Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

Abstract

We study the well-posedness of the fractional differential equations with infinite delay (P 2): \({D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}\), where A is a closed operator in a Banach space \({X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}\) and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P 2) on Lebesgue-Bochner spaces \({L^p(\mathbb{T}, X)}\) and periodic Besov spaces \({{B} _{p,q}^s(\mathbb{T}, X)}\) .