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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)