Nonresonance Problems for Differential Inclusions in Separable Banach Spaces

Let X be a real separable Banach space. The boundary value problem x′ ∈ A(t)x + F (t, x), t ∈ R+, Ux = a, (B) is studied on the infinite interval R+ = [0,∞). Here, the closed and densely defined linear operator A(t) : X ⊃ D(A)→ X, t ∈ R+, generates an evolution operator W (t, s). The function F : R+×X → 2X is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either F is bounded and W (t, s) is compact for t > s, or F is compact and W (t, s) is equicontinuous. The mapping U : Cb(R+,X)→ X is a bounded linear operator and a ∈ X is fixed. The nonresonance problem is solved by using Ma’s fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in Cb(R+,X).