Abstract This paper is related to the function space formed by weighted Sp-pseudo S-asymptotic periodicity and their applications. Initially, the translation invariance and completeness of the function space are investigated. Additionally, the composition theorem and convolution operator generated by Lebesgue integrable functions are presented. Finally, existence and uniqueness of solutions with weighted Sp-pseudo S-asymptotic periodicity for two classes of Volterra equations are proved by using the results obtained above, and some concrete examples are given. The methods mainly include Minkowski’s inequality, convolution inequality, contraction mapping principle, and especially the generalized Minkowski’s inequality. Our results extend some known results on asymptotic periodicity.

中文翻译：

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加权 S-伪 S-渐近周期性及其在 Volterra 积分方程中的应用

摘要 本文研究了加权Sp-伪S-渐近周期形成的函数空间及其应用。首先，研究了函数空间的平移不变性和完整性。此外，还介绍了由 Lebesgue 可积函数生成的合成定理和卷积算子。最后，利用上述结果证明了两类Volterra方程的加权Sp-伪S-渐近周期解的存在唯一性，并给出了一些具体例子。方法主要有闵可夫斯基不等式、卷积不等式、收缩映射原理，特别是广义的闵可夫斯基不等式。我们的结果扩展了渐近周期性的一些已知结果。