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A Generalization of Bochner’s Theorem and Its Applications in the Study of Impulsive Differential Equations

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Abstract

In this paper, we reveal the deep relation between Stepanov and piecewise continuous almost periodic functions and apply it to the study of almost periodic impulsive differential equations. Under the quasi-uniform continuity condition, the equivalence of Stepanov and piecewise continuous almost periodic functions is firstly established, which provides both a generalization of Bochner’s theorem and a powerful tool to investigate piecewise continuous almost periodic functions. As applications, the module containment for piecewise continuous almost periodic solutions to linear impulsive differential equations is studied.

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Acknowledgements

The authors thank the anonymous referees for their valuable comments and suggestions. The author R. Yuan is supported by The National Nature Science Foundation of China (Grant No. 11771044).

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  1. Chern Institute of Mathematics, Nankai University, Tianjin, People’s Republic of China

    Liangping Qi

  2. School of Mathematical Sciences, Beijing Normal University, Beijing, People’s Republic of China

    Rong Yuan

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  1. Liangping Qi
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Qi, L., Yuan, R. A Generalization of Bochner’s Theorem and Its Applications in the Study of Impulsive Differential Equations. J Dyn Diff Equat 31, 1955–1985 (2019). https://doi.org/10.1007/s10884-018-9641-7

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  • DOI: https://doi.org/10.1007/s10884-018-9641-7

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