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Periodic Solutions of Second Order Degenerate Differential Equations with Finite Delay in Banach Spaces

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Abstract

The purpose of this paper is to give necessary and sufficient conditions for the \(L^p\)-well-posedness (resp. \(B_{p,q}^s\)-well-posedness) for the second order degenerate differential equation with finite delay: \((Mu)''(t)=Au(t)+Gu'_t+Fu_t+f(t),(t\in [0,2\pi ])\) with periodic boundary conditions \(Mu(0)=Mu(2\pi )\), \((Mu)'(0)=(Mu)'(2\pi )\), where A, M are closed linear operators in a Banach space X satisfying \(D(A)\subset D(M)\), F and G are delay operators on \(L^p([-2\pi ,0];X)\) (resp. \(B_{p,q}^s([-2\pi ,0];X)\)).

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Acknowledgements

The authors are most grateful to the anonymous referees for carefully reading the manuscript and providing valuable comments and suggestions. This work was supported by the NSF of China (Grant Nos. 11401063, 11571194, 11731010, 11771063), the Natural Science Foundation of Chongqing (Grant No. cstc2017jcyjAX0006), Science and Technology Project of Chongqing Education Committee (Grant No. KJ KJ1703041), the University Young Core Teacher Foundation of Chongqing (Grant No. 020603011714), Talent Project of Chongqing Normal University (Grant No. 02030307-00024).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Shangquan Bu

  2. School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China

    Gang Cai

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  1. Shangquan Bu
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  2. Gang Cai
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Correspondence to Gang Cai.

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Communicated by Winfried Sickel.

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Bu, S., Cai, G. Periodic Solutions of Second Order Degenerate Differential Equations with Finite Delay in Banach Spaces. J Fourier Anal Appl 25, 32–50 (2019). https://doi.org/10.1007/s00041-017-9560-8

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  • DOI: https://doi.org/10.1007/s00041-017-9560-8

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