The classical uniform asymptotic stability result for a system of functional differential equations
(
1
)
x
′
=
F
(
t
,
x
t
)
\begin{equation}\tag {$1$} x’ = F(t,{x_t})\end{equation}
calls for a Liapunov functional
V
(
t
,
ϕ
)
V(t,\phi )
satisfying
W
(
|
ϕ
(
0
)
|
)
⩽
V
(
t
,
ϕ
)
⩽
W
1
(
|
ϕ
(
0
)
|
)
+
W
2
(
|
|
|
ϕ
|
|
|
)
,
V
(
1
)
′
⩽
−
W
3
(
|
ϕ
(
0
)
|
)
W(|\phi (0)|) \leqslant V(t,\phi ) \leqslant {W_1}(|\phi (0)|) + {W_2}(|||\phi |||),{V’_{(1)}} \leqslant - {W_3}(|\phi (0)|)
, and
|
f
(
t
,
x
t
)
|
|f(t,{x_t})|
bounded for
|
|
|
x
t
|
|
|
|||{x_t}|||
bounded. We show that it is not necessary to require
|
f
(
t
,
x
t
)
|
|f(t,{x_t})|
bounded. Here,
|
|
|
⋅
|
|
|
||| \cdot |||
is the
L
2
{L^2}
-norm.