ON BOUNDED WEAK AND PSEUDO-SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS HAVING TRICHOTOMY WITH AND WITHOUT DELAY IN BANACH SPACES

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 [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] We give an existence theorem for bounded weak and pseudo-solutions of the nonlinear differential equations [Formula: see text] 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 [Formula: see text] where fd : [a,b] × CE([-d,0]) → E is weakly weakly sequentially continuous function, [Formula: see text] is strongly measurable and Bochner integrable operator on [a,b] and θtx(s) = x(t + s) for all s ∈ [-d,0].

Moments of vector-valued functions and measures

There are investigated conditions under which the elements of a normed vector space are the moments of a vector-valued measure, and of a Bochner integrable function, respectively, both with values in a Banach space.

Ultrabornological Bochner integrable function spaces

If (Ω, Σ, μ) is a finite measure space and X is a normed space such that X* has the Radon-Nikodym property with respect to μ, we show first that each space Lp(μ, x), 1 < p < ∞, is ultrabornological whenever μ is atomless. When μ is arbitrary, we prove later on that the space Lp(μ, X) is ultrabornological if X* has the Radon-Nikodym property with respect to μ and X is itself an ultrabornological space.