... Qn4,Qn1,Qn2are continuous on PC0,K,X.Next, for the compactness of Qn4we refer to the proof of 4, Theorem 3.1. For Qn1and any bounded subset W of PC0,K,X, we haveQn1ut Ttu0− ... can be used to prove the compactness of D|h,tiin PCh, ti,X for i 3, ,p,p 1, where tp1 K. This completes the proof.Proof of Theorem 3.1. For un∈ D, n ≥ 1, letunt⎧⎨⎩unt,t∈δ, ... properly cited. Of concern is the existence of solutions to nonlocal impulsive Cauchy problems for evolutionequations. Combining the techniques of operator semigroups, approximate solutions, noncompactmeasures...