MAXIMAL REGULARITY FOR INTEGRO-DIFFERENTIAL EQUATION ON PERIODIC TRIEBEL-LIZORKIN SPACES

Abstract

We study maximal regularity on Triebel-Lizorkin spaces $\mathrm{F} _{p,q}^s(\mathbb T, X)$ for the integro-differential equation with infinite delay: ($P_2$): $u'(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), \ (0\leq t \leq2\pi$) with the periodic condition $u(0)=u(2\pi)$, where $X$ is a Banach space, $a\in {\mathrm L}^1(\mathbb R_+)$ and $f$ is an $X$-valued function. Under a suitable assumption (H3) on the Laplace transform of $a$, we give a necessary and sufficient condition for ($P_2$) to have the maximal regularity property on $\mathrm{F} _{p,q}^s(\mathbb T, X)$.