Abstract
We give existence theorems for weak and strong solutions with trichotomy of the nonlinear differential equation x(t) = L(t)x(t) + f(t,x(t)), t is an element of R where {a"'(t): t a R} is a family of linear operators from a Banach space E into itself and f: R x E -> E. By L(E) we denote the space of linear operators from E into itself. Furthermore, for a < b and d > 0, we let C([-d, 0],E) be the Banach space of continuous functions from [-d, 0] into E and f (d) : [a, b] x C([-d, 0],E) -> E. Let L: [a,b] L(E) be a strongly measurable and Bochner integrable operator on [a, b] and for t a [a, b] define tau (t) x(s) = x(t + s) for each s a [-d, 0]. We prove that, under certain conditions, the differential equation with delay x(t) = (L) over cap (t)x(t) + f(d)(t,T(t)x), if t is an element of[a,b], has at least one weak solution and, under suitable assumptions, the differential equation (Q) has a solution. Next, under a generalization of the compactness assumptions, we show that the problem (Q) has a solution too.