Abstract

In this paper, we study the local and global existence, and uniqueness of mild solution to initial value problems for fractional semilinear evolution equations with compact and noncompact semigroup in Banach spaces. In particular, we derive the form of fundamental solution in terms of semigroup induced by resolvent and ψ-function from Caputo fractional derivatives. These results generalize previous work where the classical Caputo fractional derivative is considered. Moreover, we prove the Mittag-Leffler–Ulam–Hyers stability result. Finally, we give examples of time-fractional heat equation to illustrate the result.

中文翻译：

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半线性ψ-Caputo分数阶演化方程的温和解的存在唯一性和稳定性

摘要

在本文中，我们研究了Banach空间中具有紧型和非紧型半群的分数半线性演化方程初值问题的温和解的局部性和全局性以及唯一性。特别是，我们从Caputo分数导数中得出了由可分解物和ψ函数引起的半群的基本解的形式。这些结果概括了以前的工作，其中考虑了经典的Caputo分数导数。此外，我们证明了Mittag-Leffler-Ulam-Hyers的稳定性结果。最后，我们以时间分数热方程为例来说明结果。