Existence of classical solutions to nonautonomous nonlocal parabolic problems

We obtain a new existence theorem for classical solutions to nonautonomous equations with nonlocal initial conditionsu′(t)=A(t)u(t)+f(t,u(t)),t∈(s,T],u(s)+g(u)=u0,in a Banach space X, where T>s⩾0, f,g are given X-valued functions, and A(t) is a sectorial operator (not necessarily densely defined) in X for each t∈[0,T]. Both Banach's contraction principle and Schauder's fixed point theorem, as well as the theory of evolution families and interpolation spaces, are employed in our approach. A concrete example is shown to illustrate the existence theorem.