Strong solutions of a neutral type equation with finite delay

Abstract

This paper is concerned to study the existence and uniqueness of solution of neutral type differential equations, by using the maximal regularity property of the first-order abstract Cauchy problem with finite delay on Lebesgue spaces defined at the line. The main tools that we use to achieve our goals are an operator-valued version of Miklhin’s Fourier multiplier theorem, weighted Sobolev spaces on the real line and fixed point arguments.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    H. Amann. Linear and Quasilinear Parabolic Problems. Volume I: Abstract Linear Theory. Monographs in Mathematics, vol 89., Birkhäuser, Basel-Boston-Berlin, 1995.

  2. 2.

    H. Amann. Operator-valued Fourier multipliers, vector-valued Besov spaces and applications. Math. Nachr. 186 (1997), 5–56.

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    W. Arendt, M. Duelli. Maximal \(L^p\)-regularity for parabolic and elliptic equations on the line. J. Evol. Equations, 6 (2006), 773–790.

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    W. Arendt, C. J.K. Batty, S. Bu. Fourier multipliers for Hölder continuous functions and maximal regularity. Studia Math. 160 (2004), 23–51.

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    W. Arendt, S. Bu. The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240 (2002), 311–343.

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    W. Arendt, S. Bu. Operator-valued Fourier multiplier on periodic Besov spaces and applications. Proc. Edim. Math. Soc. 47 (2) (2004), 15–33.

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    W. Arendt, S. Bu. Tools for maximal regularity. Math. Proc. Cambridge Ph. Soc. 134 (2) (2003), 317–336.

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    W. Arendt, C. Batty, M. Hieber, F. Neubrander. Vector-Valued Laplace transforms and Cauchy problems, Monogr. Math., vol. 96, Birkhäuser, Basel, 2001.

  9. 9.

    S. Bu, Y. Fang. Periodic solutions of delay equations in Besov spaces and Triebel-Lizorkin spaces. Taiwanese J. Math. 13 (2009), 3, 1063–1076.

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Ph. Clément, J. Prüss, An operator-valued transference principle and maximal regularity on vector-valued \(L_p\) -spaces, Evolution equations and their applications in physical and life sciences (Bad Herrenalb, 1998), 67–87, Lecture Notes in Pure and Appl. Math., 215, Dekker, New York, 2001.

  11. 11.

    R. Datko. Linear autonomous neutral differential equations in a Banach space. J. Diff. Equations, 25(2):258–274, 1977.

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    R. Denk, M. Hieber, J. Prüss, \(R\) -boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166 (788), 2003.

  13. 13.

    M. Girardi, L. Weis. Operator-valued Fourier multiplier theorems on Besov spaces.Math. Nach. 251 (2003) 34–51.

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    M. Girardi, L. Weis. Operator-valued Fourier multiplier theorems on \(L_p(X)\) and geometry of Banach spaces. J. Funct. Anal. 204 (2) (2003), 320–354.

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    J.K. Hale. Functional Differential Equations. Appl. Math. Sci., 3, Springer-Verlag, 1971.

  16. 16.

    J.K. Hale, S.M. Verduyn. Introduction to Functional-Differential Equations. Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993.

  17. 17.

    E. Hernández. Existence results for partial neutral functional integrodifferential equations with unbounded delay. J. Math. Anal. Appl., 292(1):194–210, 2004.

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    H. Henríquez, E. Hernández. Existence of periodic solutions of partial neutral functional-differential equations with unbounded delay. J. Math. Anal. Appl., 221(2):499–522, 1998.

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    P. Kunstmann, L. Weis. Maximal \(L_p\) -regularity for parabolic equations, Fourier multiplier theorems and \(H^\infty \) -functional calculus, Functional analytic methods for evolution equations, 65–311, Lecture Notes in Math., 1855, Springer, Berlin, 2004.

  20. 20.

    V. Lakshmikantham, L. Wen, B. Zhang. Theory of differential equations with unbounded delay. Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, v 298, 1994.

  21. 21.

    C. Lizama, V. Poblete. Maximal regularity of delay equations in Banach spaces. Studia Math. 175 (1) (2006), 91–102.

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    C. Lizama, V. Poblete. Periodic solutions of fractional differential equations with delay. J. Evol. Equations, 11 (2011), 57–70.

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    A. Mielke. Über maximale \(L^p\)-Regularität für Differentialgleichungen in Banach- und Hilbert-Räumen. Math. Ann. 277 (1987), no. 1, 121–133.

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    S. Schweiker. Asymptotics, Regularity and Well-Posedness of First- and Second-Order Differential Equations on the Line. PhD Thesis, 2000, Universität Ulm.

  25. 25.

    G. Webb. Functional differential equations and nonlinear semigroups in \(L^p\)-spaces. J. Differential Equations 29 (1976), 71–89.

    Article  MATH  Google Scholar 

  26. 26.

    L. Weis. Operator-valued Fourier multiplier theorems and maximal \(Lp\)-regularity. Math. Ann. 319, (2001) 735–758.

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    J. Wu. Theory and Applications of Partial Differential Equations. Appl. Math. Sci. 119, Springer Verlag, 1996.

  28. 28.

    M. Wu, Y. He, J. She. Stability analysis and robust control of time-delay systems. Science Press Beijing, Beijing; Springer-Verlag, Berlin, 2010.

    Book  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Juan C. Pozo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Juan C. Pozo: Partially supported by FONDECYT Grant 11160295. Felipe Poblete: Partially supported by FONDECYT Grant 11181263. Verónica Poblete: Partially supported by Enlace ENL016/15 y PAIFAC.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Poblete, V., Poblete, F. & Pozo, J.C. Strong solutions of a neutral type equation with finite delay. J. Evol. Equ. 19, 361–386 (2019). https://doi.org/10.1007/s00028-019-00478-9

Download citation

Mathematics Subject Classification

  • Primary 34K40
  • Secondary 34G10
  • 42B15