Annali di Matematica Pura ed Applicata

, Volume 105, Issue 1, pp 221–239 | Cite as

Invariant sets for perturbed semigroups of linear operators

  • Robert H. MartinJr.


Let E be a Banach space and consider the initial value problem (*) u’(t)=Au(t)+ +B(t,u(t)), t≥0, u(0)=z; where A is the generator of a linear contraction semigroup and B: [0, ∞)×E→E is continuous. The main results of this paper deal with criteria insuring that a closed subset Ω of E is invariant for (*)—that is, z∈Ω implies that a solution u to (*) satisfies u(t)∈Ω for all t≥0.


Banach Space Linear Operator Closed Subset Contraction Semigroup Linear Contraction 
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.


  1. [1]
    V. Barbu,Continuous perturbations of nonlinear m-accretive operators in Banach spaces, Boll. Un. Mat. Italiana,6 (1972), pp. 270–278.zbMATHMathSciNetGoogle Scholar
  2. [2]
    P. L. Butzer -H. Berens,Semi-Groups of Operators and Approximation, Springer-Verlag, New York, 1967.Google Scholar
  3. [3]
    H. Cartan,Calcul Differential, Paris, 1967.Google Scholar
  4. [4]
    M. G. Crandall -T. Liggett,Generation of semi-groups of nonlinear transformations on general Banach spaces, Amer. J. Math.,113 (1971), pp. 265–298.MathSciNetGoogle Scholar
  5. [5]
    D. L. Lovelady,A Hammerstein-Volterra integral equation with a linear semigroup convolution kernal, Indiana Univ. Math. J. (to appear).Google Scholar
  6. [6]
    R. H. Martin Jr.,Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc.,179 (1973), pp. 399–414.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [7]
    R. H. Martin Jr.,Approximation and existence of solutions to ordinary differential equations in Banach space, Funk. Ekvac.,16 (1973), pp. 195–211.zbMATHGoogle Scholar
  8. [8]
    M. Nagumo,Über die Laga der Integralkuruen gewohnlicher Differentialglerchugen, Proc. Phys.-Math. Soc. Japan,24 (1942), pp. 551–559.zbMATHMathSciNetGoogle Scholar
  9. [9]
    N. Pavel,Approximate solutions of Cauchy problems for some differential equations on Banach spaces (to appear).Google Scholar
  10. [10]
    I. Segal,Nonlinear semi-groups, Ann. of Math.,78 (1963), pp. 339–364.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    G. F. Webb,Continuous nonlinear perturbations of linear accretive operators in Banach spaces, J. Funct. Anal.,10 (1972), pp. 191–203.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata 1975

Authors and Affiliations

  • Robert H. MartinJr.
    • 1
  1. 1.RaleighUSA

Personalised recommendations