Functional-differential equations of the form
\[
u
˙
(
t
)
=
−
∫
0
t
A
(
t
−
τ
)
g
(
u
(
τ
)
)
d
τ
+
f
(
t
,
u
(
t
)
)
\dot u(t) = - \int _0^t {A(t - \tau )g(u(\tau ))d\tau + f(t,u(t))}
\]
are considered. Here
u
(
t
)
u(t)
is to be an element of a Hilbert space
H
,
A
(
t
)
\mathcal {H},A(t)
a family of bounded symmetric operators on
H
\mathcal {H}
and g an operator with domain in
H
\mathcal {H}
. g may be unbounded. A is called strongly positive if there exists a semigroup exp St, where S is symmetric and
(
S
ξ
,
ξ
)
≦
−
m
‖
ξ
‖
2
,
m
>
0
(S\xi ,\xi ) \leqq - m{\left \| \xi \right \|^2},m > 0
, such that
A
∗
=
A
−
exp
{A^ \ast } = A - \exp
St is positive, that is,
\[
∫
0
T
(
v
(
t
)
,
∫
0
t
A
∗
(
t
−
τ
)
v
(
τ
)
)
d
τ
≧
0
,
\int \nolimits _0^T \left ( {v(t),\int _0^t {{A^\ast }(t - \tau )v(\tau )} } \right )d\tau \geqq 0,
\]
for all smooth
v
(
t
)
v(t)
. It is shown that if A is strongly positive, and g and f are suitably restricted, then any solution which is weakly bounded and uniformly continuous must tend weakly to zero. Examples are given of both ordinary and partial differential-functional equations.