Advertisement

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
Article
  • 92 Downloads

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

Keywords

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

Mathematics Subject Classification

34G10 34K30 43A15 47D06 

Notes

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).

References

  1. 1.
    Amann, H.: Operator-valued Fourier multipliers, vector-valued Besov spaces, and applications. Math. Nachr. 186, 5–56 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arendt, W., Bu, S.: The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240, 311–343 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Arendt, W., Bu, S.: Operator-valued Fourier multipliers on periodic Besov spaces and applications. Proc. Edinb. Math. Soc. 47, 15–33 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arendt, W., Batty, C., Hieber, M., Neubrander, F.: Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser, Basel (2001)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bourgain, J.: A Hausdorff-Young inequality for \(B\)-convex Banach spaces. Pac. J. Math. 101, 255–262 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourgain, J.: Vector-valued singular integrals and the \(H^{1}\)-BMO duality. In: Probability Theory and Harmonic Analysis (Cleveland, OH, 1983). Monographs Festbooks Pure Applied Mathematics, vol. 98, pp. 1–19. Dekker, New York (1988)Google Scholar
  7. 7.
    Bu, S.: Well-posedness of second order degenerate differential equations in vector-valued function spaces. Studia Math. 214(1), 1–16 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bu, S., Fang, Y.: Periodic solutions of delay equations in Besov spaces and Triebel-Lizorkin spaces. Taiwan. J. Math. 13(3), 1063–1076 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bu, S., Fang, Y.: Maximal regularity of second order delay equations in Banach spaces. Sci. China Math. 53(1), 51–62 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Favini, A., Yagi, A.: Degenerate Differential Equations in Banach Spaces. Pure and Applied Mathematics, vol. 215. Dekker, New York (1999)zbMATHGoogle Scholar
  11. 11.
    Lizama, C.: Fourier multipliers and periodic solutions of delay equations in Banach spaces. J. Math. Anal. Appl. 324, 921–933 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Lizama, C., Ponce, R.: Periodic solutions of degenerate differential equations in vector-valued function spaces. Studia Math. 202(1), 49–63 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lizama, C., Ponce, R.: Maximal regularity for degenerate differential equations with infinite delay in periodic vector-valued function spaces. Proc. Edinb. Math. Soc. 56(3), 853–871 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Pisier, G.: Sur les espaces de Banach qui ne contiennent pas uniformément de \(\ell _1^n\). C. R. Acad. Sci. Paris A 277(1), 991–994 (1973)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Poblete, V.: Maximal regularity of second-order equations with delay. J. Differ. Equ. 246(1), 261–276 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Poblete, V., Pozo, J.C.: Periodic solutions of an abstract third-order differential equations. Studia Math. 215(3), 195–219 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Poblete, V., Pozo, J.C.: Periodic solutions of a fractional neutral equations with finite delay. J. Evol. Equ. 14(2), 417–444 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Schmeisser, H.J., Triebel, H.: Topics in Fourier Analysis and Function Spaces. Wiley, Chichester (1987)zbMATHGoogle Scholar
  19. 19.
    Weis, L.: Operator-valued Fourier multiplier theorems and maximal \(L_p\)-regularity. Math. Ann. 319(4), 735–758 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

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

Personalised recommendations