On maximal regularity and semivariation of $\alpha$-times resolvent families
On maximal regularity and semivariation of $\alpha$-times resolvent families
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发表时间：
2010-07
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通讯作者：
Fubo Li;Miao Li
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作者：
Fubo Li;Miao Li
设$1< \alpha <2$和$A$是Banach空间$X$上的$\alpha$次预解族$\{S_\alpha（t）\}_{t \ge 0}$的生成元。证明了分数阶Cauchy问题${\bfD} t^\alpha u（t）= Au（t）+f（t）$，$t \in [0，r]$; $u（0），u '（0）\in D（A）$在$C（[0，r];X）$上具有极大正则性当且仅当$S\alpha（\cdot）$在$[0，r]$上具有有界半变差.
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 $[0,r]$.