Abstract

In this paper, we introduce the concept of square-mean piecewise almost automorphic function. By using the theory of semigroups of operators and the contraction mapping principle, the existence of square-mean piecewise almost automorphic mild solutions for linear and nonlinear impulsive stochastic evolution equations is investigated. In addition, the exponential stability of square-mean piecewise almost automorphic mild solutions for nonlinear impulsive stochastic evolution equations is obtained by the generalized Gronwall–Bellman inequality. Finally, we provide an illustrative example to justify the results.

中文翻译：

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一类脉冲随机演化方程的平方均值分段几乎自同构温和解

摘要

在本文中，我们介绍了均方分段几乎自构函数的概念。利用算符半群理论和压缩映射原理，研究了线性和非线性脉冲随机演化方程的均方分段近似自同构温和解的存在性。此外，通过广义的Gronwall-Bellman不等式，获得了非线性脉冲随机演化方程的平方均值分段几乎自形温和解的指数稳定性。最后，我们提供一个说明性的例子来证明结果的合理性。