On maximal regularity and semivariation of $α$-times resolvent families

Details On maximal regularity and semivariation of $α$-times resolvent families On maximal regularity and semivariation of $α$-times resolvent families

2010-07-26

arxiv.org/abs/1007.4455v1

Mathematics

Functional Analysis

7 pages

Scientific paper

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 ${\bf D}_t^\alpha u(t) = Au(t)+f(t)$, $t \in [0,r]$;
$u(0), u'(0) \in D(A)$ has maximal regularity on $C([0,r];X)$ if and only if
$S_\alpha(\cdot)$ is of bounded semivariation on $[0,r]$.

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