way that (a) x~ the" history of x up to time zero", belongs to the equivalence class I) in~= 11 (ie equals I? after calling the same functions# t,'/'2 with [[# 1-# 211= 0), and (b) the equation (1.1) holds for each z in [0, t]. When a function x:[0, t]--+ V and the" initial history" b are specified, u,, the equivalence class in 1I to which x" belongs, is uniquely determined [by the formula (3.2) given in Section 3] for each, in [0, t]; hence the question of whether x:[0, t]-+ V is a solution up to t of (1.1) with initial history t) reduces to the