Well‐posedness of degenerate fractional differential equations with finite delay in complex Banach spaces.

We study the well‐posedness of the degenerate fractional differential equations with finite delay: Dα(Mu)(t)+cDβ(Mu)(t)$D^\alpha (Mu)(t) + cD^\beta (Mu)(t)$=Au(t)+Fut+f(t),(0≤t≤2π)$= Au(t) + Fu_t + f(t),(0\le t\le 2\pi)$ on Lebesgue–Bochner spaces Lp(T;X)$L^p(\mathbb {T}; X)$ and periodic Besov spaces Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, where A$A$ and M$M$ are closed linear operators in a complex Banach space X$X$ satisfying D(A)⊂D(M)$D(A)\subset D(M)$, c∈C$c\in \mathbb {C}$ and 0<β<α$0 < \beta < \alpha$ are fixed, ut(s)=u(t+s)$u_t(s) = u(t+s)$ when 0≤t≤2π,−2π≤s≤0$0\le t\le 2\pi , -2\pi \le s\le 0$, and the delay operator F$F$ is a bounded linear operator from Lp([−2π,0];X)$L^p([-2\pi ,0]; X)$ (resp. Bp,qs([−2π,0];X)$B_{p,q}^s([-2\pi ,0]; X)$) into X$X$. Using known operator‐valued Fourier multiplier theorems on Lp(T;X)$L^p(\mathbb {T}; X)$ and Bp,qs(T;X)$B_{p,q}^s(\mathbb {T}; X)$, we completely characterize the Lp$L^p$‐well‐posedness and the Bp,qs$B_{p,q}^s$‐well‐posedness of above equations. We also give concrete examples that our abstract results may be applied. [ABSTRACT FROM AUTHOR]