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arXiv 1809.04743math.FA
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积分分式预解算子函数生成元的逆

Inverses of generators of integrated fractional resolvent operator functions

Miao Li, Javier Pastor, Sergey Piskarev

发表信息：
Fract. Calc. Appl. Anal. 21(2018), at https://www.degruyter.com/view/j/fca
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摘要：

本文研究了积分预解算子函数生成元设置中的逆生成元问题。证明了：若算子$A$是调和$β$ -次积分$α$ -预解算子函数（$(α,β)$）的生成元，且它是单射的，则逆算子$A^{-1}$对所有$γ> β+1/2,$都是调和$(α,γ)$的生成元.本文还考虑了解析预解算子函数，证明了$A^{-1}$是所有$δ<α.$的回火$(δ,0)$的生成元，并给出了$(α,β)$的最佳衰减率为$t\to \infty$.这些结果适用于分数阶Cauchy问题在分数阶导数中未解的情形。

英文摘要：

This paper is devoted to the inverse generator problem in the setting of generators of integrated resolvent operator functions. It is shown that if the operator $A$ is the generator of a tempered $β$-times integrated $α$-resolvent operator function ($(α,β)$-ROF) and it is injective, then the inverse operator $A^{-1}$ is the generator of a tempered $(α,γ)$-ROF for all $γ> β+1/2,$ by means of an explicit representation of the integrated resolvent operator function based in Bessel functions of first kind. Analytic resolvent operator functions are also considered, showing that $A^{-1}$ is in addition the generator of a tempered $(δ,0)$-ROF for all $δ<α.$ Moreover, the optimal decay rate of $(α,β)$-ROFs as $t\to \infty$ is given. These result are applied to fractional Cauchy problem unsolved in the fractional derivative.

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