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Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces

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Abstract

In this paper, we study the well-posedness of the third-order differential equation with finite delay (P3): αu‴ (t) + u″(t) = Au(t) + Bu′ (t) + Fut + f(t)(t ∈ \(\mathbb{T}\):= [0, 2π]) with periodic boundary conditions u(0) = u(2π), u′(0) = u′(2π), u″ (0) = u″(2π), in periodic Lebesgue–Bochner spaces Lp(\(\mathbb{T}\);X) and periodic Besov spaces Bp,qs(\(\mathbb{T}\);X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠{0}, α ≠ 0 is a fixed constant and F is a bounded linear operator from Lp([−2π, 0];X) (resp. Bp,qs([−2π, 0];X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [−2π, 0]. Necessary and sufficient conditions for the Lp-well-posedness (resp. Lp-well-posedness) of (P3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.

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Authors and Affiliations

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

    Shang Quan Bu

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

    Gang Cai

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  1. Shang Quan Bu
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  2. Gang Cai
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Correspondence to Gang Cai.

Additional information

Supported by the NSF of China (Grant Nos. 11571194, 11731010 and 11771063), the Natural Science Foundation of Chongqing (Grant No. cstc2017jcyjAX0006), Science and Technology Project of Chongqing Education Committee (Grant No. 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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Bu, S.Q., Cai, G. Periodic Solutions of Third-order Differential Equations with Finite Delay in Vector-valued Functional Spaces. Acta. Math. Sin.-English Ser. 35, 105–122 (2019). https://doi.org/10.1007/s10114-018-8001-1

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  • DOI: https://doi.org/10.1007/s10114-018-8001-1

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