Journal of Fourier Analysis and Applications

, Volume 25, Issue 1, pp 32–50 | Cite as

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

  • Shangquan Bu
  • Gang CaiEmail author


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 AM 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. \(B_{p,q}^s([-2\pi ,0];X)\)).


Degenerate differential equations Delay equations Well-posedness Lebesgue–Bochner spaces Besov spaces Fourier multipliers 

Mathematics Subject Classification

34G10 34K30 43A15 47D06 



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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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Department of Mathematical SciencesTsinghua UniversityBeijingChina
  2. 2.School of Mathematical SciencesChongqing Normal UniversityChongqingChina

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