In this paper, we consider a Cauchy problem for fractional evolution equations with Hilfer’s fractional derivative on semi-infinite interval. An elementary fact shows that semi-infinite interval is not compact, the classical Ascoli-Arzelà theorem is not valid. In order to establish the global existence criteria, we first generalize Ascoli-Arzelà theorem into the semi-infinite interval. Next, we introduce a new concept of mild solutions based on cosine/sine family and probability density function and obtain several existence results of mild solutions on semi-infinite interval.

中文翻译：

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半无限区间上具有 Hilfer 分数阶导数的分数阶演化方程的 Cauchy 问题

在本文中，我们考虑了半无限区间上具有 Hilfer 分数阶导数的分数阶演化方程的 Cauchy 问题。一个基本事实表明，半无限区间不是紧致的，经典的 Ascoli-Arzelà 定理是无效的。为了建立全局存在准则，我们首先将 Ascoli-Arzelà 定理推广到半无限区间。接下来，我们引入了一种基于余弦/正弦族和概率密度函数的温和解的新概念，得到了半无限区间上温和解的若干存在性结果。