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

AbstractLet 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]). 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)– 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)