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Nonlinear perturbation of linear evolution equations in a banach space

  • W. E. Fitzgibbon
Article

Summary

Let X be a Banach space and {A(t)|t ε [0, T]} a family of closed linear, densely defined m-accretive operators in X. This paper is concerned with the additive perturbation of {A(t)|t ε [0, T]} by a continuous family of nonlinear accretive operators {B(t)|t ε [0, T]}. Namely solutions are provided for the integral equation u(t, τ, x) = W(t, τ)x −\(\mathop \smallint \limits_\tau ^t \)W(t, s)B(s) · · u(s, τ, x)ds, u(τ, τ, x) = x where W(t, s) is the linear evolution operator associated with the linear differential equation v'(t, s, x) + A(t)v(t, s, x) = 0, v(s, s, x) = x.

Keywords

Differential Equation Banach Space Integral Equation Evolution Equation Evolution Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [1]
    F. E. Browder,Nonlinear equations of evolution, Ann. of Math.,80 (1964), pp. 485–523.CrossRefzbMATHMathSciNetGoogle Scholar
  2. [2]
    M. G. Crandall -A. Pazy,Nonlinear evolution equations in Banach spaces, Israel J. Math.,11 (1972), pp. 57–94.MathSciNetzbMATHGoogle Scholar
  3. [3]
    W. A. Coppel,Stability and asymptotic behavior of differential equations, P. C. Health and Company, Boston (1965).zbMATHGoogle Scholar
  4. [4]
    W. E. Fitzgibbon,Time dependent perturbations of time dependent linear accretive operators, Notices Amer. Math. Soc.,20 (1973), Abstract 701-47-17.Google Scholar
  5. [5]
    J. A. Goldstein,Abstract evolution equations, Trans. Amer. Math. Soc.,141 (1969), pp. 159–185.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    J. A. Goldstein,Semigroups of operators and abstract Cauchy problems, Lecture Notes Series, Tulane University (1970).Google Scholar
  7. [7]
    T. Kato,Integration of the equation of evolution in a Banach space J. Math. Soc. Japan,5 (1953), pp. 208–304.zbMATHMathSciNetGoogle Scholar
  8. [8]
    T. Kato,Linear evolution equation of hyperbolic type, J. of Fac. Sci. Univ. of Tokyo,17, pp. 241–258.Google Scholar
  9. [9]
    T. Kato,Linear evolution equations of hyperbolic type II, J. Math. Soc. Japan,25 (1973).Google Scholar
  10. [10]
    T. Kato,On linear differential equations in a Banach space, Comm. Pure App. Math.,9 (1956).Google Scholar
  11. [11]
    T. Kato -H. Tanabe,On equations of evolution in a Banach space, Osaka Math. J.,14 (1962), pp. 107–133.MathSciNetzbMATHGoogle Scholar
  12. [12]
    D. L. Lovelady,A Hammerstein Volterra integral equation with a linear semigroup convolution kernel (to appear Indiana Journal).Google Scholar
  13. [13]
    D. L. Lovelady,On the generation of linear evolution equations (to appear Duke Math. J.).Google Scholar
  14. [14]
    R. H. Martin,A global existance theorem for autonomous differential equations in a Banach space, Proc. Amer. Math. Soc.,26 (1970), pp. 307–314.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    R. H. Martin,The logarithmic derivative and equations of evolution in a Banach space, J. Math. Soc. Japan,22 (1970), pp. 411–429.zbMATHMathSciNetCrossRefGoogle Scholar
  16. [16]
    K. Mauro -N. Yamada,A remark on an integral equation in a Banach space, Proc. Japan. Acad.,49 (1973), pp. 13–17.MathSciNetCrossRefGoogle Scholar
  17. [17]
    I. Segal,Nonlinear semigroups, Annals of Math.,78 (1963), pp. 339–364.CrossRefzbMATHGoogle Scholar
  18. [18]
    G. F. Webb,Continuous nonlinear perturbation of linear accretive operators in Banach spaces, Journ. Func. Anal.,10 (1972), pp. 191–203.CrossRefzbMATHMathSciNetGoogle Scholar
  19. [19]
    G. F. Webb,Nonlinear perturbation of linear accretive operators in Banach spaces, Israel J. Math.,12 (1972), pp. 237–248.zbMATHMathSciNetGoogle Scholar
  20. [20]
    K. Yosida,Functional Analysis, Springer-Verlag, New York, 2nd Ed. (1968).Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1975

Authors and Affiliations

  • W. E. Fitzgibbon
    • 1
  1. 1.Houston

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