In this paper, we mainly study the existence, uniqueness, and conditional stability of bounded and periodic solutions for a class of noninstantaneous impulsive linear and semilinear equations with evolution family and exponential dichotomy. We utilize the weak ${}^{*}$ convergence analysis in the conjugate space and the Banach‐Alaoglu theorem to derive the existence result, and then we use the principle of compressed image to prove the uniqueness. In addition, we study the conditional stability of periodic solution with the help of the Grownwall‐Coppel inequality. Finally, we present an example of a noninstantaneous impulsive partial differential equation, which is transferred into an abstract impulsive evolution equation.

中文翻译：

---

非瞬时脉冲演化方程的有界性，周期性和条件稳定性

本文主要研究一类具有演化族和指数二分法的非瞬时脉冲线性和半线性方程的有界和周期解的存在性，唯一性和条件稳定性。我们利用弱者${}^{*}$在共轭空间中进行收敛性分析，并利用Banach-Alaoglu定理推导存在性结果，然后利用压缩图像原理证明其唯一性。此外，借助Grownwall-Coppel不等式，我们研究了周期解的条件稳定性。最后，我们给出一个非瞬时脉冲偏微分方程的例子，该方程被转换成抽象的脉冲演化方程。