Abstract
In the present work we consider E is a Banach space, E* is its dual space and L(E) is the space of continuous linear operators from E to itself. A function x: ℝ → E is said to be a pseudo-solution of the equation
$$
\dot{x}=A(t)x+f(t,x) \quad t\in \mathbb{R},
$$
where A:ℝ → L(E) is strongly measurable and Bochner integrable function on every finite subinterval of ℝ with f:ℝ × E → E is only assumed to be weakly weakly sequentially continuous or Pettis-integrable and the linear equation
$$
\dot x(t)=A(t)x(t)
$$
has a trichotomy with constants α ≥ 1 and σ > 0, if x is absolutely continuous function and for each x* ∈ E* there exists a negligible set ℵx* such that for each t ∉ ℵx*, then we have
$$
x^*(\dot x (t))=x^*(A(t)x+f(t,x)).
$$
We give an existence theorem for bounded weak and pseudo-solutions of the nonlinear differential equations
$$
\dot{x}=A(t)x+f(t,x), \quad t\in \mathbb{R}.
$$
Let T, r, d > 0, Br = {x > E: ‖x‖ ≤ r} and CE([-d,0]) be the Banach space of continuous functions from [-d,0] into E. Finally we prove an existence result for the differential equation with delay
$$
\dot{x}=\hat{A}(t)x+f^{d}(t,\theta _{t}x)\quad\text{ if }t\in [a,b]
$$
where fd : [a,b] × CE([-d,0]) → E is weakly weakly sequentially continuous function,
$\hat A: [a, b]\to L(E)$
is strongly measurable and Bochner integrable operator on [a,b] and θtx(s) = x(t + s) for all s ∈ [-d,0].