Well-posedness of fractional degenerate differential equations with infinite delay in vector-valued functional spaces

Abstract

We study the well-posedness of degenerate fractional differential equations with infinite delay $(P_\alpha ): D^\alpha (Mu)(t) =Au(t)+\int _{-\infty }^t a(t-s)Au(s)\,ds+f(t)$, $0\leq t\leq 2\pi $, in Lebesgue-Bochner spaces $L^p(\mathbb {T}; X)$ and Besov spaces $B_{p,q}^s(\mathbb {T}; X)$, where $A$ and $M$ are closed linear operators on a Banach space~$X$ satisfying $D(A)\subset D(M)$, $\alpha >0$ and $a\in L^1(\mathbb {R}_+)$ are fixed. Using well known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of $(P_\alpha )$ in the above vector-valued function spaces on $\mathbb {T}$.