This paper is mainly focused upon the pseudo $S$-asymptotically Bloch type periodicity and its applications. Firstly, a new notion of pseudo $S$-asymptotically Bloch type periodic functions is introduced, and some fundamental properties on pseudo $S$-asymptotically Bloch type periodic functions are established. Then, the notion and properties of weighted pseudo $S$-asymptotically Bloch type periodic functions are similarly presented. Finally, the obtained results are applied to investigate the existence and uniqueness of pseudo $S$-asymptotically Bloch type periodic mild solutions for some semi-linear evolution equations in Banach spaces.

中文翻译：

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伪$ S $-渐近Bloch型周期及其在某些演化方程中的应用

本文主要关注伪S $-渐近Bloch型周期及其应用。首先，介绍了伪$ S $-渐近Bloch型周期函数的新概念，建立了伪$ S $-渐近Bloch型周期函数的一些基本性质。然后，类似地给出了加权伪$ S $-渐近Bloch型周期函数的概念和性质。最后，将所得结果用于研究Banach空间中某些半线性演化方程的伪$ S $-渐近Bloch型周期温和解的存在性和唯一性。