Well†posedness of fractional degenerate differential equations with finite delay on vector†valued functional spaces

We study the well†posedness of the fractional degenerate differential equations with finite delay (Pα):Dα(Mu)(t)=Au(t)+Fut+f(t),(0≤t≤2π,α>0) on Lebesgue–Bochner spaces Lp(T;X), periodic Besov spaces Bp,qs(T;X) and periodic Triebel–Lizorkin spaces Fp,qs(T;X), where A and M are closed linear operators on a Banach space X satisfying D(A)⊂D(M), F is a bounded linear operator from Lp([−2π,0];X) (resp. Bp,qs([−2π,0];X) and Fp,qs([−2π,0];X)) into X, where ut is given by ut(s)=u(t+s) when s∈[−2π,0] and t∈[0,2π]. Using known operator†valued Fourier multiplier theorems, we give necessary or sufficient conditions for the well†posedness of (Pα) in the above three function spaces.