On bounded solutions of differential inclusions in Banach spaces

Abstract

Let X be a separable Banach space and t ϵ R +. We prove the existence and an asymptotic property of bounded solutions of the differential inclusion x ̇ (t) ϵ A(t)x(t) + F(t, x(t)) . The orientor field F(·,·) satisfies Carathéodory-type conditions and a regularity condition in terms of the Hausdorff measure of noncompactness β(·). In the process of proving our main theorem we also obtain some new results about measurable multifunctions of independent interest.