Well‐posedness of fractional integro‐differential equations in vector‐valued functional spaces.

We study the well‐posedness of the fractional differential equations with infinite delay (Pα):Dαu(t)=Au(t)+∫−∞ta(t−s)Au(s)ds+∫−∞tb(t−s)Bu(s)ds+f(t),(0≤t≤2π),on Lebesgue–Bochner spaces Lp(T;X) and Besov spaces Bp,qs(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A)∩D(B)≠{0},  α>0 and a,b∈L1(R+). Under suitable assumptions on the kernels a and b, we completely characterize the well‐posedness of (Pα) in the above vector‐valued function spaces on T by using known operator‐valued Fourier multiplier theorems. We also give concrete examples where our abstract results may be applied. [ABSTRACT FROM AUTHOR]