On existence solutions and solution sets of differential equations and differential inclusions with delay in Banach spaces

Let CE([−d, 0]) (resp. CB(0, T)([−d, 0]) be the Banach space of continuous functions from [−d, 0] into a Banach space E (resp. into B(0, T)), where B(0, T) = {x ∈ E : ∥x∥ ⩽ T  } and let C∈CE([-d,0])C∈CE([-d,0]). In this paper we prove an existence theorem for the differential equation with delay(P)x˙(t)=fd(t,θ¯tx),t∈[0,T],x=C,on[-d,0],where θ¯t:CB(0,T)([-d,t])→CE([-d,0]) is such that θ¯tx(s)=x(t+s) for all s ∈ [−d, 0] and for all x ∈ CB(0,T)([−d, t]) while fd is a function from [0, T] × CB(0, T)([−d, 0]) into E  . By using (RE,N,p)(RE,N,p)– measure of noncompactness and under a generalization of the compactness assumptions, we prove an existence theorem and give some topological properties of solution sets of the problem(Q)x˙(t)∈A(t)x(t)+Fd(t,θtx),t∈[0,T],x=C,on[-d,0],where Fd : [0, T] × CE([−d, 0]) → Pfc(E), Pfc(E) is the set of all nonempty closed convex subsets of E while θt : CE([−d, t]) → CE([−d, 0]) defined by θtx(s) = x(t + s) ∀ x ∈ CE([−d, t]), ∀s ∈ [−d, 0] and {A(t) : 0 ⩽ t ⩽ b  } is a family of densely defined closed linear operators generating a continuous evolution operator S(t,s)S(t,s).