AN INVARIANCE PRINCIPLE IN THE THEORY OF STABILITY

functions cp on [-r,O] to Rn with l{cpll =
max (Icp(~)l ; -r 5 T s 01. Convergence in c is uniform conver-
gence on [-r,O]. A function x defined on [-r,m) to Rn to
said to be a solution of (9) satisfying - the initial condition cp
at time t = 0 if there is an a > 0
such that %(t) = f( "t )
for all t in [O,a) and x = cp . Remember x = cp means
0 0
X(T) = cp(z), -r 5 T 5 0. At t = 0, k is the right hand deriv-
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ative.
The existence uniqueness theorems are quite similar to
those for ordinary differential equations. If f is locally
Lipschitzian on C, then for each cp in C there is one and only
one solution of (9) and the solution depends continuously on cp .
The solution can also be extended in C
for t > 0 as long as it
remains bounded.
As in Section 2, we will always speak as though
solutions are defined on [-r,m). The space C is now the state
space of (9) and through each point cp of C there is the motion
or flow x starting at cp defined by the solution x(t) of (9)
t
satisfying at time t = 0 the initial condition cp; x t' 0 5 t