First Page

Differential and Integral Equations, Volume 8, Number 5, May 1995, pp. 1157 – 1166.
INTEGRAL SOLUTIONS OF LOCALLY LIPSCHITZ CONTINUOUS
FUNCTIONAL DIFFERENTIAL EQUATIONS
Janet Dyson
Mansfield College, Oxford, England
Rosanna Villella-Bressan
Dipartimento di Matematica Pura e Applicata, Università di Padova, Padova, Italy
(Submitted by: Glenn Webb)
Abstract. A local existence and uniqueness result for the functional differential equation in a Banach
space X
(FDE)
x (t) = f (t)x(t) + g(t)xt ,
x0 = , x(0) = h, { , h} ⇤ L1 ( R, 0; X) ⇥ X
is obtained, for the case where the operators f (t) satisfy only a local dissipativity condition and the
operators g(t) are only locally Lipschitz continuous. This is done by relating (FDE) to the evolution
equation in L1 ( R, 0; X) ⇥ X
(E)
u (t) = A(t)u(t),
u(0) = { , h},
where
D(A(t)) = {{ , h} ⇤ L1 ( R, 0; X) ⇥ X;
⇤ W 1,1 ( R, 0; X), h ⇤ D(f (t)), (0) = h}
A(t){ , h} = { , f (t)h + g(t) }.
It is shown that if u(t) is the limit solution of (E), then u(t) = {xt , x(t)}, where x(t) is the integral
solution of (FDE).
0. Introduction. In this paper we continue to study the relationship between the
functional differential equation in a Banach space X
x⌅ (t) = f (t)x(t) + g(t)xt ,
x0 = ↵,
x(0) = h,
0 ⇤ t ⇤ T,
{↵, h}
L1 ( R, 0; X) ⇥ X,
(FDE)
where R > 0 is the delay and xt L1 = L1 ( R, 0; X) is defined pointwise by xt (⇧) = x(t+⇧),
and the evolution equation in L1 ( R, 0; X) ⇥ X
u⌅ (t) = A(t)u(t),
0 ⇤ t ⇤ T,
u(0) = {↵, h},
where
D(A(t)) = {{↵, h}
L1 ⇥ X; ↵
W 1,1 ( R, 0; X), h
A(t){↵, h} = {↵ , f (t)h + g(t)↵},
⌅
Received February 1993.
AMS Subject Classifications: 34K30.
1157
D(f (t)), ↵(0) = h}
(E)