In this paper, we study the differentiability of mild solutions for a class of fractional abstract Cauchy problem. We consider the fractional derivative of order $\alpha \in (1,2)$ in the sense of Caputo. In the first part, we establish the existence of classical solutions for the homogeneous Cauchy problem in terms of the α-resolvent family corresponding to the problem and, in the second part, we establish the existence of classical solutions for the inhomogeneous abstract Cauchy problem in terms of regularity properties of the α-resolvent family and of the forcing function.

中文翻译：

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分数阶抽象柯西问题解的存在性

在本文中，我们研究了一类分数次抽象柯西问题的温和解的可微性。我们考虑阶数的分数导数$\alpha \in （\mathrm{1个}，\mathrm{2个}）$在Caputo的意义上。在第一部分中，我们根据与该问题相对应的α-溶剂族建立了均匀柯西问题的经典解的存在，在第二部分中，我们建立了非均匀抽象柯西问题的经典解的存在。α-溶剂家族的正则性质和强迫函数。