On maximal regularity and semivariation of $\alpha$-times resolvent families

Abstract

Let $1<\alpha <2$ and $A$ be the generator of an $\alpha$-times resolvent family $\{S_{\alpha }(t){\}}_{t\ge 0}$ on a Banach space $X$. It is shown that the fractional Cauchy problem ${\mathbf{D}}_t^{\alpha }u(t)=Au(t)+f(t)$, $t\in [0,r]$; $u(0),u^{\prime }(0)\in D(A)$ has maximal regularity on $C([0,r];X)$ if and only if $S_{\alpha }(\cdot )$ is of bounded semivariation on $[0,r]$.

7 pages