We prove an existence theorem for the equation x = f(t, xt), x(Θ) = φ(Θ), where xt(Θ) = x(t + Θ), for −r ≤ Θ < 0, t ∈ Ia, Ia = [0, a], a ∈ R+ in a Banach space, using the Henstock-KurzweilPettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose tha...

The constructive definition usually begins with a function f, then by the process of using Riemann sums and limits, we arrive the definition of the integral of f, ∫ b a f. On the other hand, a descriptive definition starts with a primitive F satisfying certain condition(s) such as F ′ = f and F is absolutely continuous if f is Lebesgue integrable, and F is generalized absolutely continuous if f...