Mild Well-posedness of Abstract Differential Equations

  • Valentin Keyantuo
  • Carlos Lizama


We obtain spectral conditions that characterize mild well-posed inhomogeneous differential equations in a general Banach space X. L p periodic solutions of first and second-order equations are considered. The results are expressed in terms of operator-valued Fourier multipliers. Our approach provides a unified framework for various notions of strong and mild solutions. Applications to semilinear equations of second order in Hilbert spaces are given.


Hilbert Space Banach Space Strong Solution Mild Solution Solution Operator 
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  1. [1]
    H. Amann. Linear and Quasilinear Parabolic Problems. Monographs in Mathematics. 89. Basel: Birkhäuser Verlag, 1995.Google Scholar
  2. [2]
    W. Arendt. Semigroups and evolution equations: functional calculus, regularity and kernel estimates. Evolutionary equations. Vol. I, 1–85, Handb. Differ. Equ., North-Holland, Amsterdam, 2004.Google Scholar
  3. [3]
    W. Arendt, C.J.K. Batty, M. Hieber and F. Neubrander. Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. 96. Basel: Birkhäuser Verlag, 2001.Google Scholar
  4. [4]
    W. Arendt, C. Batty, S. Bu. Fourier multipliers for Hölder continuous functions and maximal regularity. Studia Math. 160 (2004), 23–51.MATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    W. Arendt, S. Bu. The operator-valued Marcinkiewicz multiplier theorem and maximal regularity. Math. Z. 240 (2002), 311–343.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    J. Bergh, J. Löfström. Interpolation Spaces. An Introduction. Die Grundlehren der Mathematischen Wissenschaften, 223. Springer-Verlag, Berlin-New York, 1976.Google Scholar
  7. [7]
    P.L. Butzer, H. Berens. Semi-Groups of Operators and Approximation. Die Grundlehren der Mathematischen Wissenschaften, 145, Springer Verlag, 1967.Google Scholar
  8. [8]
    I. Cioranescu, C. Lizama. Spectral properties of cosine operator functions. Aequationes Mathematicae 36 (1988), 80–98.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    G. Da Prato, P. Grisvard. Sommes d’opérateurs linéaires et équations différentielles opérationnelles. J. Math. Pures Appl. 54 (1975), 305–387.MathSciNetMATHGoogle Scholar
  10. [10]
    R. Denk, M. Hieber, J. Prüss. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc. 166 (2003), no. 788.Google Scholar
  11. [11]
    H.O. Fattorini. Second Order Linear Differential Equations in Banach Spaces. North Holland, Amsterdam, 1985.MATHGoogle Scholar
  12. [12]
    M. Girardi, L. Weis. Criteria for R-boundedness of operator families. Evolution equations, 203–221, Lecture Notes in Pure and Appl. Math., 234, Dekker, New York, 2003.Google Scholar
  13. [13]
    V. Keyantuo, C. Lizama. Periodic solutions of second-order differential equations in Banach spaces, Math. Z. 253 (2006), 489–514.MATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    E. Schüler. On the spectrum of cosine functions. J. Math Anal. Appl. 229 (1999), 376–398.MATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    O.J. Staffans. Periodic L 2-solutions of an integrodifferential equation in Hilbert space, Proc. Amer. Math. Soc. 117(3) (1993), 745–751.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    C.C. Travis, G.F. Webb. Cosine families and abstract nonlinear second-order differential equations. Acta Math. Acad. Sci. Hungar. 32 (1978), 75–96.MATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    H. Triebel. Fractals and spectra. Related to Fourier analysis and function spaces. Monographs in Mathematics, 91. Birkhäuser Verlag, Basel, 1997.Google Scholar
  18. [18]
    L. Weis. Operator-valued Fourier multiplier theorems and maximal L p-regularity. Math. Ann. 319 (2001), 735–758.MATHCrossRefMathSciNetGoogle Scholar

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© Birkhäuser Verlag Basel/Switzerland 2007

Authors and Affiliations

  • Valentin Keyantuo
    • 1
  • Carlos Lizama
    • 2
  1. 1.Department of Mathematics Faculty of Natural SciencesUniversity of Puerto RicoUSA
  2. 2.Departamento de Matemática Facultad de CienciasUniversidad de Santiago de ChileSantiagoChile

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